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Electronic Theses, Treatises and Dissertations The Graduate School

2007

Numerical Study of Spin-Fermion Modelsfor Diluted Magnetic Semiconductors andHigh Tc CupratesYucel Yildirim

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THE FLORIDA STATE UNIVERSITY

COLLEGE OF ARTS AND SCIENCES

NUMERICAL STUDY OF SPIN-FERMION MODELS FOR DILUTED

MAGNETIC SEMICONDUCTORS AND HIGH TC CUPRATES

By

YUCEL YILDIRIM

A Dissertation submitted to the

Department of Physics

in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

Degree Awarded:

Summer Semester, 2007

The members of the Committee approve the Dissertation of Yucel YILDIRIM defended

on May 30, 2007.

Nicholas E. BonesteelProfessor Directing Dissertation

Naresh DalalOutside Committee Member

Oskar VafekCommittee Member

Jorge PiekarewiczCommittee Member

Peng XiongCommittee Member

Approved:

David Van Winkle, ChairDepartment of Physics

Joseph Travis, Dean, College of Arts and Sciences

The Office of Graduate Studies has verified and approved the above named committee members.

ii

ACKNOWLEDGEMENTS

There are many people to whom I owe a debt of thanks for their support during my

doctoral study.

I especially want to thank my advisor, Professor Adriana Moreo, for her guidance, support

and encouragement during my research and studies. Her perpetual energy and enthusiasm

in research have motivated me. In addition, she was always accessible and willing to help

her students with their research. As a result, research life became smooth and rewarding for

me.

Professor Elbio Dagotto has helped and guided me with my research and teaching duties.

To him I want to express also my deepest gratitude. Professors Nicholas Bonesteel, Peng

Xiong, Jorge Piekarewicz, Narash Dalal and Oskar Vafek deserve a special thanks as my

thesis committee members and advisors. I was delighted to interact with Dr.Gonzalo Alvarez.

Thanks to him, I expanded my skills and experience in computational physics. My deepest

gratitude goes to my family for their unflagging love and support throughout my life. The

generous support from Computational Science Division at ORNL is greatly appreciated.

They kindly provided most of the computer resources used in this work. Last but not least,

I would like to thank my friends at the National High Magnetic Field Lab, the Florida State

University, ORNL and the University of Tennessee: Cengiz Sen, Ozgur Polat, Khaled Al-

Hassanieh, Florentin Popescu, Fabian Heidrich-Meisner,Ivan Gonzalez, Roger Melko, Ivan

Sergienko , Carlos Busser and Horacio Aliaga for their continued support and friendship.

iii

TABLE OF CONTENTS

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. DILUTED MAGNETIC SEMICONDUCTORS . . . . . . . . . . . . . . . . . 42.1 Band Structure of III − V Semiconductors . . . . . . . . . . . . . . . . 72.2 Ferromagnetism in DMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Theoretical Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Models for DMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3. MODELING DMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Multiple Orbital Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Realistic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4 Numerical Results In Finite Systems . . . . . . . . . . . . . . . . . . . . 42

4. DEEP IMPURITIES IN DMS . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1 Phenomenological Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.3 Coulomb attraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5. Effect of Adiabatic Phonons on Striped and Homogeneous Ground States ForHigh Tc Cuprates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.1 Electron-phonon interactions in high TC cuprates . . . . . . . . . . . . . 705.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.3 Influence of Phonons on Striped States . . . . . . . . . . . . . . . . . . . 735.4 Influence of Phonons on Homogeneous States and generation of Stripes . 865.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6. NUMERICAL TECHNIQUES . . . . . . . . . . . . . . . . . . . . . . . . . . 906.1 Conventional MC Algorithm Applied to Spin Fermion Models . . . . . . 90

7. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

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A. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

v

LIST OF FIGURES

2.1 Temperature dependence of remnant magnetization and inverse paramagneticsusceptibility for Ga0.947Mn0.053As sample with Tc = 110 K; From [1]. . . . . 5

2.2 Temperature dependence of remnant magnetization and inverse paramagneticsusceptibility for Ga0.91Mn0.09As sample with Tc = 173 K; inset: hysteresisloop for the same sample at 172 K. From [2]. . . . . . . . . . . . . . . . . . 6

2.3 Curie temperatures evaluated for various III-V (a) as well group IV and II-VIsemiconducting compounds (b) containing 5% of Mn in 2+ charge state and3.5× 1020 holes per cm3. Material parameters adopted for the calculation aredisplayed in Appendix C Ref. [3]. . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Lattice structure of Zinc-blende type semiconductors. . . . . . . . . . . . . . 8

2.5 Left: GaAs band structure and relevant critical point transitions reproducedfrom Ref. [4]. The Conduction bands are labeled as Γ7 and Γ8 based onsymmetry, while the lowest conduction band is labeled Γ6. The valence bandhave been labeled as H.H. for heavy-hole, L.H. for light-hole, and S.O. forsplit-off. Taken from Ref. [5], Mn d filled (d5/d4) and empty (d5/d4) levelsare shown in gray, and the acceptor Mn A is dashed-Gray. The dispersion ofthe Mn acceptor level is also taken from Ref. [5]. The L point corresponds tothe 111 direction and the X point to the 001 direction. Right:a cartoon-likedescription of the band structure. . . . . . . . . . . . . . . . . . . . . . . . . 9

2.6 First Brillouin zone for the face-centered cubic lattice. Special symmetrypoints and directions are labeled. . . . . . . . . . . . . . . . . . . . . . . . . 9

2.7 Top panel: Substitutional MnGa and interstitial MnI in GaAs. Bottom panel:two eg 3d-orbitals and three t2g 3d-orbitals of Mn. . . . . . . . . . . . . . . 11

2.8 Validity of picture in different regimes. . . . . . . . . . . . . . . . . . . . . . 13

3.1 Tc versus the carrier concentration p, at various J2/J1, obtained with theDMFT technique. Here, x = 0.05, W1/W2 = 1, and J1/W1 = 0.5. The insetshows the corresponding DOS at T = 0. . . . . . . . . . . . . . . . . . . . . 22

vi

3.2 Results obtained with the DMFT approximation: (a) Tc vs. J2/J1, atW1/W2 = 1, and J1/W1 = 0.5, for the values of p indicated in (b). Atp = 0.02 and J2/J1 = 0, a finite Tc/W1 ∼ 0.0037 is caused by the l=1 band.At p = 0.05, Tc is not zero for J2/J1 ∈ (0.6, 1.35), and it increases significantlydue to band overlap. The case p = 0.07 corresponds to the first IB completelyfilled. (b) Tc vs. J1/W1, at W1/W2 = 1 and J1/W1 = J2/W2, for the p’sindicated. (c) Tc vs. p at different ratios W2/W1, fixing J1 = J2 = 2. In allthe frames x = 0.05. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 Tc versus p at different ratios J1/W1, obtained using DMFT. The parametersx, W1, W2, and J1 are as in Fig. 3.1. The inset contains the DOS at T = 0. . 24

3.4 (a) MC magnetization (in absolute value) vs. temperature (T ), with J1=J2=4and t1=t2=1, for the hole densities indicated. The dashed line is the exactasymptotic high-temperature value M∞, which tends to 0 only in the bulklimit. In this paper, the Tc on the 53 cluster was (arbitrarily) defined as the Twhere M reaches ∼5% of the 1−M∞ value (indicated by arrows in (a) and (b)).Other criteria lead to very similar conclusions. (b) Magnetization vs. T forp=1, t1=t2, and several (equal) magnetic couplings J ; (c) Curie temperaturevs. hole density for J=4 and different ratios of the band hoppings; (d)Curie temperature versus the (equal) magnetic interactions J for several holedensities and equal band hoppings t1 and t2. Results for 53 (63) lattices areindicated by open (filled) symbols. Error bars due to the disorder (up to 7samples) are only shown for a few points for clarity. . . . . . . . . . . . . . 26

3.5 (a) Tc vs. ph obtained with MC on a 53 (63) lattice with t1=t2=1 for thevalues of J2/J1 indicated by the open (filled) symbols. J1 is fixed to 4, i.e.when the IB are about to separate from the valence band for band 1 (insetFig. 5(b)). For J2 < J1 (e.g. J2/J1=0.4 curve), Tc is regulated by the IBin band 1, since the IB for band 2 is deep into the valence band. In thiscase, a single-band behavior is observed, with Tc maximized for p=0.5. ForJ2 > J1, both IB play a role. For J2 ≫ J1 (see J2/J1=2), the two IB donot overlap, and for 0 ≤ p ≤ 1 Tc is determined by the band-2 IB reaching amaximum at p=0.5 and vanishing at p=1. For larger p, Tc is controlled nowby the IB 1, and it raises again passing through a maximum at p=1.5 andvanishing at p=2. For J2/J1=1.25, the two IB overlap and we observe residuallocal maxima at p=0.5 and 1.5, related to the single band physics, and a newlocal maximum at p=1 due to IB overlap for the corresponding value of thechemical potential. Tc at p=0.5 is boosted by the partial IB overlap as well.(b) Tc vs. p for t2/t1=1 and several Js. Inset: low-T DOS. . . . . . . . . . . 27

3.6 Band structure for GaAs obtained by diagonalizing the Realistic Systemdescribed by Eq. 3.30 (continuous lines) with J = 0. The dashed red linesindicate the results for the Luttinger-Kohn (LK) Hamiltonian. . . . . . . . . 42

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3.7 Band structure for GaAs obtained by diagonalizing the Realistic Systemdescribed by Eq. 3.30 (continuous lines). The symbols indicate resultsobtained with our real space code in the finite lattices with N cubes alongeach spatial direction. The number of momentum states available inside theFBZ is given by 4N3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.8 Band structure for GaAs obtained diagonalizing Eq. 3.14 (dashed bluelines).The red lines indicate the results for the Luttinger-Kohn model. Theblack lines are our results shifted so that the bottom of the valence band is at 0. 45

3.9 Temperature of dependence of magnetization for x=8.5% and several p’s using256 sites lattices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.10 Temperature of dependence of magnetization for different lattice sizes forx = 8.5%, p ≈ 0.75, and J = 1.2eV . . . . . . . . . . . . . . . . . . . . . . . . 46

3.11 MC calculated TC vs. p, at x=8.5% and J = 1.2eV . Notice that N = L3

where L is the size of the unit cell L=Lx=Ly=Lz . . . . . . . . . . . . . . . 47

3.12 (a) Curie temperature vs. J , for x=8.5% and p≈0.75. The MC results areindicated by circles, while the continuous line is the MF prediction.[6] Inset:MC results for larger values of J to observe the crossover toward a localizedpicture. Vertical lines indicate the experimentally acceptable range of J .(b) MC calculated TC vs. p, at x=8.5%. The blue dots are experimentalresults,[7, 8, 9] and the solid line is the MF prediction. . . . . . . . . . . . . 48

3.13 (a) Magnetization M vs. T , for x=8.5% and several p’s (indicated), using a256 sites lattice (open symbols). Averages over 5 Mn-disorder configurationsare shown. (b) Same as (a), but for x=3%. Close circles are results for a 500sites lattice. The magnetization is measured as M =

√M · M, with M the

vectorial magnetization. As a consequence, for fully disordered spins, M isstill nonzero due to the SI

2=1 contributions, causing a finite value at largetemperatures (M(T → ∞) = 1/

√xNGa) unrelated to ferromagnetism. Thus,

we plotted M = (M−M(T → ∞))/(1 −M(T → ∞)), i.e. the backgroundwas substracted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.14 (a) M (defined as in Fig. 3.13) vs T for different lattice sizes for x = 8.5%,p ≈ 0.75, and J = 1.2eV ; (b) M vs. T for the same parameters as in (a)on a 256 sites lattice with (without) spin-orbit interaction indicated by thesquares (circles); (c) Charge density normalized to the MF value (see text),for x = 8.5%, p≈ 0.75, T=10K, on a 256-sites cluster for J=1.2eV . The colorintensity is proportional to the charge density (see scale). (d) Same as (c),but for J=12eV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

viii

3.15 (a) Density-of-states, for x=8.5%, p≈0.75, and several J ’s. The dashedvertical lines indicate the position of the chemical potentials. (b) Same as(a) but for x = 3%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.16 Magnetization as a function of temperature for large values of Js. . . . . . . 53

3.17 Temperature dependence of magnetization for different III-V type materialsat x = 8.5% and ph = 0.72 in 43 lattices. . . . . . . . . . . . . . . . . . . . . 54

4.1 a) Temperature dependence of magnetization at x=8.5% for GaAs, obtainedwith 6-orbital (black curve) and 8-orbital (red curve) models. b) Same forGaN but with different values of Vd . . . . . . . . . . . . . . . . . . . . . . . 60

4.2 Density of states (DOS) for different values of Vd considering weak p − dhybridization for Mn doped GaN. . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 Magnetization versus temperature and Density od states (DOS) for strongp − d hybridization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4 The density of states (DOS) for J=1.2 eV and various values of Vd. Thevertical lines indicate the position of the chemical potential. . . . . . . . . . 63

4.5 The density of states (DOS) for J=7 eV and V=0 (black line); for an on-siteCoulomb attraction V=-3.5 eV (blue line); and for a Coulomb attraction withon site intensity V and next nearest neighbors intensity Vnn=V=-3.5 eV (redline). The vertical lines indicate the position of the chemical potential. . . . 64

5.1 View of the La2CuO4 structure. Cu atoms are represented by red, La atomsby green and O atoms by blue balls. The six coordinated copper and theCuO2 planes are shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2 View of the Y Ba2Cu3O7 structure. Cu are represented by red, Y atoms bygreen, Ba atoms by gold and O atoms by blue balls. This structure has twoCuO2 planes in the elementary cell, separated by an Y atom. . . . . . . . . . 67

5.3 Schematic phase diagram of the high temperature superconductors. . . . . . 68

5.4 (a) MC snapshot of an 8×8 lattice at 〈n〉=0.875 for λ = 0 and γ = 0. The sizeof the circles is proportional to the electronic density; the shaded circles havecharge density larger than the average, i.e., ni ≥ 〈n〉 = 0.875. The arrowsrepresent the projection of the localized spins in the plane x − y; the linesindicate lattice distortions (see text); (b) same as (a) but for λ = 1 and modeQ(2); (c) same as (b) but for λ = 2. . . . . . . . . . . . . . . . . . . . . . . . 75

5.5 (a) MC snapshot of an 8 × 8 lattice at 〈n〉=0.875, for mode Q(1), λ = 2 andγ = 0; (b) same as (a) but for mode Q(2); (c) same as (a) but for mode Q(3). 76

ix

5.6 (a) The charge structure factor N(q) for various values of λ for the sameparameters as in Fig. 5.4; (b) The density of states N(ω) for several valuesof the diagonal electron-phonon coupling λ, for the same parameters as in(a).The phonon mode is Q(2). . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.7 (a) Spectral weight in the density of states N(ω) at ω = 0 as a function ofthe diagonal electron-phonon coupling λ for several values of the electronicdensity 〈n〉 and mode Q(2); (b) Drude weight as a function of the diagonalelectron-phonon coupling λ for several values of the electronic density 〈n〉 andmode Q(2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.8 (a) MC snapshot of an 8 × 8 lattice at 〈n〉=0.8 λ = 0 and γ = 0; (b) sameas (a) but for λ = 1 with mode Q(2); (c) same as (b) for λ = 2; (d) Magneticstructure factor for the parameters in (b). . . . . . . . . . . . . . . . . . . . 79

5.9 (a) MC snapshot of an 8 × 8 lattice at 〈n〉=0.8, for mode Q(2), λ = 1 andγ = 0, after 2600 measuring sweeps; (b) same as (a) but after 3750 measuringsweeps; (c) same as (a) but after 4250 measuring sweeps; (d) same as (a) butafter 5000 measuring sweeps. . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.10 Study of the effect of off-diagonal couplings. (a) MC snapshot of an 8 × 8lattice at 〈n〉=0.75, λ = 0 and γ = 0; (b) same as (a) but for γ = 0.1 andmode Q(2); (c) same as (b) for γ = 0.2; (d) same as (b) but for γ = 0.6 . . . 81

5.11 (a) Charge structure factor for different values of γ (strength of the off-diagonal EPI) and for λ = 0 on a 8 × 8 lattice at 〈n〉=0.75, for mode Q(2);(b) the magnetic structure factor for the same parameters as in (a); (c) theDrude weight for the same parameters as in (a); (d) the density of states forthe same parameters as in (a). . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.12 (a) MC snapshot of an 8 × 8 lattice at 〈n〉=0.875, for mode Q(2), λ = 1 andγ = 0; (b) same as (a) but for γ = 0.1; (c) same as (a) but for γ = 0.2; (d)same as (a) but for γ = 0.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.13 (a) MC snapshot of an 8 × 8 lattice at 〈n〉=0.875, for mode Q(2), λ = 6 andγ = 0. The dashed lines separate CDW domains; (b) same as (a) but forγ = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.14 (a) Spectral functions along the path q = (0, 0)− (π, 0) − (π, π) on a 12 × 12lattice at 〈n〉=0.8, λ = 0 and γ = 0; (b) same as (a) but along the diagonaldirection in the Brillouin zone; (c) same as (a) but for λ = 2 and mode Q(3);(d) same as (b) but for λ = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.15 (a) Spectral functions along the path q = (0, 0)− (π, 0) − (π, π) on a 12 × 12lattice at 〈n〉=0.8, λ = 2 and γ = 0.4 for the half-breathing mode; (b) sameas (a) but along the diagonal direction in the Brillouin zone. . . . . . . . . . 86

x

5.16 (a) MC snapshot of an 8 × 8 lattice at 〈n〉=0.75, J = 1.5, λ = 0 and γ = 0;(b) same as (a) but for λ = 2 with mode Q(2); (c) same as (b) with mode Q(1)

. The dashed line indicates the stripes. . . . . . . . . . . . . . . . . . . . . . 88

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ABSTRACT

In this dissertation, Spin-Fermion (SF) models for diluted magnetic semiconductors

and high temperature superconducting cuprates are constructed and studied with unbiased

numerical techniques. A microscopic model to describe magnetically doped III-V semi-

conductors is proposed. This model includes the appropriate lattice geometry, as well

as, magnetic, spin-orbit, and Coulomb interactions and contains no free parameters. Its

study using state-of-the-art numerical techniques provides results in excellent agreement

with experimental data for Mn doped GaAs. For the first time, Curie-Weiss behavior of

the magnetization is obtained numerically and the values of the Curie temperature are

reproduced in a wide range of Mn doping and compensations. We observed that for

x ≥ 3%, the holes are doped into the valence band and uniformly distributed in the material.

This could support the “valence band” scenario regarding this material. Phononic degrees

of freedom, which are often neglected in studies of high Tc cuprates, are considered in a

numerical study of a spin-fermion model. Both diagonal and off-diagonal electron-phonon

interactions are considered. While diagonal terms tend to stabilize ordered structures such

as stripes, the off-diagonal terms introduce disorder making this structures more dynamical.

Our results indicate that phonons play a role in the stabilization of stripe-like states.

xii

CHAPTER 1

INTRODUCTION

Over the last several decades, many materials exhibiting complex properties have been

discovered[10], such as high Tc cuprates, manganites, heavy fermions, and diluted magnetic

semiconductors. During the 20th century great progress was made in understanding the

behavior of many solids such as simple metals, insulators and semiconductors. In these

materials the electrons only weakly interact with each other or their surrounding ions.

But materials such as transition metal oxides with open d and f electron shells, where

electrons occupy narrow orbitals, have properties that are harder to explain. For instance,

in transition metals, such as vanadium, iron, and their oxides, electrons experience strong

Coulomb repulsion because of their spatial confinement in those orbitals. Such strongly

interacting or “correlated” electrons cannot be described within a single-electron picture

such as Fermi liquid theory[11, 12], since the inter-electronic interactions are strong enough

as to alter the free electron nature of the system. The influence of an electron on the others

is simply too pronounced for each to be treated independently.

The effect of correlations on materials properties is often profound. The interplay of the

d and f electron internal degrees of freedom, such as spin, charge, and orbital moment can

trigger complex ordering phenomena at low temperatures. That interplay makes strongly

correlated electron systems extremely sensitive to small changes in external parameters, such

as temperature, pressure, or doping.

The dramatic effects can range from huge changes in the resistivity across the metal-

insulator transition in vanadium oxide[13] and considerable volume changes across phase

transitions in actinides and lanthanides[14], to exceptionally high transition temperatures in

superconductors with copper-oxygen planes[15]. In materials called ”heavy fermion systems”,

mobile electrons at low temperature behave as if their masses were a thousand of times the

1

mass of a free electron in a simple metal. Some strongly correlated materials display a

very large thermoelectric response; others show a great sensitivity to changes in an applied

magnetic field giving rise to the phenomenon known as ”colossal magneto-resistance”[16].

Such properties make the prospects for developing applications from correlated-electron

materials exciting. However, the richness of the phenomena and the marked sensitivity

to microscopic details make their experimental and analytical study all the more difficult.

Most advances in the field have been driven by the interplay of experimental discoveries

with the development of theoretical frameworks such as band theory[17], Fermi liquid

theory[18], BCS[19]. Semiconductors were understood and gave rise to our current electronic

technology. Superconductors and ferromagnets also had an impact at the industrial level.

Band theory, which imagines electrons behaving like extended plane waves, is a good

starting point to understand materials made up of weakly correlated electrons, such as silicon

or aluminum. This theory helps to capture the delocalized nature of electrons in metals.

Fermi liquid theory[18] describes the transport of conduction electrons in momentum space

and provides a simple but rigorous conceptual picture of the spectrum of excitations in a solid.

In calculation of the various microscopic properties of such solids, the accurate quantitative

techniques, for example Density Functional theory (DFT),[20] allow us to compute the

total energy of some materials. However, the independent-electron model and the DFT

method are not accurate enough when applied to strongly correlated materials. The failure

of band theory was first noticed in insulators such as nickel oxide and manganese oxide,

which have relatively low magnetic-ordering temperatures but large insulating gaps. Band

theory incorrectly predicts them to be metallic.[21, 22]

In an ongoing theoretical endeavor a variety of models were introduced to take into

account the strong electronic, magnetic and phononic interactions in order to understand

these materials. Amongst the simple models are the Kondo[23] (it successfully addresses

the experimentally observed non-zero low-temperature minimum found in the resistivity

ρ(T ) of some metals with magnetic impurities), Anderson[24] (describes electrons in a

transition metal, including the interactions between the electrons) and Hubbard[25] (ad-

dresses Coulomb interaction between electrons at a simple level) models. These models

are simple but pose a significant challenge in solving them. While it is possible to find

analytical solutions[26, 27] to some of these models in one dimension, approximations are

necessary in higher dimensions. As a result, new theoretical approaches have to be developed.

2

Usually, the standard many-body techniques that we described do not work for predicting

the behavior of these models. Perturbation theory is inappropriate, because the electron-

electron interactions are very strong, and mean field theory takes not into account the subtle

inter-electron correlations which are induced.

Over the years, a great variety of non-perturbative techniques has been developed in

order to deal with such problems: quantum field theoretical approaches, Bethe ansatz[28, 29],

dynamical mean field theories[21, 30], conformal field theory[31, 32], renormalization group

theory[33, 34], slave bosons[35, 36], 1/d and 1/N expansions (d is the dimensionality

and N is the number of degrees of freedom). In addition, various numerical techniques

such as exact diagonalization, Lanczos[15, 37] and Monte Carlo[15, 38, 39, 37], have been

developed. Numerical approaches have to be applied to finite size systems and most of them

provide exact results. They are crucial to provide guidance to approximate schemes that

would handle the thermodynamic limit. The combined application of the many techniques

named above and the comparison with experimental data is what allows us to develop

an understanding of strongly correlated materials. The approaches mentioned above have

been used to provide insights and quantitative descriptions of heavy fermions[40], metal-

insulator transitions[41, 42, 43], transport in one-dimensional conductors[44], integer and

fractional quantum Hall systems[45] and many different properties of high temperature

superconductors[46, 47, 48, 48, 49]. However, many important issues remain to be solved in

these and other systems such as magnates and diluted magnetic semiconductors.

This thesis focuses on the application of numerical techniques to diluted magnetic

semiconductors (DMS) and High Tc superconductors, which are materials with strong

correlations. In Chapter 2, an introduction to diluted magnetic semiconductors is presented.

Chapter 3 is devoted to the modeling of DMS. In Chapter 4, we discuss electron-phonon

interaction in High TC cuprates in the context of a Spin-Fermion model. The numerical

techniques applied to the models will be presented in Chapter 5. We conclude in Chapter 6

with a summary. Finally, the change of basis matrix used in Chapter 3 will be provided in

the Appendix.

3

CHAPTER 2

DILUTED MAGNETIC SEMICONDUCTORS

The past three decades have witnessed the rapid advancement of solid state electronics,

including the integration of circuit elements into one semiconductor chip, i.e. integrated

circuit devices. Magnetic devices are not part of circuits. Semiconductors and magnetic

materials are two very basic components of electronic industries. Semiconductor devices are

used to transport information that can be reliably controlled by external fields, and magnetic

devices are used to store information.

The realization of materials that combine semiconducting behavior with magnetism in a

single device has long been a dream of material physics.

One of the hot topics today is to control the spin of electrons, holes, nuclei, or ions to

gain new functionalists in both analog and digital electronics. The charge, mass and spin of

electrons form the foundation of present information technology. The integrated circuits and

the high frequency devices made of semiconductors, used for information processing, use only

the charge of electrons while the storage of information is done by magnetic recording using

spin of electrons in a ferromagnetic metal [50, 51]. But tomorrow’s information technology

may see magnetism (spin), and semi-conductivity (charge) combined in devices that exploits

both charge and ’spin’ to process, and store the information. We may then be able to use

the capability of mass storage and processing of information in the same device.

In order to develop such a device, an obstacle that must be overcome is the realization

of reliable injection of spins into semiconductors. The convenient source of polarized spins

should have a structure compatible with the one of the semiconductors used in devices.

Thus, a semiconductor that is ferromagnetic at room temperature could fulfill this function.

One of the approaches to drive a semiconductor ferromagnetic is to introduce magnetic

ions such as Mn, Cr, Co and Fe into non-magnetic semiconductors. Only a small amount of

4

Figure 2.1: Temperature dependence of remnant magnetization and inverse paramagneticsusceptibility for Ga0.947Mn0.053As sample with Tc = 110 K; From [1].

magnetic ions can be successfully doped in the semiconductors. Hence they are called diluted

magnetic semiconductors (DMS)[52]. The metastability of the DMS (III,Mn)V compounds

and low solubility of magnetic ions in these materials were the major obstacles for synthesis

of this kind of DMS. Beyond a certain doping level, surface segregation, and in extreme

cases phase separation, would occur and impede further incorporation of magnetic ions into

the crystals. Therefore, preparation of alloys to make DMS semiconductors is an extremely

difficult task. The first positive result was obtained by Ohno and collaborators[53] who

reported ferromagnetism at a critical temperature Tc = 7.5 K in epitaxial films of InMnAs

grown by Molecular Beam Epitaxy (MBE) in 1992 for 1.3%(x = 1.3%) Mn concentration.

In 1996 Ohno et al.,[54] were able to dope Mn ions into GaAs host semiconductor achieving

a TC = 60 K which opened the way to the possibility room temperature DMS. Since then

various experimental groups have tried to increase Tc. In 1998 the Tohoku University group

announced a jump of Tc in p-type (Ga, Mn)As to 110 K[1] by increasing Mn concentration

to 5.3%. Fig. 2.1 shows the temperature dependence of magnetization and susceptibility of

this work. The goal of breaking the 110 K record in (Ga,Mn)As remained elusive for nearly

four years. Only recently has progress in MBE growth and in the development of post-

growth annealing techniques [55, 56, 57, 58, 7, 59] made it possible to push Tc in (Ga,Mn)As

up to 173 K [2, 60]. Fig. 2.2 shows the experimental result taken from Ref. [2, 60] where

temperature dependence of the magnetization is presented. The current Tc record could be

5

Figure 2.2: Temperature dependence of remnant magnetization and inverse paramagneticsusceptibility for Ga0.91Mn0.09As sample with Tc = 173 K; inset: hysteresis loop for the samesample at 172 K. From [2].

broken in (Ga,Mn)As if a higher concentration of substitutional Mn ions could be achieved

with low compensation since the Curie temperature Tc is believed to increase with the

amount of substitutional Mn and also with carrier concentration[61]. Other efforts focus

on finding alternative materials with stronger magnetic exchange coupling between the holes

and magnetic impurities.

Dietl et al.[3] estimated that ZnO and GaN would exhibit ferromagnetism above room

temperature upon doping with Mn, provided that the hole density is sufficiently high.

Fig. 2.3 presents the calculated values of the Curie temperature TC for different types of

semiconductors with 5% of Mn. The balance between achieving high Curie temperatures

and the desire for low, and therefore gateable, carrier densities, is one of the major issues in

the fabrication of these materials.

In order to offer guidance to the experimental effort, one of the key question that needs

to be addressed is the origin of ferromagnetism(FM) in these materials. Theoretical models

are still very controversial and in the following sections we are going to describe the proposed

sources of the mechanism.

6

10 100 1000

CdSe

CdS

ZnS

CdTe

ZnO

ZnTe

ZnSe

GeSi

CURIE TEMPERATURE [K]

InN

InAs

InP

GaSb

GaPGaAs

GaN

AlAs

AlP

Figure 2.3: Curie temperatures evaluated for various III-V (a) as well group IV and II-VIsemiconducting compounds (b) containing 5% of Mn in 2+ charge state and 3.5× 1020 holesper cm3. Material parameters adopted for the calculation are displayed in Appendix CRef. [3].

2.1 Band Structure of III − V Semiconductors

The lattice structure of GaAs is zincblende[62] shown in Fig. 2.4. It consists of two

interpenetrating face-centered cubic Bravais lattices, displaced along the body diagonal of

the cubic cell by one quarter the length of the diagonal. In this arrangement, Ga and As

atoms occupy each sublattice respectively. The atomic structure of Ga is [Ar]3d104s2p1 and

[Ar]3d104s2p3 for As. In GaAs the Ga and As atoms bond covalently. Both atoms share the

electrons that they have in their 4s and 4p shells. Ga donates 3 electrons while As donates 5.

7

Figure 2.4: Lattice structure of Zinc-blende type semiconductors.

The hybridized orbitals have character sp3. Several numerical techniques allow to obtain the

band structure generated by this hybridization such as empirical pseudo-potential method

(EPM) or the empirical tight-binding method (ETBM) [63, 64, 65] The corresponding band

structure is shown in Fig. 2.5[66]. The right panel of the figure shows a real sp3 band

structure calculation of GaAs and a cartoon-like description is presented in the left panel.

There is a conduction band that is dominated by s-orbital (singlet) and a valence band with

p character which is composed of 3 orbitals. The 3p states at the top of the valence band

(Γ point) should be degenerate but the spin-orbit interaction removes the degeneracy[63].

As a result, the valence band is composed of a heavy-hole and light-hole bands that are

degenerate at the Γ point and a split-off band at a distance ∆SO (or Eso) from the top of

the valence band at the Γ point. For GaAs, ∆so = 0.34 eV , but it strongly depends on the

material and it is 0.017 eV for GaN , 0.08 eV for GaP and 0.74 eV for GaSb. Fig. 2.6 shows

the Brillouin zone for the zinc-blende structure. The symbols Γ, X, L,... represents special

symmetry points and directions within the Brillouin zone. The Γ point corresponds to the

condition k=0, i.e. the center of the Brillouin zone.

8

Figure 2.5: Left: GaAs band structure and relevant critical point transitions reproducedfrom Ref. [4]. The Conduction bands are labeled as Γ7 and Γ8 based on symmetry, whilethe lowest conduction band is labeled Γ6. The valence band have been labeled as H.H. forheavy-hole, L.H. for light-hole, and S.O. for split-off. Taken from Ref. [5], Mn d filled (d5/d4)and empty (d5/d4) levels are shown in gray, and the acceptor Mn A is dashed-Gray. Thedispersion of the Mn acceptor level is also taken from Ref. [5]. The L point corresponds tothe 111 direction and the X point to the 001 direction. Right:a cartoon-like description ofthe band structure.

Figure 2.6: First Brillouin zone for the face-centered cubic lattice. Special symmetry pointsand directions are labeled.

9

2.2 Ferromagnetism in DMS

In this section, we will be describe how ferromagnetism develops in DMSs. An important

initial question that should be addressed is what is the effect of substituting Ga by Mn in

GaAs.

The atomic structure of the elements in (Ga,Mn)As is [Ar]3d104s2p1 for Ga, [Ar]3d54s2

for Mn, and [Ar]3d104s2p3 for As. This circumstance correctly suggests that the most stable

and, therefore, most common position of Mn in the GaAs host lattice is on the Ga site where

its two 4s-electrons can participate in crystal bonding in much the same way as the two Ga

4s-electrons.[61] When Mn replaces Ga a p-electron is introduced which means that a hole

for Mn is effectively doped into the system. If the 2s electrons of the Mn are shared in

the sp3 bonds with As then the Ga3+ center is replaced by a Mn2+ one. This less positive

center will moderately attract the doped hole. In addition, the 5 electrons in the d shell of

Mn produce a localized spin 5/2. Some theoreticians predict that it also could be possible

that the d levels of Mn hybridize with the sp levels of As and one of the electrons in the 3d

shell of Mn gets promoted to the sp3 levels leaving the hole in the 3d shell and localized spin

2 (instead of 5/2). However, experiments such as electron paramagnetic resonance (EPR)

and optical measurements[67, 68, 69] indicate that in GaAs even for very few amounts of

Mn the only impurity level observed corresponding to the d5-Mn2+ ionized state, indicating

that indeed the strongly localized 3d5 electrons with total spin S =5/2 are a good starting

point for theoretical models in GaAs.

It is generally accepted that ferromagnetism in (Ga,Mn)As is due primarily to coupling

between the localized magnetic moments that is mediated by the holes.

While the Mn neutral impurity with the hole in the d shell may be important for some

DMS’s it is not observed experimentally in GaAs.[67] Electron spin resonance measurements,

which support the picture of divalent Mn in GaAs[70] indicate that trivalent d4 configurations

for Mn impurities may be present in GaP[71]. For GaN it has also been reported that Mn

could be a divalent state when electrons are doped[72], but in a trivalent state when holes

are doped to the system.

Due to the non-equilibrium growth process, As antisites and Mn-interstitials MnI occur

compensating some doped holes and therefore reducing the free carrier concentration with

respect to the substitutional-Mn density which means the ratio of Mn impurity concentration

10

Figure 2.7: Top panel: Substitutional MnGa and interstitial MnI in GaAs. Bottom panel:two eg 3d-orbitals and three t2g 3d-orbitals of Mn.

x with respect to hole concentration ph, ph/x = p less than unity. Possible explanations that

have been proposed for the compensation in (GA, Mn)As are the presence of As antisite

defects or MnI interstitials that act as donors and tend to passivate substitutional Mn

acceptors, reducing the number of holes[73]. The substitutional MnGa, and the less common

interstitial MnI, positions are illustrated in Fig. 2.7. Annealing procedures, at temperatures

slightly lower than growth temperature, have shown to give a reduction of the amount of

compensation.[74, 55, 7, 75, 76, 73] The initial procedure has now been modified by different

groups [55, 7, 75] and the carrier concentration can actually be tuned precisely through

resistance-monitored annealing.[55]

11

2.3 Theoretical Scenarios

In order to understand the properties of DMS, it is important to identify the most important

interactions and include them in a microscopic models. Experiments indicate that there is

an antiferromagnetic exchange between the localized spins and the spins of the doped holes.

It is believed that this is the interaction that induces ferromagnetism. Additional effects

that need to be considered are; the kinetic energy of the holes, Coulomb interactions, the

antiferromagnetic Mn − Mn superexchange interactions, hole-hole interactions,positional

disorder of the Mn ions and the random potential associated with the compensation

mechanism and anisotropy in the Mn-hole exchange interactions due to spin-orbit coupling.

The common point in all theoretical approaches is to include the Mn-hole exchange

interaction, since it drives the ferromagnetism. Much of the debate among theorists, as it will

be discussed below, is due to the different aspects that arose from a variety of approximate

treatments of this interaction. There are two extreme starting points usually considered

by theorists; low and high doping regimes. In Fig. 2.8 a simple picture is presented. A

bound state (112 meV for GaAs and 1400 meV for GaN) is formed when a single Mn ion

is doped into the host semiconductor. At small values of Mn concentration, (x) the holes

are localized and the system is insulator. With increasing doping the wave functions of the

localized holes start to overlap and an impurity band (IB) develops. The carriers move in the

impurity band which depends strongly on the disordered positions of the impurities. Some

researchers assume that this situation prevails up to x ≈ 10% and use this scenario, which

is known as the IB scenario, to describe DMS.[77, 78]

At higher Mn concentrations, the impurity band gradually merges with the valence

band[79] and the impurity states delocalize. The holes distribution becomes uniform and

they are effectively doped into the valence band (VB). Many researcher assume that this

situations occurs at x ≥ 1% and have developed the so called valence band scenario to

describe DMS.[1, 6, 80]

The crossover from impurity-band to valence-band scenario is difficult to study. There

are controversial theoretical end experimental results. The Brillouin-function character of

the magnetization curve data[55, 81] confirms that the valence band picture is valid in

this regime. However, some groups[82] observed the impurity band in the crossover. On

the other hand some researchers[83] suggest that the experimental observation of the in a

12

} Eb=0.1 eV (GaAs)

1.4 eV (GaN)

VBVB VB

bound state

(1−Mn)

IB

fEf

E

insulating regime (x < 1%)

low doping (small x) higly doped (large x)

metalic regime

Figure 2.8: Validity of picture in different regimes.

GaMnAs sample is an indicator of the presence of the interstitials impurities. The crossover is

controlled not only by the Mn density but (because of the importance of Coulomb interaction

screening) also by the carrier density. There is a stark distinction between the compensation

dependence predicted by impurity-band and valence-band pictures. When the impurity-band

picture applies, ferromagnetism does not occur in the absence of compensation[84, 85, 86],

because the impurity band is filled. Given this, we can conclude from experiment that the

impurity band picture does not apply to optimally annealed (weakly-compensated) samples

which exhibit robust ferromagnetism.

2.4 Models for DMS

In this section, some of the approaches that have been used to describe DMS will be

presented.

2.4.1 First Principle Approaches

Quantum mechanics provides a reliable way to calculate what electrons and atomic nuclei

do in any situation. The behaviour of electrons in particular governs most of the properties

13

of materials. This is true for a single atom or for assemblies of atoms in condensed matter,

because quantum mechanics describes and explains chemical bonds. Therefore we can

understand the properties of any material from first-principles, that is, based on fundamental

physical laws and without using free parameters, by solving the Schrodinger equation for the

electrons in that material. However, we rapidly run into difficulty because electrons interact

strongly with each other. The alarming consequence is that exact solutions exist only for a

single electron in simple potentials: solving the Schodinger equation for the hydrogen atom

is a classic undergraduate task, but solving it for helium requires a computational approach.

The problem of interacting electrons in condensed-matter physics, one manifestation of the

many-body problem, is the defining challenge of the subject. For practical calculations

on condensed matter, most first-principles approaches recast the problem from one where

electron interactions are explicit to one where the interactions are represented by an effective

potential acting on apparently independent electrons. The interactions are hidden in the

effective potential, and one deals with one electron at a time. The result is a set of one-

electron Schodinger-like equations:

An early approach was developed by Hartree. He set Veff to the average of the Coulomb

potential between an electron and all others in the system, giving what is now called the

Hartree potential. An electron experiencing this potential is said to move in the mean field of

the other electrons. Of course this is an approximation, and for two reasons. In the real case

the interaction depends explicitly on the position of the other electrons. Something is missed

when the interaction is averaged to form Veff . Also, electrons are fermions, and they obey

the Pauli exclusion principle and Fermi statistics. This gives rise to an effective interaction,

called the exchange interaction, which is not accounted for. The Hartree approach neglects

exchange and correlation, and as one may guess it gives rather poor results. Adding Fermi

statistics to Hartrees method yields the Hartree-Fock approach. The effective potential is

now non-local, and arises from the demand that the total wavefunction be antisymmetric

upon exchange of any two electrons. The exchange interaction is treated exactly, but the

method remains inherently approximate because it neglects correlation. Nonetheless it has

enabled advances in quantitative theory and structural studies of molecules and solids, and

remains the platform on which highly accurate quantum-chemical theories are built.

First principles calculations are the simple empirical rule to explain the magnetic states

of DMSs. For the purpose of describing the overall trend in the magnetism of DMS, first-

14

principles calculations represent a powerful and efficient tool, because they do not need any

parameters obtained from experiments.

One of the first principle approach that is free of phenomenological parameters is density

functional theory (DFT), which is an important tool for studying the microscopic origin

of ferromagnetism through calculations of electronic, magnetic, and structural ground-state

properties.[87] The main technical challenge in DFT theory applications is the development of

numerical methods that provide accurate solutions of single-body Schrodinger equations.[20,

87]. A simple and widely successful approximation is the local spin density approximation

(LSDA) [88].

The problem of solving LSDA equations with enough accuracy remains a challenge. In

DMSs the degrees of freedom that are important for ferromagnetism, The orientations of

the Mn local moments and other length scales in these materials, like the Fermi wavelengths

of the valence band carriers complicate numerical implementation of the LSDA technique.

This property limits the number of independent magnetic degrees of freedom that can be

included in a DFT simulation of DMS materials.

A local-density-approximation (LDA) of the DFT, combined with disorder-averaging

coherent-potential approximation (CPA) or supercell approach, has been used successfully

to address physical parameters of (III,Mn)V DMS that are derived from total-energy

calculations, such as the lattice constants,[89] and formation and binding energies of various

defects.[90, 91] Supercell calculations have usually studied interactions between Mn moment

orientations by comparing the energies of parallel spin and opposite spin orientation states

in supercells that contain two Mn atoms. If the Mn-Mn spin interaction has a range larger

than a couple of lattice constants, this induces a problem for the supercell approach.

The CPA approach can estimate the energy cost of flipping a single spin in the

ferromagnetic ground state, which is proportional to the mean-field-approximation for the

critical temperature of the effective Heisenberg model [92], and in this sense is limited in its

predictive powers when mean-field theory is not reliable.

LSDA predictions for spectral properties, like the local DOS, are less reliable than

predictions for total energy related properties. This is especially true for states above the

Fermi energy, and is manifested by a notorious inaccuracy in predicted semiconductor band

gaps. From a DFT theory point of view, this inconsistency arises from attempting to address

the physics of quasiparticle excitations using ground-state DFT. In Mn-doped DMSs, the

15

LSDA also fails to account for strong correlations that suppress fluctuations in the number

of electrons in the d-shell. One generally accepted consequence is that the energy splitting

between the occupied and empty d-states is underestimated in SDF theory, leading to an

unrealistically large d-state local DOS near the top of the valence band and to an overestimate

of the strength of the p − d exchange.[87, 93]

In general first principle calculations neglect spin-orbit interaction and have a difficulty

dealing with long range interactions. Moreover, it has lack of quantitative predictability.

2.4.2 Microscopic Tight-Binding Approximation

A practical approach, that circumvents some of the complexities of this strongly-correlated

many-body problem is the microscopic tight-binding (TB) band-structure theory. In this

model, it is assumed that the electrons are tightly bound to their nuclei as in the atoms. Next

the atoms are brought together. When their separations become comparable to the lattice

constant in solids, their wave functions will overlap. The electronic wave functions in the

solid are approximated by linear combinations of the atomic wave functions. This approach

is known as the (TB) approximation approach. In the covalently bonded semiconductor there

are two kinds of electronic states. Electrons in the conduction bands are delocalized and so

can be approximated well by nearly free electrons. The valence electrons are concentrated

mainly in the bonds and so they can retain more of their atomic character. The valence

electron wave function should be very similar to bonding orbitals found in molecules. In

addition to being a good approximation for calculating the valence band structure, the

TB method has the advantage that the band structure can be defined in terms of a small

number of overlap parameters. These overlap parameters have a physical interpretation as

representing interactions between electrons on adjacent atoms. More generally, local changes

of the crystal potential at Mn and other impurity sites are represented by shifted atomic

levels. A proper parametrization of these shifts, with the Hubbard correlation potential that

favors single occupancy of the localized d-orbitals, and the Hund potential forcing the five

d-orbital spins to align, and of the hopping amplitudes between neighboring atoms provides

the correct band gap for the host III-V semiconductor and an appropriate exchange splitting

of the Mn d-levels. The TB model is a semi-phenomenological theory, however, it shares the

virtue with first principles approaches of treating disorder microscopically. The decoherence

of Bloch quasiparticle states or effects of doping and disorder on the strength of the sp − d

16

exchange coupling and effective Mn-Mn interaction are among the problems that have been

analyzed using this model.[94, 95, 96, 97, 98] A disadvantage of the tight-binding model

approach is that normally neglects Coulomb interaction effects which influence the charge

and spin densities over several lattice constants surrounding the Mn ion positions and hard

to reach large system sizes..

2.4.3 k · p Hamiltonian Theories

In optical experiments one typically determines both energy gaps and oscillator strengths of

the of the transitions. Thus it can be advantage if the optical matrix elements can also be

used as inputs in the band structure calculation. In the k ·p method the band structure over

the entire Brillouin zone can be extrapolated from the zone center energy gaps and optical

matrix elements. The k · p method is, therefore, particularly convenient for interpreting

optical spectra. In addition, using this method one can obtain analytic expression for the

band dispersion and effective masses around high-symmetry points.

In the metallic regime, where the largest critical ferromagnetic temperatures are achieved

(for doping levels above 1.5%), semi-phenomenological models that are built on Bloch states

for the band quasiparticles, rather than localized basis states appropriate for the localized

regime, [99] provide the natural starting point for a model Hamiltonian which reproduces

many of the observed experimental effects. Recognizing that the length scales associated with

holes in the DMS compounds are still long enough, a k·p envelope function description of the

semiconductor valence bands is appropriate. Since for many properties, e.g. anomalous Hall

effect and magnetic anisotropy, it is necessary to incorporate intrinsic spin-orbit coupling

in a realistic way, the six-band (or multiple-band, in general) Kohn-Luttinger (KL) k · pHamiltonian that includes the spin-orbit split-off bands is desirable.[3, 61] The approximation

of using the KL Hamiltonian to describe the free holes is based primarily in the shallow

acceptor picture demonstrated by the experiments [67, 70, 100] in (Ga,Mn)As and (In,Mn)As

and must be reexamined for any other DMS materials that this model is applied to.

The effective Hamiltonian considered within this model is

H = HKL + Jpd

∑

i,I

SI · siδ(ri − RI), (2.1)

where HKL is the six-band (multiple-band) Kohn-Luttinger (KL) k · p Hamiltonian,[3,

17

101] the second term is the short-range antiferromagnetic kinetic- exchange interaction

between local spin SI at site RI and the itinerant hole spin si at site ri.The kinetic part of the

Hamiltonian which is well defined in k space produces correct band structure. However,in

the second term of the Hamiltonian defined in real space it is assumed that the holes are

uniformly distributed which means that the second term is considered at a MF level and the

disorder in the Mn doping is neglected.

The k · p approach has the advantage that it focuses strongly on the magnetic degrees

of freedom introduced by the dilute moments. This approach makes it possible to use

standard electron-gas theory tools to account for hole-hole Coulomb interactions [102, 103].

The envelope function approximation is simply extended to model magnetic semiconductor

heterostructures, like superlattices or quantum wells [104, 105, 106, 107]. This strategy

will fail if the p− d exchange is too strong and the Mn acceptor level is correspondingly too

spatially localized or too deep in the gap. For example, Mn-doped GaP and GaN compounds

are likely less favorable for this approach than (III,Mn)As and (III,Mn)Sb compounds.

Generally speaking, this approach may be applicable only for metallic weakly hybridized

systems (e.g. optimally doped GaMnAs), no good for deep impurity levels and disorder is

not treated correctly.

2.4.4 Impurity Band and Polaronic Models

There has also been theoretical work on (III,Mn)V DMS materials based on still simpler

models in which holes are assumed to hop between Mn acceptor sites, where they interact

with the Mn moments via phenomenological exchange interactions [77, 108, 109, 110, 111].

These models have the advantage of approaching the physics of the insulating dilute Mn

limit, and can also be adapted to include the holes that are localized on other ionized

defects besides the Mn acceptors through dynamical mean field (or CPA) techniques. The

free-parameter nature of this phenomenological approach and their oversimplified electronic

structure allows one to make only qualitative predictions.

Hamiltonians used in these studies have a form of (or similar to),

H = −t∑

<ij>,σ

c†iσ cjσ + J∑

I

SI · σI , (2.2)

where c†iσ creates a hole at site i with spin σ, the hole spin operator σI = c†Iααβ cIβ, and αβ

are the Pauli matrices. Through nearest-neighbor hopping, the carriers can hop to any site

18

of square or cubic lattice. The interaction term is restricted to a randomly selected but fixed

set of sites, denoted by I.

Other related models assume that the Mn acceptors are strongly compensated so that

the density of localized holes is much smaller than the density of Mn ions, leading to a

polaronic picture in which a single hole polarizes a cloud of Mn spins [112, 113]. The free-

parameter nature of these phenomenological models means that they have only qualitative

predictive power. They are not appropriate for the high Tc (Ga,Mn)As materials which are

heavily doped by weakly compensated Mn acceptors and are metallic. On the other hand,

the impurity band models may represent a useful approach to address experimental magnetic

and transport properties of ferromagnetic (Ga,Mn)P where holes are more strongly localized

[84, 85, 86].

All the proposed theoretical models that we discussed to explore the properties of DMSs

have some strengths and weakness. Hence, it is very obvious that the models discussed above

are not enough to study these systems. The need for a microscopic model that correctly

account for correct lattice structure, band dispersion, disorder and spin-orbit interaction is

inevitable. The following sections will be devoted to construct and study such a model.

19

CHAPTER 3

MODELING DMS

3.1 Introduction

In the previous chapter, we described some of the Hamiltonians, as well as the techniques

that have been used to study DMS. The models that include all the relevant 6 orbitals in the

valence band and the spin-orbit interactions, in addition to the magnetic exchange, have only

been studied approximately, mostly using mean-field theory which neglects the disordered

distribution of the magnetic impurities. Since the uniform distribution of the doped holes is

assumed, it is clear that the approach is biased towards the ”Valence Band” regime.[6, 1, 61]

On the other hand, the studies which take into account the non-uniform hole distribution

have been performed on very simple systems lacking the correct lattice geometry, realistic

number of orbitals, dispersion of the valence band, and spin-orbit interaction. One of the

main goals of this thesis is to develop a realistic model for DMS and study it with unbiased

techniques. This chapter is divided into several sections: a phenomenological multi-band

orbital model with unbiased methods that allow the consideration of disorder is presented

in Section 3.2. And the correct valence band structure, lattice geometry, disorder and spin-

orbit interaction are part of the Hamiltonian presented in Section 3.3. Finally, numerical

results are presented in Section 3.4

3.2 Multiple Orbital Model

In Chapter 2, we described the phenomenological single orbital tight binding model that has

been successfully studied with unbiased numerical techniques. But due to its simplicity this

model[110, 114] fails to reproduce important properties of DMS: first, Tc is maximum for

50% hole concentration (p = 0.5) and zero for the uncompensated case(p = 1) which does

not agree with experimental results[6]. This may be the result of considering a single orbital

20

band when it is well known that at least two bands are important in DMS. A simple way of

addressing this problem is to consider a multi-orbital Hamiltonian.

It is known that in Mn-doped GaAs, the Mn ions substitute for Ga cations and contribute

itinerant holes to the valence band. The Mn ions have a half-filled d-shell which acts as a

S = 5/2 local moment. Due to a strong spin-orbit (SO) interaction, the angular momentum

L of the p-like valence bands mixes with the hole spin degree of freedom s and produces low-

and high-energy bands with angular momentum j = 1/2 and 3/2, respectively. A robust

SO split between these bands causes the holes to populate the j = 3/2 state, which itself

is split by the crystal field into a mj = ±3/2 band with heavy holes and a mj = ±1/2

band with light holes. This is the reason why we choose to study two bands since this is

the relevant number of orbitals in most III-V DMS. Since we do not work in a (j, mj) basis

our Hamiltonian does not capture the orbital mixing in the Hund term.[114] However, we

roughly consider the diagonal SO effects in the magnetic interactions by allowing different

values of J in the two orbitals considered. The simple two-band model for DMS proposed

here is given by the Hamiltonian

H= −∑

l,ij,α

tl(c+l,i,αcl,j,α + H.c.)−

∑

l,I

JlSI · sl,I , (3.1)

where l=1,2 is the band index (not to be confused with angular momentum), i, j label

sites (nearest neighbors for the hopping term), cl,i,α creates a hole at site i in the band l,

sl,i =∑

α,β c+l,i,ασαβ cl,i,β is the spin- operator of the mobile hole (σ = Pauli vector), α and

β are spin indices, Jl is the coupling between the core spin and the electrons of band l,

and SI is the spin of the localized Mn ion at randomly selected sites I, assumed classical

in the MC simulations. tl is the hopping term in band l. The inter-band hopping t12

(= t21) is zero at the nearest neighbor level in cubic lattices [115]. Even if t12 is explicitly

added, the conclusions are similar as reported here [116]. While real DMS materials have

zincblende (ZB) structures, in this first nonperturbative study of a multiband DMS model

the simplicity of a cubic lattice allows us to focus on the dominant qualitative tendencies,

a first step toward future quantitative studies with realistic ZB lattices. The model will be

studied using DMFT[117] and MC techniques.

21

0 0.025 0.050 0.075 0.100 0.125 0.150 0.1750

0.002

0.004

0.006

0.008

0.010

ω

J2/J1=1.5

J2/J1=1.25

J2/J1=1.125

J2/J1=1

J2/J1=0.875

J2/J1=0.75

J2/J1=0.675

J2/J1=0.5

TC/W

1

p

-4.0 -3.5 -3.0 -2.5 -2.0 -1.50

0.1

0.2

0.3

0.4

de

nsity o

f sta

tes

Figure 3.1: Tc versus the carrier concentration p, at various J2/J1, obtained with the DMFTtechnique. Here, x = 0.05, W1/W2 = 1, and J1/W1 = 0.5. The inset shows the correspondingDOS at T = 0.

3.2.1 DMFT Results

In Fig. 3.1, we show Tc vs. p, for different ratios J2/J1 and at fixed W2/W1=1 and

J1/W1 = 0.5, situation corresponding to the existence of a well-defined l=1 IB (although

p≤x in real DMS, in this paper the case p>x will also be studied for completeness, as done in

Ref.[30, 118]). The inset shows the total interacting DOS evolution as a function of J . The

IB’s overlap if |J2/W2 − J1/W1| < 0.5. If the IB do not overlap, then each one determines

Tc separately, causing the double-peak structure observed for some J2/J1 ratios. The band

with the largest Jl/Wl is filled first, for smaller µ’s. At all p’s, it is found that Tc is maximum

when J2/J1 = 1, namely when the IB’s fully overlap. The dependence of Tc on J2/J1 at

fixed p is in Fig. 3.2(a). The maximum value is achieved when the bands fully overlap (i.e.

at J2/J1=1). Once the bands decouple, the value for Tc matches one-band model results.

22

0 0.5 1.0 1.5 20

0.002

0.004

0.006

0.008

0.010(b)(a)

p=0.07

p=0.05

p=0.02

TC/W

1

J2/J1

0 0.2 0.4 0.6 0.8 1.0 1.2

J1/W1

0 0.025 0.050 0.075 0.100 0.125 0.150

0.002

0.004

0.006

0.008

W2/W1=1

W2/W1=0.75

W2/W1=0.625

W2/W1=0.5

W2/W1=0.35

(c)

TC/W

1

p

Figure 3.2: Results obtained with the DMFT approximation: (a) Tc vs. J2/J1, atW1/W2 = 1, and J1/W1 = 0.5, for the values of p indicated in (b). At p = 0.02 andJ2/J1 = 0, a finite Tc/W1 ∼ 0.0037 is caused by the l=1 band. At p = 0.05, Tc is notzero for J2/J1 ∈ (0.6, 1.35), and it increases significantly due to band overlap. The casep = 0.07 corresponds to the first IB completely filled. (b) Tc vs. J1/W1, at W1/W2 = 1and J1/W1 = J2/W2, for the p’s indicated. (c) Tc vs. p at different ratios W2/W1, fixingJ1 = J2 = 2. In all the frames x = 0.05.

Let us consider now how changes in bandwidths influence Tc. In Fig. 3.2(c), we show

Tc vs. p parametric with W2/W1, at fixed J1/W1 = 0.5 (intermediate coupling), and with

J2/J1 = 1 the relative position of the bands is fixed by shifting numerically the l=2 valence

band such that both valence bands start at the same energy, to mimic the degeneracy of

the light and heavy bands of GaAs at Γ. At small W2/W1 the second IB shall be located

in a region of ω smaller (i.e. farther from the valence bands), than the energy interval

occupied by the l = 1 IB. Hence, the l=2 IB will be the first to be filled. Decreasing J2/W2,

the second band moves to the right on the ω axis, towards the location of the first band.

While the bands are still separated, each gives its own contribution to Tc. The curves with

23

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.500

0.002

0.004

0.006

0.008

0.010

J1/W1=0.75

J1/W1=0.5

J1/W1=0.375

J1/W1=0.25

J1/W1=0.1875

J1/W1=0.125

J1/W1=0.0625

TC/W

1

p

-4.0 -3.5 -3.0 -2.5 -2.0 -1.50

0.2

0.4

0.6

0.8

1.0

de

nsity o

f sta

tes

ω

Figure 3.3: Tc versus p at different ratios J1/W1, obtained using DMFT. The parametersx, W1, W2, and J1 are as in Fig. 3.1. The inset contains the DOS at T = 0.

W2/W1 = 0.35 and 0.5 correspond to decoupled IB, while those with W2/W1 = 0.625 and

0.75 correspond to partially overlapping bands. Again, Tc is maximized at all fillings when

the bands fully overlap (W2/W1=1), in good agreement with Ref.[119]. Although W2/W1=1

is not realistic in DMS, Mn doping materials with a relatively small heavy-light mass ratio

will favor a higher Tc.

Once established that Tc is maximal for all p when J2/J1=1 and W2/W1=1, let us analyze

Tc vs. p when J1/W1 varies. The results for Tc, and total interacting DOS, are in Fig.3.3. At

small coupling J1/W1 ≪ 0.33, Tc is small, flat, and much extended on the p axis, qualitatively

similar to the one-band results.[30, 118] However, at intermediate coupling, Tc is nonzero in

the range from p = 0 to p = 2x, adopting a parabolic form with the maximum at p ∼= x,

in contrast with the one-band model which gives a null Tc when p ∼= x. The explanation

24

is straightforward: at p = x in the one-band model the IB is fully occupied leading to a

vanishing Tc, but for the same p in the two-band model both bands are half filled, which

ultimately leads to the highest value for Tc. The Tc dependence on the ratio J1/W1 at some

fillings p is displayed in Fig.3.2(b). At small coupling, Tc correctly increases quadratically

with J1/W1, but at strong coupling Tc incorrectly continues growing, result which will be

improved upon by the MC simulations shown below.

3.2.2 Monte Carlo Results

The Hamiltonian Eq. (3.1) was also studied numerically using MC techniques which are

described in Chapter 6 similar to those applied to Mn-oxide investigations.[109, 110, 120]

The fermionic sector is treated exactly, while a MC simulation is applied to the classical

localized spins. Cubic lattices with 53 and 63 sites were investigated. These lattice sizes

have been shown to be sufficient for the comparison with DMFT results and to unveil general

trends. In addition, the figures show only small variations for the Tc estimations using the

two lattices. One may suspect that actually the number of Mn spins may regulate the size

effects, rather than the number of sites. For the small values of x used in our study, the

number of Mn spins is also very small and serious size effects could be expected. However,

in practice this does not seem to occur, and moreover the comparison with DMFT shows

similar results using both techniques[117]. Perhaps in the small J’s regime, the delocalized

nature of the carriers smears the effects caused by the actual location of the Mn spins. These

issues deserve further study, but for our purposes of unveiling trends in the multi-parameter

space of DMS materials, the lattices here used are sufficient. Returning to the numerical

data ( as it explained in Chapter. 6), the spin magnetization is the order parameter that was

used to detect the ferromagnetic transition.[120] All quantities are in units of t1=1, and the

density of magnetic impurities is x ≈ 0.065. In Fig. 3.4(a), typical magnetization curves are

presented at several carrier densities p, and for J1=J2=J=4 and t2=1. In excellent agreement

with DMFT, it was observed that the estimated Tc is the highest for p=1. The value of J

used maximizes the critical temperature, and it was confirmed that it corresponds to the

case where the IB’s are about to become separated from the valence band. In Fig. 3.4(b),

it is shown how Tc increases with J up to J=4, in agreement with DMFT. The strong

coupling behavior is nevertheless different since Tc decreases at large J . This is caused by

hole localization in strong coupling, [110] beyond the capabilities of DMFT. The dependence

25

0.0 0.1 0.2 0.3T

0.4

0.6

0.8

1.0

M

p=0.25

p=0.5

p=1.0

p=1.625

0.0 0.1 0.2 0.3T

0.4

0.6

0.8

1.0

M

J=2J=4J=10J=16

0.0 0.5 1.0 1.5 2.0 2.5p

0.00

0.05

0.10

0.15

0.20

0.25

Tc

t2=0.35

t2=0.625

t2=0.75

t2=1.0

0 5 10 15J

0.00

0.05

0.10

0.15

0.20

0.25

Tc

p=0.25

p=0.5

p=1.0

p=1.5

(a) (b)

(c) (d)

Figure 3.4: (a) MC magnetization (in absolute value) vs. temperature (T ), with J1=J2=4and t1=t2=1, for the hole densities indicated. The dashed line is the exact asymptotic high-temperature value M∞, which tends to 0 only in the bulk limit. In this paper, the Tc onthe 53 cluster was (arbitrarily) defined as the T where M reaches ∼5% of the 1−M∞ value(indicated by arrows in (a) and (b)). Other criteria lead to very similar conclusions. (b)Magnetization vs. T for p=1, t1=t2, and several (equal) magnetic couplings J ; (c) Curietemperature vs. hole density for J=4 and different ratios of the band hoppings; (d) Curietemperature versus the (equal) magnetic interactions J for several hole densities and equalband hoppings t1 and t2. Results for 53 (63) lattices are indicated by open (filled) symbols.Error bars due to the disorder (up to 7 samples) are only shown for a few points for clarity.

of Tc on the ratio of band hoppings is in Fig. 3.4(c), varying p. These results are again in good

qualitative agreement with DMFT (Fig. 2(c)). The maximum Tc for all values of p occurs

when t2/t1=1. However, when t2 is very different from t1 the development of magnetism is

regulated by only one of the IB and the results are similar to those obtained with a single

band model, as shown in the curves for t2=0.35 and 0.625 in Fig. 3.4(c). For t2/t1 closer

to 1, a partial overlap of the IB occurs and a hump in Tc develops at p=1 (see curve for

t2=0.75).

In Fig. 3.4(d) we show that, once t2/t1 is optimized, a similar finite J maximizes Tc for

26

0.0 0.5 1.0 1.5 2.0 2.5p

0.00

0.05

0.10

0.15

0.20

0.25

Tc

J2/J1=2.0J2/J1=1.25J2/J1=1.125J2/J1=1.0J2/J1=0.4

0 1 2 3 4p

0.00

0.05

0.10

0.15

0.20

0.25

Tc J=2

J=4J=6J=8J=10J=16

-12 -9 -6ω

DO

S

(a) (b)

Figure 3.5: (a) Tc vs. ph obtained with MC on a 53 (63) lattice with t1=t2=1 for the valuesof J2/J1 indicated by the open (filled) symbols. J1 is fixed to 4, i.e. when the IB are aboutto separate from the valence band for band 1 (inset Fig. 5(b)). For J2 < J1 (e.g. J2/J1=0.4curve), Tc is regulated by the IB in band 1, since the IB for band 2 is deep into the valenceband. In this case, a single-band behavior is observed, with Tc maximized for p=0.5. ForJ2 > J1, both IB play a role. For J2 ≫ J1 (see J2/J1=2), the two IB do not overlap, and for0 ≤ p ≤ 1 Tc is determined by the band-2 IB reaching a maximum at p=0.5 and vanishingat p=1. For larger p, Tc is controlled now by the IB 1, and it raises again passing througha maximum at p=1.5 and vanishing at p=2. For J2/J1=1.25, the two IB overlap and weobserve residual local maxima at p=0.5 and 1.5, related to the single band physics, and anew local maximum at p=1 due to IB overlap for the corresponding value of the chemicalpotential. Tc at p=0.5 is boosted by the partial IB overlap as well. (b) Tc vs. p for t2/t1=1and several Js. Inset: low-T DOS.

several p’s. In all cases, the optimal J best balances the weak coupling behavior, with mobile

holes not much affected by the interaction with the Mn ions, and the strong coupling region

where the hole spins strongly align with the Mn spins, becoming localized. This “sweet spot”

is achieved when the IB’s are about to be separated from the valence bands.

Tc vs. p, at several ratios J2/J1 and for t1=t2 is presented in Fig. 3.5(a). In agreement

with DMFT (Fig. 3.3), Tc is maximized for all values of ph if J2=J1, with the highest value

at p=1. Overall, there is an excellent agreement with DMFT, as described in the caption

27

(this agreement is even quantitative for the optimal Tc and J coupling, once the factor√

2d

(d=dimension), use to rescale the hopping in DMFT, is considered). Tc vs. p for several

J1=J2=J is in Fig. 3.5(b). At small J , again the MC results resemble those obtained with

DMFT (Fig. 3.3. For, e.g., J=2 the IB are not formed yet (inset). In this regime, Tc remains

finite, although small, even for p larger than 2. Increasing J , a nonzero Tc is obtained only

for p between 0 and 2, due to IB formation. Tc reaches a maximum at J=4.

3.2.3 Conclusions

We have carried out the first study of a multiband model for DMS using a powerful

combination of nonperturbative techniques, DMFT and MC. We found the parameter regime

that maximizes Tc. This happens at intermediate couplings and for all hole densities when

J1/J2=1 and t1/t2=1. The maximum Tc is obtained at p ∼= x, in contrast with the one-band

model which has a vanishing Tc at the same doping. In addition, Tc at filling p ∼= x/2 in the

one-band case is smaller than with two bands by a factor ∼2. In view of the simplicity of the

main results, it is clear that adding an extra band to the calculations (which is relevant for

system with negligible SO, but considerably raises the CPU cost) will only lead to a further

increase in Tc when all the IB overlap.

The excellent agreement DMFT-MC is somewhat surprising due to the fact that Monte

Carlo considers the influence of the random location of the Mn sites much better than DMFT.

However, at small and intermediate J’s, the carriers can be sufficiently delocalized that a

smearing effect may occurs and considering the quenched disorder only in average appears

to be sufficient. Certainly at large J’s the MC and DMFT methods give totally different

answers, with MC capturing the correct localization result.

The general qualitative picture presented here can be used to search for DMS with even

higher Tc’s than currently known. Our results suggest that semiconductors with the smallest

heavy to light hole mass ratio, such as AlAs, could have the highest Tc if the couplings J could

be tuned to its optimal value. The present effort paves the way toward future nonperturbative

studies of DMS models using realistic ZB lattices, and points toward procedures to further

increase the Curie temperatures.

28

3.3 Realistic Model

In Section 3.2, we showed how consideration of multiple orbitals fixed some of the problems

that single orbital models had for capturing the properties of DMS. However, some important

behavior of these materials have been still not reproduced. This is not surprising because

the two-band model neglects many important characteristics of DMS: the lattice structure

used in this model is not realistic (cubic lattice structure instead of FCC), the hoppings

parameters were arbitrary, and the spin-orbit interaction was not considered. The calculated

magnetization curves never showed Curie-Weiss behavior and with arbitrary parameters it

was impossible to determine whether an impurity band was present or not for realistic

materials such as (Ga,Mn)As. Therefore, we are going to construct a more realistic

Hamiltonian that reproduces the top of the valence band (which is relevant for hole doping)

of the III-V compounds and, at the same time, handles the Hund coupling between the spin

of the holes and the spin of the randomly distributed magnetic impurities by considering

the correct lattice geometry and spin-orbit effect. The band structure of Zinc-Blende

type semiconductors has been accurately obtained using a variety of different approaches

like pseudo-potential methods, tight-binding techniques and by careful consideration of the

symmetries involved through the application of group theory. [121] However, many of these

calculations involve a large number of orbitals per site making them impossible to simulate

with present day computers. The successful Luttinger-Kohn model [122], that describes the

top of the valence band of semiconductors with diamond structure, has existed for many

years, and it has been used in mean-field and approximated studies of DMS but it has

not lend itself to numerical simulations because it is defined in momentum rather than in

coordinate space.

The goal of this section is to provide a realistic tight-binding Hamiltonain in real space

that reproduces the top of the valence band of Zinc-Blende type semiconductors with the

smallest possible number of degrees of freedom per site.

3.3.1 Tight Binding Approach

Although we are going to provide a Hamiltonian that can be applied to any semiconductor

with Zinc-Blende type structure, we are going to use Mn doped GaAs as our example. The

(Ga,Mn)As compounds have nominal atomic structures [Ar]3d104s2p1 for Ga, [Ar]3d54s2 for

29

Mn, and [Ar]3d104s2p3 for As. In GaAs the Ga and As atoms bond covalently. Both atoms

share the electrons that they have in their 4s and 4p shells. Ga shares 3 electrons while As

shares 5. The hybridized orbitals have character sp3. Although the s and p orbitals have to

be considered in order to obtain the correct band structure, we are interested in light doping

of the valence band. It is well known that around the Γ point the valence band of GaAs has

j = 3/2 character which arise from the original p orbitals. Thus, in order to construct the

simplest model that captures this feature we are going to consider the p orbitals only. The

three p orbitals px, py and pz in each ion can be populated with particles with spin up or

down.

To study GaAs we should consider two interpenetrating fcc lattices separated by a

distance (a0/4, a0/4, a0/4). The Ga atoms seat in one of the fcc lattices and the As

atoms in the other. Each atom has 4 nearest neighbors of the opposite species located

at (a0/4, a0/4, a0/4), (a0/4,−a0/4,−a0/4), (−a0/4, a0/4,−a0/4) and (−a0/4,−a0/4, a0/4).

Since we are only interested in obtaining the valence band we are going to consider the

bonding combinations of the Ga and As p orbitals.[121, 123, 124] This leads to an effective

fcc lattice for the Ga ions with three p bonding orbitals at each site that can be occupied

by particles with spin up or down. Thus, working in this |p, α > basis there are 6 states

per site of the fcc lattice. We will consider the nearest neighbor hopping of holes in this

lattice to construct the effective tight-binding Hamiltonian. The twelve nearest neighbors

are located at (±a0/2,±a0/2, 0), (±a0/2, 0,±a0/2), (0,±a0/2,±a0/2) considering the 4 sign

combinations for the three sets of points provided.

In order to calculate the hoppings we follow Slater and Koster.[125] The nearest neighbors

in our effective fcc lattice are the second nearest neighbors in the original diamond structure.

From Table I in Ref. [125]we see that the relevant overlap integrals in this case are

Exx = l2(ppσ) + (1 − l2)(ppπ)

Exy = lm[(ppσ) − (ppπ)]

Exz = ln[(ppσ) − (ppπ)]. (3.2)

For the 12 nearest neighbors in the fcc lattice (p, q, r) are given by two of the indices

taking the value ±1 and the remaining one 0. We will label the 12 sites by (µ, ν) with µ and

ν taking the values ±x, ±y,and ±z. Since l = p√p2+q2+r2

with (p, q, r) = (1, 1, 0), etc.(we are

30

following Slater’s notation). So that l, m and n are equal to 0 or ±1/√

2. Then the hoppings

to the twelve neighbors are:

− tµνaa = Exx(µ, ν) =

1

2[(ppσ) + (ppπ)] = −t‖xx,

−tµνaa = Exx(µ, ν) = (ppπ) = −t⊥xx,

−tµνab = Exy(µ, ν) = ±1

2[(ppσ) − (ppπ)] = ∓txy, (3.3)

with the minus (plus) sign for the case in which µ and ν have the same (opposite) sign.

Also notice that the inter-orbital hopping is only possible when (µ, ν) and ab are in the same

plane, i.e., there is no perpendicular inter-orbital hopping.

3.3.2 The model neglecting spin-orbit interaction

The hoppings calculated above allow us to write the following Hamiltonian for holes in the

valence band interacting with localized spin of randomly doped Mn2+ ions:

H =∑

i,µ,ν,α,a,b

tµνab(c†i,α,aci+µ+ν,α,b + h.c.) + J

∑

I,a

sIa · SI, (3.4)

where c†i,α,a creates an electron at site i = (ix, iy, iz) in orbital a with spin projection α,

sIa=

∑

αβ c†I,α,aσαβcI,β,a is the spin of the mobile hole, the Pauli matrices are denoted by σ,

SI is the localized Mn spin 5/2 at site I (only a small fraction of the total number of sites

N since Mn replaces a small number of Ga). tµνa,b are the hopping amplitudes for the holes

that were defined in Section 3.3.1, and J > 0 is an antiferromagnetic (AF) coupling between

the spins of the mobile and localized degrees of freedom. The density 〈n〉 of itinerant holes

is controlled by a chemical potential µ. The sites i are in the fcc sublattice and the vectors

µ, ν indicate the 12 nearest neighbors of each site i by taking the values ±x, ±y, and ±z,

with µ 6= ν.

Notice that we basically have only three different hoppings: two intraorbital ones t⊥aa and

t‖aa and the interorbital ones tab which each have the same absolute value (but not always the

same sign) for all combination of orbitals and neighbors. The interorbital hoppings that have

the sign reversed are the ones towards sites labeled by µ and ν with opposite signs. Also,

31

notice that the interorbital hoppings occur only in the planes defined by the two orbitals, i.e.,

it vanishes in the direction perpendicular to the plane were the two orbitals are. This can

be seen in the expressions provided in Eq. 3.2. In order to obtain material specific values of

the hopping parameters we are going to write the Hamiltonian matrix in momentum space

for the undoped case, i.e. Eq. 3.4 with J = 0, by using Table II or III in Ref. [125]. It is

given by:

Tx −txysxsy −txysxsz

−txysxsy Ty −tx,ysysz

−tx,ysxsz −tx,ysysz Tz

, (3.5)

for spin up and a similar block for spin down. Here, Tx = 4t‖xx(cxcy + cxcz) + 4t⊥xxcycz,

Ty = 4t‖xx(cxcy + cycz) + 4t⊥xxcxcz, and Tz = 4t

‖xx(cxcz + cycz) + 4t⊥xxcxcy with ci = cos(aki),

si = sin(aki), where a = a0/2 in Slater’s notation (a0 is the Zinc-Blende lattice constant)

and ki are the momentum components.

Luttinger and Kohn [122]studied the movement of holes in the valence band of semi-

conductors with diamond lattice symmetry. Working in momentum space they found an

expression for the Hamiltonian matrix that describes the top of the valence band i.e., the

neighborhood of the Γ point. In the |p, α > basis the matrix has the form:

Ak2x + B(k2

y + k2z) Ckxky Ckxkz

Ckxky Ak2y + B(k2

x + k2z) Ckykz

Ckxkz Ckykz Ak2z + B(k2

x + k2y)

(3.6)

There is a similar block for spin down. Where A, B and C are constants that can be

defined in terms of the Luttinger parameters γ1, γ2, and γ3[126] and the lattice parameter a.

The accepted values for GaAs are (γ1, γ2, γ3) = (6.85, 2.1, 2.9).[61] The constants are given

by:

A = − ~2

2m(γ1 + 4γ2),

B = − ~2

2m(γ1 − 2γ2),

C = −6~

2

2mγ3, (3.7)

where m is the mass of the bare electron. Remembering that Eq. 3.6 is an approximation

which is valid for the top of the valence band (i.e. around the Γ point), we can expand the

32

cosines and sines in Eq. 3.5 and hence obtain the matrix shown in Eq. 3.6 if a constant term

along the diagonal that just shifts the bottom of the valence band to 8t‖xx +4t⊥xx instead of 0

is disregarded. Comparing the coefficients we obtain expressions for the hoppings in terms

of the Luttinger parameters and the lattice constant a0 = 2a [127]:

t‖x,x =~

2

8ma2(γ1 + 4γ2) =

~2

2ma02(γ1 + 4γ2),

t⊥x,x =~

2

8ma2(γ1 − 2γ2) =

~2

2ma02(γ1 − 2γ2),

tx,y =3~

2

4ma2γ3 =

3~2

ma02γ3. (3.8)

Because we want to write a tight-binding Hamiltonian for holes that will dope the bottom

of the band, we will reverse the signs of the hoppings since the band obtained with Eq. 3.6

gets reflected with respect to zero by reversing the signs of A, B, and C. [128]. Then, for

GaAs a0=5.64 A [121], and from Eq. 3.8 we obtain:

t‖x,x = −1.82eV,

t⊥x,x = 1.20eV,

tx,y = 2.08eV. (3.9)

For completeness, we will also provide the hoppings for GaN . The lattice constant is

given by a = 4.42A [129] and the Luttinger parameters are γ1 = 2.463, γ2 = 0.647 and

γ3 = 0.975.[3, 121] With these values the hopping parameters for GaN are:

t‖x,x = − ~2

2ma2(γ1 + 4γ2) = −0.976eV,

t⊥x,x = − ~2

2ma2(γ1 − 2γ2) − tx, x‖ = 0.524eV,

tx,y =3~

2

ma2γ3 = 1.13eV. (3.10)

It is important to notice that the phenomenological evaluation of the hopping parameters

given by Eq. 3.8 requires three independent parameters. Whereas, if we look at the expression

for the hoppings given in Eq. 3.3 in terms of overlap integrals, it would appear as if

only two parameters were necessary and the three hoppings should be interrelated. The

phenomenological evaluation is more accurate though, because it considers the influence of

the neglected bands in the shape of the valence band at Γ.

33

3.3.3 Spin-Orbit interaction

The spin-orbit interaction is a well-known phenomenon that manifests itself in lifting the

degeneracy of one-electron energy levels in atoms, molecules, and solids. This interaction

is a relativistic effect that scales with the atomic number. Thus, for semiconductors with

heavier elements such as GaAs, it is expected that effect would be important. In atomic

physics HSO=λ~l.~s. It mixes the angular momentum (l=1 for the p orbitals) with the holes

spin degrees of freedom producing states with j = 3/2 and j = 1/2. In a cubic lattice

with the diamond symmetry, Luttinger and Kohn showed that the states with j = 1/2 get

separated from the ones with j = 3/2 which are the relevant states at the top of the valence

band. As a result, only 4 states per site, instead of 6, become relevant when the spin-orbit

interaction is considered.[122]. However, since the spin-orbit spliting ∆SO of III-V materials

range from 0.017 eV for GaN to 0.75 eV for GaSb we are going to keep the 6-orbitals.

To take into account the spin-orbit interaction we will have to make a change of base from

|p, α > to |j, mj >. This change of base has been studied by Kohn and Luttinger.[122] The

Luttinger-Kohn matrix (Eq. 3.6) in the |j, mj > base is presented in Eq.(A8) of Ref. [61].

|1〉 ≡ |j = 3/2, mj = 3/2〉

|2〉 ≡ |j = 3/2, mj = −1/2〉

|3〉 ≡ |j = 3/2, mj = 1/2〉

|4〉 ≡ |j = 3/2, mj = −3/2〉

|5〉 ≡ |j = 1/2, mj = 1/2〉

|6〉 ≡ |j = 1/2, mj = −1/2〉 (3.11)

The basis (3.11) is related to the orbital angular momentum (ml = 1, 0,−1) and spin

34

(σ =↑, ↓) eigenstates by

|1〉 = |ml = 1, ↑〉

|2〉 =1√3|ml = −1, ↑〉 +

√

2

3|ml = 0, ↓〉

|3〉 =1√3|ml = 1, ↓〉 +

√

2

3|ml = 0, ↑〉

|4〉 = |ml = −1, ↓〉

|5〉 = − 1√3|ml = 0, ↑〉 +

√

2

3|ml = 1, ↓〉

|6〉 =1√3|ml = 0, ↓〉 −

√

2

3|ml = −1, ↑〉 (3.12)

or

|1〉 = − 1√2

(

|X, ↑〉 + i|Y, ↑〉)

|2〉 =1√6

(

|X, ↑〉 − i|Y, ↑〉)

+

√

2

3|Z, ↓〉

|3〉 = − 1√6

(

|X, ↑〉 + i|Y, ↑〉)

+

√

2

3|Z, ↑〉

|4〉 =1√2

(

|X, ↓〉 − i|Y, ↓〉)

|5〉 = − 1√3

(

|X, ↓〉 + i|Y, ↓〉)

− 1√3|Z, ↑〉

|6〉 = − 1√3

(

|X, ↑〉 − i|Y, ↑〉)

+1√3|Z, ↓〉 (3.13)

Applying the same change of basis to T (Eq. 3.5) we obtain the 6×6 matrix T ′ = MTM−1,

where M is the change of basis matrix provided in the Appendix.

H =

Hhh −c −b 0 b√2

c√

2

−c∗ Hlh 0 b − b∗√

3√2

−d

−b∗ 0 Hlh −c d − b√

3√2

0 b∗ −c∗ Hhh −c∗√

2 b∗√2

b∗√2

− b√

3√2

d∗ −c√

2 Hso 0

c∗√

2 −d∗ − b∗√

3√2

b√2

0 Hso

(3.14)

with

35

Hhh = 4t‖x,xcxcy + (2t⊥x,x + t‖x,x)(cxcz + cycz)

Hlh =2

3(5t‖x,x + t⊥x,x)(cycz + cxcz) +

4

3(t‖x,x + 2t⊥x,x)cxcy

Hso = 4(2t

‖x,x + t⊥x,x)

3(cxcy + cycz + cxcz) + ∆so

b =−4√

3tx,y(sxsz + isysz)

c =2√3(t‖x,x − t⊥x,x)(cxcz − cycz) − i

4√3tx,ysxsy

d = 4

√2

3(t‖x,x − t⊥x,x)cxcy −

(t‖x,x − t⊥x,x)

3√

2(cycz + cxcz). (3.15)

The six band model Kohn-Luttinger Hamiltonian, H, in the representation of vectors

(3.11) has the same form, but the coefficients are given by:

Hhh =~

2

2m

[

(γ1 + γ2)(k2x + k2

y) + (γ1 − 2γ2)k2z

Hlh =~

2

2m

[

(γ1 − γ2)(k2x + k2

y) + (γ1 + 2γ2)k2z

Hso =~

2

2mγ1(k

2x + k2

y + k2z) + ∆so

b =

√3~

2

mγ3kz(kx − iky)

c =

√3~

2

2m

[

γ2(k2x − k2

y) − 2iγ3kxky

]

d = −√

2~2

2mγ2

[

2k2z − (k2

x + k2y)

]

. (3.16)

where ∆so is the energy gap between the split-off orbital and the top of the VB. Expanding

ci and si in Eq. 3.15 up to k2 and replacing the hoppings by their expressions in terms of γi

(Eq. 3.8) we found that Eq. 3.15 is identical to Eq. 3.16.

If ∆so is very large, the two j = 1/2 orbitals can be neglected close to the Γ point and

many authors consider the 4 orbital model with j = 3/2 given by 4× 4 section of the matrix

shown in Eq. 3.14.

36

3.3.4 Hoppings between |j, mj > states in real space

Now we need to detemine the hoppings in real space between the orbitals characterized by

j = 32

and mj = ±32

and ±12

and j = 12

with mj = ± 12.

The next step is to apply the base transformation into the basis |j, mj > to the c operators

that appear in Eq. 3.4,

c†i,σ,x = σ(−1√

2c†i, 3

2,σ 3

2

+1√6c†i, 3

2,−σ 1

2

) − 1√3c†i, 1

2,−σ 1

2

c†i,σ,y = −i(1√2c†i, 3

2,σ 3

2

+1√6c†i, 3

2,−σ 1

2

− σ√3c†i, 1

2,−σ 1

2

)

c†i,σ,z =

√

2

3c†i, 3

2,σ 1

2

− σ

√

1

3c†i, 1

2,σ 1

2

ci,σ,x = σ(−1√

2ci, 3

2,σ 3

2

+1√6ci, 3

2,−σ 1

2

) − 1√3ci, 1

2,−σ 1

2

ci,σ,y = (1√2ci, 3

2,σ 3

2

+1√6ci, 3

2,−σ 1

2

) − iσ√3ci, 1

2,−σ 1

2

ci,σ,z =

√

2

3ci, 3

2,σ 1

2

− σ

√

1

3ci, 1

2,σ 1

2

(3.17)

Replacing these operators in Eq. 3.4 and rearranging the terms we find that the

intraorbital hoppings are given by:

tx,y

σ 3

2,σ 3

2

= t‖xx

tx,y

σ 1

2,σ 1

2

=t‖xx + 2t⊥xx

3

ty,z

σ 3

2,σ 3

2

= tx,z

σ 3

2,σ 3

2

=t‖xx + t⊥xx

2

ty,z

σ 1

2,σ 1

2

= tx,z

σ 1

2,σ 1

2

=5t

‖xx + t⊥xx

6

tµ,ν

σ 1

2,σ 1

2

=2t

‖xx + t⊥xx

3(3.18)

Now let’s consider the inter-orbital hoppings between orbitals with j = 3/2:

37

tx,zσa,−σa′ = ty,z

σa,−σa′ =t⊥xx − t

‖xx√

12

tx,ya,−a′ = (tx,y

a,−a′)∗ = ty,z

σ 1

2,σ 3

2

= (ty,z

σ 1

2,σ 3

2

)∗ =i√3tµ,νxy

tx,zσa,σa′ = − σ√

3tµ,νxy

tx,yσa,σa′ = 0

tµ,νσa,−σa = 0. (3.19)

In the above expression a 6= a′. There are no interorbital hoppings between the two j = 1/2

orbitals i.e.,

tµ,νσa,−σa = 0. (3.20)

a = 12

represents mj for the orbitals with j = 12. The interorbital hoppings between

j = 3/2 and j = 1/2 orbitals are given by:

tx,z

σ 1

2,σ 3

2

= tx,z

σ 3

2,σ 1

2

= −tµ,νxy√6

ty,z

σ 3

2,σ 1

2

= (ty,z

σ 1

2,σ 3

2

)∗ = −itµ,νxy√6

tx,z

σ 1

2,−σ 1

2

= tx,z

σ 1

2,−σ 1

2

=tµ,νxy√2

ty,z

σ 1

2,−σ 1

2

= ty,z

σ 1

2,−σ 1

2

= iσtµ,νxy√2

tx,y

σ 1

2,σ 1

2

= tx,y

σ 1

2,σ 1

2

= −σ

√2(t⊥xx − t

‖xx)

3

tx,z

σ 1

2,σ 1

2

= ty,z

σ 1

2,σ 1

2

= tx,z

σ 1

2,σ 1

2

= ty,z

σ 1

2,σ 1

2

= σ(t⊥xx − t

‖xx)

3√

2. (3.21)

3.3.5 Hund Coupling:

Finally let us consider the expression for the Hund coupling term in Eq. 3.4 in the new basis.

It should be noticed that the Mn ions replace Ga, so they will be present in a subset of the

points of the fcc lattice that we consider. The spin operators in the |j, mj > base are given

by[61]:

38

sx =

0 0 12√

30 1√

60

0 0 13

12√

3− 1

3√

20

12√

313

0 0 0 13√

2

0 12√

30 0 0 − 1√

61√6

− 13√

20 0 0 −1

6

0 0 13√

2− 1√

6−1

60

sy = i

0 0 − 12√

30 − 1√

60

0 0 13

− 12√

3− 1

3√

20

12√

3−1

30 0 0 − 1

3√

2

0 12√

30 0 0 − 1√

61√6

13√

20 0 0 1

6

0 0 13√

21√6

−16

0

sz =

12

0 0 0 0 0

0 −16

0 0 0 −√

23

0 0 16

0 −√

23

00 0 0 −1

20 0

0 0 −√

23

0 −16

0

0 −√

23

0 0 0 16

(3.22)

Then,

JSI · sIα → J(Sx

I sxI + Sy

I syI + Sz

I szI)), (3.23)

(3.24)

where

sIα = c†IsαcI (3.25)

(3.26)

with

c†I =(

c†i, 3

2, 32

c†i, 3

2,− 1

2

c†i, 3

2,− 12

c†i, 3

2,− 3

2

c†i, 1

2, 12

c†i, 1

2,− 1

2

,)

(3.27)

39

where,

SxI = sinθicosφi

SyI = sinθisinφi

SxI = cosθi, (3.28)

since the localized spin 5/2 of the Mn is approximated by a classical spin. Now c†I,σα creates

a hole at site I with absolute value of mj = α (α = 3/2, 1/2 or 1/2) and sign of mj given by

σ.

The values of J are obtained from the experimental data and are material dependent. In

the notation of Ref. [130], J in Eq. 3.4 is given by βN0 where β has units of eV nm3 and N0 is

the concentration of cation sites and proportional to 4a−30 where a0 is the lattice parameter

of the material. β is considered to depend only on the characteristics of the parent material

and it is the same for all III-V semiconductors. Notice that β is called Jpd by other authors

[61]. Dietl estimates J for GaN assuming that the accepted values of GaAs is accurate and

given by J = 4βa−30 = −1.2eV [6, 130]. He also assumes that βN0 ≈ a−3

0 for all III-V

compounds. Using the parameters for GaAs we can estimate that

βIII−V =−1.2eV

N0

= −1.2eV a30 =

−1.2 × (0.564)3eV nm3 = −0.215eV nm3.

Then J for a general III-V material M is given by

JM =βIII−V

a30

. (3.29)

Thus J for GaN is given by

JGaN =βIII−V

a30GaN

=−0.215

(0.4423)eV = −2.49eV,

while

JGaAs = −1.2eV.

However, in the literature the value of J for GaAs ranges between −0.89eV ≤ J ≤−3.34eV [61, 130, 131] which corresponds to −0.04eV nm3 ≤ βIII−V ≤ −0.15eV nm3 , thus

40

for GaN we obtain −1.85eV ≤ J ≤ −6.93eV . The sign depends on the definition. It has

to be antiferromagnetic which means that we need to take it as a positive number in our

Hamiltonian.

Notice that using the parameters provided in this section, numerical calculation of the

temperature provides values directly in eV (no rescaling by the hoppings needed), so by

multiplying the result by 11,604 the temperature(T) in K is obtained.

3.3.6 Total Hamiltonian

We have constructed a real-space fcc-lattice Hamiltonian whose kinetic-energy term maps

into the Luttinger-Kohn model,[122] when k-space Fourier transformed and at k → 0. As

a consequence, the hopping amplitudes are functions of (tabulated) Luttinger parameters,

and thus they are precisely known.[126] In particular the hoppings used in our study are all

of order 1 eV . To incorporate the spin-orbit (SO) interaction, we work in the |j, mj〉 basis,

where j can be 3/2 or 1/2 (since we consider the p orbitals, l=1, relevant at the Γ point of

the GaAs valence band). Consequently, there are 6 possible values for mj, indicating that

this is a fully 6-orbital approach, arising from the 3 original p orbitals and the 2 hole spin

projections. The Hamiltonian is formally given by

H =1

2

∑

i,µ,ν,α,α′,a,b

(tµναa,α′bc

†i,αaci+µ+ν,α′b + h.c.) + ∆SO

∑

i,α

c†i,α 1

2

ci,α 1

2

+ J∑

I

sI · SI, (3.30)

where a, b take the values 12, 3

2(for j=3/2), or 1

2(for j=1/2), and α and α′ can be 1 or

−1. The Hund term describes the interaction between the hole spins sI (expressed in the

|j, mj〉 basis, these are 6 × 6 matrices; see [61, 116]) and the spin of the localized Mn ion

SI. The latter is considered classical (|SI|=1), since it is large S = 5/2.[132, 133] The

classical limit corresponds to lim~→0,S→∞ ~S = ~0 = 6.58× 10−16 eV s. Since the parameter

J , experimentally measured, is proportional to ~ via the Bohr magneton, it results that

J = K~ where K is a constant. Thus, lim~→0,S→∞ JS = lim~→0,S→∞ K~S = K~0 = J

(see Ref.[134, 135]). µ + ν are the 12 vectors indicating the 12 nearest-neighbor (NN)

sites of each ion located at site i, while I are random sites in the fcc lattice. ∆SO is

the spin-orbit interaction strength[121] (in GaAs ∆so=0.341 eV). The hopping parameters,

tµναa,α′b, are complex numbers, whose real and imaginary parts are functions of the Luttinger

parameters.[116]

41

3.4 Numerical Results In Finite Systems

3.4.1 Band Dispersion for the Non-Interacting Case

In this section, we address the study of our tight-binding Hamiltonian. We know that

even in real materials there are discrete momentum values since the samples are finite. In

reciprocal space the momentum values form a cubic structure with δki=2π/a0Ni, where

i = x, y, or z and Ni is the number of unit cubic cells in the sample along the i direction.

Since in laboratory samples Ni is of the order of Avogadro’s number we can replace ki by

a continuous variable. However, this is not true in the small finite samples that can be

studied numerically. When J = 0 we can diagonalize Eq. 3.30 using continuous values of k.

The results for GaAs along high symmetry directions in the first Brillouin zone (FBZ) are

indicated by the dashed red lines in Fig. 3.6.

Figure 3.6: Band structure for GaAs obtained by diagonalizing the Realistic Systemdescribed by Eq. 3.30 (continuous lines) with J = 0. The dashed red lines indicate theresults for the Luttinger-Kohn (LK) Hamiltonian.

As expected, the bottom of the band is at Γ and we observe the heavy hole, light hole and

42

split-off bands along high symmetry lines in the Brillouin zone shown in the figure. Since we

use the ”hole” language, here the top of the electronic valence band appears as the bottom

of the hole valence band. The lines in red are the eigenvalues of Eq.(A8) in Ref.[61]i.e., the

Luttinger-Kohn model results. In order to check the agreement with our results around the

point Γ, we shift our curves by the values 8t‖x,x +4t⊥x,x so that the bottom of our valence band

is at zero. The shifted curves are indicated in black in the figure. It can be seen that the

agreement between the curves obtained with our tight-binding model and Luttinger-Kohn

model is excellent at the bottom of the valence band.

3.4.2 Momentum discretization

To decide whether this approximation can be used to describe DMS, we present some

numerical results. When J 6= 0 and finite number of magnetic impurities are considered, the

diagonalization of Eq. 3.30 has to be performed in real space. Numerically, we study clusters

that contain N cubes of side a0 along each of the three spacial directions. Since there are 4

ions associated to each site of the cube in an fcc lattice, the total number of Ga ions in the

numerical simulations is given by NGa=4N3. This means that there is an equal number of

points inside the first Brillouin zone FBZ in momentum space. As we already mentioned,

the discrete lattice in momentum space is cubic. The side of the smallest cube is given by

b = 2π/Na0, which is the size of the mesh corresponding to N cubic cells along each of the

three spatial directions. The first Brillouin zone has the shape of a truncated octahedron

which is defined by the well known high symmetry points, such as L=(π/a0, π/a0, π/a0),

X=(2π/a0, 0, 0), U=(2π/a0, 2π/a0, 2π/a0). Points along the high symmetry directions inside

the FBZ for discrete systems obtained with a code written in real space are shown in Fig. 3.7.

3.4.3 Four band approximation

In the case of III-V semiconductors with strong spin-orbit interaction, i.e., large ∆so such

as in the case of GaAs and GaSb, only the j = 3/2 orbitals constitute the top of the

valence band. It is customary then to consider only these 4 orbitals to study properties of

these materials that involve light hole doping [61, 122]. This approximation would certainly

simplify numerical simulations since the number of degrees of freedom is reduced. In Fig. 3.8,

the band dispersion for the four band model is shown. While the heavy hole band dispersion

is captured by the 4-orbital approach, the second band dispersion away from Γ is very

43

0

1

2

E(e

V)

N=3N=4N=5N=6N=8Infinite System (J=0)

0

5

10

15

20

25

30

L Γ X U Γ K

Γ ΓL X U K∆

SO

Figure 3.7: Band structure for GaAs obtained by diagonalizing the Realistic Systemdescribed by Eq. 3.30 (continuous lines). The symbols indicate results obtained with ourreal space code in the finite lattices with N cubes along each spatial direction. The numberof momentum states available inside the FBZ is given by 4N3.

different from the light band dispersion obtained with 6 orbitals (Fig. 3.6). In fact, the

dispersion of the second orbital resembles the one for the split-off band that should not play

a relevant role at light doping if the spin-orbit interaction is strong.

Using the unbiased Monte Carlo (MC) method described in Chapter 6 we calculated the

magnetization in the 4-orbital approximation. We studied clusters that contain N = NxNyNz

unit cells (Ni is the number of cells along the spatial direction i) of side a0 (a0=5.64A,[121]

is the GaAs cubic lattice parameter). Because in an fcc lattice there are 4 ions associated

to each cell, the total number of Ga sites is NGa=4N . Since there are 4 single fermionic

states per site, the diagonalization of a 4NGa × 4NGa matrix is needed at each step of the

MC simulation, which demands large computational efforts. The diagonalization can be

performed exactly for values of NGa up to 864. Here, we analyze lattices with Ni = 4 (256

sites).

In Fig. 3.9, we present the temperature dependence of the magnetization (defined in

44

0

20

40

60 Shifted Tight Binding

Luttinger-Kohn Bands

Tight Binding

L Γ X U Γ

Figure 3.8: Band structure for GaAs obtained diagonalizing Eq. 3.14 (dashed blue lines).Thered lines indicate the results for the Luttinger-Kohn model. The black lines are our resultsshifted so that the bottom of the valence band is at 0.

0 50 100 150 200Temperature( K)

0

0.2

0.4

0.6

0.8

Mag

net

izat

ion

p=0.36p=0.45p=0.72p=0.81p=0.91p=1.0

x=8.5%

J=1.2 eV

(Ga,Mn)As

Figure 3.9: Temperature of dependence of magnetization for x=8.5% and several p’s using256 sites lattices.

45

0 50 100 150 200 250Temperature

0

0.2

0.4

0.6

0.8

Mag

net

izat

ion

L=3, p=0.70L=4, p=0.72

x=%8.5

J=1.2 eV

Tc~ 132 K

Tc~ 100 K

Figure 3.10: Temperature of dependence of magnetization for different lattice sizes forx = 8.5%, p ≈ 0.75, and J = 1.2eV .

Chapter 6) for various values of hole concentrations p at x=8.5%. The results were

obtained averaging over at least 6 different disorder configurations. At the realistic value,

J=1.2 eV [130], a maximum Tc of 150 K was obtained for p = 1 (no compensation) as shown

in Fig. 3.9. Despite the fact that the estimated Tcs are reasonable, one of the main problems

of this model is its failure to provide the correct shape of the magnetization curves obtained

in recent experiments.[81, 2] The Curie-Weiss (CW) character of the magnetization curve is

not observed with the 4-orbital model. This is an important property of these materials that

has not been reproduced by unbiased calculations. In the next section, we are going to see

that the Hamiltonian of the more complex 6-orbital model allows us to capture this feature.

In Fig. 3.10, we plot the magnetization versus temperature for two different system sizes

which show the finite-size effect in this model. Finally, in Fig. 3.11, we present calculated

values of TC as a function of hole concentrations p at x=8.5% for 33 and 43 lattices. The trend

of TC as a function of hole concentration agrees with experimental results and theoretical

calculations. [6, 136] The next step is to consider more realistic systems to handle the

problems seen in 4-orbital models.

46

0.0 0.2 0.4 0.6 0.8 1.0p

h

25

50

75

100

125

150

175

200

225

Tc(

K)

L=4L=3

X=%8.5

J=1.2 eV

Figure 3.11: MC calculated TC vs. p, at x=8.5% and J = 1.2eV . Notice that N = L3 whereL is the size of the unit cell L=Lx=Ly=Lz

3.4.4 Results for 6-Orbitals Model

Equation 3.30 will be studied with the standard MC techniques described in Chapter 6 for

systems involving fermions and classical spins. Numerically, we analyze clusters that contain

N = NxNyNz unit cells (Ni is the number of cells along the spatial direction i) of side a0

(a0=5.64A,[121] is the GaAs cubic lattice parameter). Since in an fcc lattice there are 4 ions

associated to each cell, the total number of Ga sites is NGa=4N . Since there are 6 single

fermionic states per site, the diagonalization of a 6NGa × 6NGa matrix is needed at each

step of the MC simulation, which demands considerable computational resources for large

enough clusters. The diagonalization can be performed exactly for values of NGa up to 500.

We show below that lattices with Ni = 4 (256 sites) are large enough to study Mn dopings

x and compensations p in the range of interest, with sufficient precision for our purposes.

Nominally, there should be one hole per Mn ion, but p can be smaller than 1 due to hole

trapping defects, thus p and x are considered independent in this study.

The highest TC experimentally observed in bulk Ga1−xMnxAs is ∼ 150 K, at x=8.5%

and p≈0.7[7, 60, 2]. The system is metallic, and the magnetization vs. temperature displays

mean-field behavior.[7] In Fig. 3.12.a, we present the (MC calculated) TC as a function of the

47

0 0.5 1 1.5 2 2.5J(eV)

0

500

Tc(K

) MC MF

0 5 10 15 20J(eV)

0

1000

2000

3000T

c(K)

0 0.2 0.4 0.6 0.8 1p

x=8.5%

Tc

exp

J=1.2eV

(a) (b)

p=0.75

Figure 3.12: (a) Curie temperature vs. J , for x=8.5% and p≈0.75. The MC results areindicated by circles, while the continuous line is the MF prediction.[6] Inset: MC results forlarger values of J to observe the crossover toward a localized picture. Vertical lines indicatethe experimentally acceptable range of J . (b) MC calculated TC vs. p, at x=8.5%. The bluedots are experimental results,[7, 8, 9] and the solid line is the MF prediction.

coupling J , at x=8.5% and p≈0.75. The results shown were obtained on lattices containing

256 (Ga,Mn) ions, and using an average over at least 5 different disorder configurations (only

small differences were observed among the Mn configurations). Results for lattices with up

to 500 sites have also been calculated for some parameters (see below). At the realistic

J=1.2 eV [130], Fig. 3.12.a shows that the critical temperature is TC=155 ± 20K. Since

J is not accurately known, this excellent agreement with experiment[7] could be partially

fortuitous, but at least the results indicate that a reasonable quantitative estimation of

the real TC can be made via MC simulations of lattice models. The solid line in the

figure corresponds to the MF results.[3, 6] The quantitative MF-MC agreement at small

J provides a strong test of the reliability of the present MC approach. At J=1.2 eV, the

MF TC is ∼300 K, showing that at these couplings and densities appreciable differences

48

between MC and MF exist: the fluctuations considered in the MC approach cannot be

neglected. The inset of Fig. 3.12 demonstrates that eventually for very large values of J the

MF approximation breaks down, as expected. The MC simulations show that TC reaches a

maximum for J ≈ 12eV , of the order of the carriers bandwidth, and then it decreases due to

the tendency of holes toward strong localization. This “up and down” behavior can only be

obtained with lattice MC simulations valid at arbitrary values of J .[117, 137, 138, 109, 139]

At J∼1.2eV the system is closer to a hole-fluid than a localized regime as suggested by

the magnetization vs. T curve, displayed in Fig. 3.13.a. This curve has Curie-Weiss shape in

qualitative agreement with both experimental results [7] and previous MF calculations.[3, 61]

This qualitatively correct shape of the magnetization curve was not obtained in previous

lattice MC simulations.[117, 137, 138, 109, 139] Size effects are mild as it can be seen in

Fig. 3.14.a, where data for magnetization vs. T are presented for x = 8.5%, p ≈ 0.75,

and J = 1.2eV in lattices with (Nx, Ny, Nz) = (4, 3, 3), (4, 4, 4), (5, 4, 4), and (6, 4, 4), i.e.,

with NGa=144, 256, 320, and 384. Results for N=500 were obtained for lower doping (see

Fig. 3.13.b). Considering together the results for the different size clusters the estimated

TC ≈ 155 ± 15 K is still in agreement with experiments (and also with the 256 sites

results). Regarding the Curie-Weiss (CW) shape of the magnetization curve, we have

phenomenologically observed that the finite spin-orbit coupling plays a crucial role in this

respect. In Fig. 3.14.b we show the magnetization vs. T for x = 8.5%, p = 0.75, and

J = 1.2eV for ∆so = 0.34eV (squares) and ∆so = 0 (circles): only the nonzero SO coupling

produces CW behavior. In addition, we have noticed that the CW shape is also missing with

the 4 orbital (with j=3/2) model that results in the limit ∆so → ∞.[116] This indicates the

important role that a realistic representation of the valence band plays in properly describing

the thermodynamic observables.

The charge distribution in the cluster provides interesting information. In the HF

scenario, the charge is assumed to be uniformly distributed while in the IB picture the

charge is strongly localized. Fig. 3.14.c indicates that in the realistic regime with J = 1.2

eV, x = 8.5%, and p ≈ 0.75 the charge is fairly uniformly distributed. The slightly darker

points correspond to the sites where the Mn are located. They have charge of the order of

20% above the MF value defined as nMF =n/NGa (with n the number of holes). As shown

in Fig. 3.14.d, charge localization occurs when J is increased to large values such as 12 eV.

The dark circles at the Mn sites have charge intensities about 20 times the MF value, with

49

0 50 100 150 200 250 300T(K)

0

0.2

0.4

0.6

0.8

1

M

p=0.18

p=0.36

p=0.54p=0.72p=1.0

0 50 100 150 200T(K)

0

0.2

0.4

0.6

0.8

1

p=0.25

p=0.5

p=0.75

p=0.75p=1.0

x=8.5%x=3%

(b)

(a)

Figure 3.13: (a) Magnetization M vs. T , for x=8.5% and several p’s (indicated), using a256 sites lattice (open symbols). Averages over 5 Mn-disorder configurations are shown. (b)Same as (a), but for x=3%. Close circles are results for a 500 sites lattice. The magnetizationis measured as M =

√M · M, with M the vectorial magnetization. As a consequence, for

fully disordered spins, M is still nonzero due to the SI2=1 contributions, causing a finite

value at large temperatures (M(T → ∞) = 1/√

xNGa) unrelated to ferromagnetism. Thus,we plotted M = (M−M(T → ∞))/(1−M(T → ∞)), i.e. the background was substracted.

very little charge found away from the impurities.

The lack of charge localization effects in the ordered state is concomitant with the absence

of a notorious impurity band in the density-of-states (DOS) (Fig. 3.15). Increasing J , an IB

regime is eventually observed, given confidence that the study is truly unbiased. This occurs

for J ≈ 4eV and beyond, with the IB becoming totally detached from the valence band at

J ≈ 16eV . Figures 3.14.c, d and 3.15 show that the degree of spatial hole localization is

correlated with the development of the IB. In addition, when the holes are localized, the M

vs. T curves present substantial deviations from MF behavior shown in Fig. 3.16, with a

different concavity as that of Fig. 3.13.a.[117, 137, 138, 109, 139]

It is well known that the carrier density in DMS is strongly dependent on sample

50

0 50 100 150 200 250 300T(K)

0

0.2

0.4

0.6

0.8

1

M

∆SO

=0

∆SO

=0.34eV

0 50 100 150 200 250 3000

0.2

0.4

0.6

0.8

1

M

(4,3,3)

(4,4,4)

(5,4,4)

(6,4,4)

(b)

p~0.75x=8.5%

(a)0

19

0

19

(c)

(d)

0

20

Figure 3.14: (a) M (defined as in Fig. 3.13) vs T for different lattice sizes for x = 8.5%,p ≈ 0.75, and J = 1.2eV ; (b) M vs. T for the same parameters as in (a) on a 256 sites latticewith (without) spin-orbit interaction indicated by the squares (circles); (c) Charge densitynormalized to the MF value (see text), for x = 8.5%, p≈ 0.75, T=10K, on a 256-sites clusterfor J=1.2eV . The color intensity is proportional to the charge density (see scale). (d) Sameas (c), but for J=12eV .

preparation. Due to defects, p is much smaller than 1 in most samples. In Fig. 3.12.b,

the MC calculated TC vs p for x=8.5% and J=1.2eV is shown. TC increases with p, and

it reaches a maximum at p∼1 as in previous theoretical,[6, 117, 137, 138, 109, 139] and

experimental results[81, 7, 8, 9] also shown in the figure. The agreement between MC results

and experiments is once again quite satisfactory. The figure suggests that if p=1 were

reachable experimentally, then TC could be as high as ∼200 K. We also observed qualitative

changes in the magnetization curve varying p (Fig. 3.13.a): as p is reduced, the magnetization

changes from a Brillouin form to an approximately linear shape with T . In fact, the observed

p dependence is once again similar as found experimentally, with p modified by annealing;[7]

the Mn disorder plays a more dominant role when the number of holes is reduced. In spite

51

-18 -16 -14 -12 -10 -8 -6ω

0

500

1000

1500

2000

2500

3000

J=0.0 eVJ=1.2 eVJ=4.0 eVJ=8.0 eVJ=12.0 eVJ=16.0 eVJ=20.0 eV

-14 -12 -10 -8 -6ω

0

500

1000

1500

2000

2500

3000

J=0.0 eVJ=1.2 eVJ=4.4 eVJ=6.4 eVJ=8.0 eVJ=12.0 eVJ=16.0 eV

x=8.5% x=3%

(a) (b)

Figure 3.15: (a) Density-of-states, for x=8.5%, p≈0.75, and several J ’s. The dashed verticallines indicate the position of the chemical potentials. (b) Same as (a) but for x = 3%.

of the deviations from MF behavior at small p, we could not observe a clear IB as p was

reduced, for the coupling J used in our analysis.

Consider now the low Mn-doping regime. The latest experimental results suggest that

(Mn,Ga)As should still be metallic at x=3%.[9] A metal-insulator transition is expected at

x ≤ 1%,[140] but MC studies for such very small dopings need much larger clusters than

currently possible. Our MC simulations indicate that at low Mn-doping, the dependence of

TC with J is similar as in the higher doping case.

No clearly formed IB is observed in the x=3% DOS displayed in Fig. 3.15.b: the IB’s are

formed for similar values of J with increasing J , at both x=3% and 8.5%, as deduced from

an analysis of fermionic eigenvalues for spin-ordered configurations.[116, 141]

Finally, we study the other III-V type materials in which holes go into the p-bands with

realistic 6-band model. Fig. 3.17 shows results of magnetization versus temperature for

52

0 500 1000 1500 2000 2500 3000Temperature(K)

0

0.2

0.4

0.6

0.8

Mag

net

izat

ion

J=6.4 eVJ=12.0 eVJ=16.0 eV

x=8.5%

p=0.72

GaAs

Figure 3.16: Magnetization as a function of temperature for large values of Js.

GaAs, GaSb, GaP and GaN. The predicted Tc and and the shape of the magnetization for

each material is in a good agreement with experimental results on these compounds.

Here, a MC study of an fcc lattice model for DMS compounds, including the realistic

valence bands of GaAs, the spin-orbit interaction, and the random distribution of Mn dopants

has been presented. The use of the Cray XT3 at ORNL made this effort possible. The results

show magnetizations and TC ’s in reasonable agreement with experiments. The simulations

show that the carriers tend to spread over the entire lattice, and they reside in the valence

band at realistic couplings, qualitatively in agreement with MF [6, 61] and first principles

[142] calculations in the same parameter regime, as well as with experimental data on

annealed samples.[81, 9] However, an IB band populated by a fraction 1 − p of trapped

holes that do not participate in the transport properties is not ruled out by our results. The

MC method described here opens a new semi-quantitative window for theoretical research

on the properties of DMS materials.

53

0 100 200 300 400 500 600 700T(K)

0

0.2

0.4

0.6

0.8

1

M

GaSbGaAsGaPGaN

J=1.2 eV

x=8.5%

p=0.72

J=1.34 eVJ=0.96 eV

∆=0.76 ∆=0.08∆=0.34 ∆=0.017J=2.49 eV

Figure 3.17: Temperature dependence of magnetization for different III-V type materials atx = 8.5% and ph = 0.72 in 43 lattices.

54

CHAPTER 4

DEEP IMPURITIES IN DMS

When non-magnetic host semiconductors such as GaAs, GaN and GaSb, are doped with

a single Mn, one hole and a localized spin are nominally introduced into the system. As

mentioned in previous chapters, the interaction between the spin of the hole and the spin of

the magnetic impurity is believed to be the origin of the ferromagnetism in these materials.

There are two possible states in which holes can be introduced. In one case, holes go into

the sp3 orbitals that generate the valence band of the parent compound. Then, the Mn

impurity has a localized moment S = 5/2, formed by the electrons in the five d-orbitals (d5)

that interact weakly with holes in the host valence band, leading to the simple (d5 + h)

picture where the Mn spin couples antiferromagnetically to the spin of the holes. This case

was discussed in detail in Chapter 3. In the second scenario the holes occupy the Mn − d

orbitals (d4). In this case Mn3+ instead of Mn2+ centers are produced, the localized spin

has S=2, and a different model of FM in DMS has to be considered. In this Chapter, we are

going to propose a model that allows the study of d-level holes.

Theoretical calculations assuming (d5 + h) indicate that doping with Mn, GaP[143] or

GaN[3, 144, 145] should provide higher Tc’s. However, the approximation of the (d5 + h)

scenario for this material is the subject of strong controversy since some of the experimental

results indicate the presence of Mn3+[80] in these materials. For example, electron spin

resonance measurements, which indeed support the picture of divalent Mn in GaAs[70],

indicate that the trivalent d4 configuration for Mn impurities may be important in GaP.[146]

For GaN it has been reported that Mn is in a divalent state when electrons are doped[72], but

in a trivalent state when holes are doped to the system.[147] Hwang et al.[148], using photo-

emission and soft X-ray absorption spectroscopy, observe that in the n-type doped GaN the

Mn state is divalent, while for the non-doped one it is trivalent. Also, recent experiments by

55

Edmonds et al.[149] indicate a carrier-induced nature of the ferromagnetic exchange, but a

small, finite density of unoccupied Mn d states is found close to the Fermi level, reflecting

hybridization with the host valence bands.

Due to the crystal field, the degenerate atomic Mn-d levels are split into the triply

degenerate t2g states and the doubly degenerate eg states.[15] In (Ga,Mn)As the Mn d-levels

are deep into the VB and, thus, they can be neglected when considering microscopic models.

However, if the d-orbitals are close to the top of the valence band (or above it) as it may be

the case in (Ga,Mn)N, these orbitals have to be considered. Therefore, the study of models

considering the d-orbitals may serve as a guide for making quantitative predictions about

ferromagnetism in some DMS materials in which holes go into the Mn-d levels, such as GaN

and GaP.

4.1 Phenomenological Model

Since the position of the Mn d-level is not exactly known and the addition of the d orbitals

would make the number of degrees of freedom untreatable numerically, we are going to

propose a phenomenological approach. One extra s-orbital that will mimic the d-orbital

into which the hole is doped will be considered. This orbital will only be active at the

sites in which the Mn impurities are located. Free hopping parameters between next-nearest

neighbors Mn and from d-Mn levels to p-levels of the nearest neighbor ions will be considered.

The relative position of the d-level with respect to the top of the valence band will be another

free parameter. The Hamiltonian is given by:

H =1

2

∑

i,µ,ν,α,α′,a,b

(tµναa,α′bc

†i,αaci+µ+ν,α′b + h.c.) +

∑

<I,I′>,σ

td,d(d†I,σdI′,σ + h.c.)+

∑

<I,µ,ν>,σ,α,b

tpd(d†I,σcI+µ+ν,αb + h.c.) + ∆SO

∑

i,α

c†i,α 1

2

ci,α 1

2

− Vd

∑

I

nI + J∑

I

sI · SI, (4.1)

where the hopping parameters, tµναa,α′b, a, b, α, α′, the Hund term and spin-orbit

interaction part ∆SO

∑

i,α c†i,α 1

2

ci,α 1

2

were already discussed in detail in Chapter 3 (we will

use ∆SO=0.017 eV for GaN). The second term of the Hamiltonian describes the hopping

between nearest neighbor Mn − d level, and the third term controls the hopping between p

and d orbitals (p− d hybridization). tdd and tpd are complex numbers and will be considered

56

as free parameters. Vd moves the d-band up or down relative to the valence band. Finally, we

control the number of holes by adding the term -µ∑

i ni where µ is the chemical potential.

If J = 0 the Hamiltonian is represented by the following 8 × 8 matrix:

HKL =

ss 0 −Ee −E∗

e√3

Ef 0 −Ef√2

E∗

e

√2√

3

0 ss 0 Ef −Ee√3

−E∗e −Ee

√2√

3

Ef√2

−E∗e 0 Hhh −c −b 0 b√

2c√

2

−Ee√3

Ef −c∗ Hlh 0 b − b∗√

3√2

−d

E∗f −E∗

e√3

−b∗ 0 Hlh −c d − b√

3√2

0 −Ee 0 b∗ −c∗ Hhh −c∗√

2 b∗√2

−E∗

f√2

−E∗

e

√2√

3b∗√2

− b√

3√2

d −c√

2 Hso 0Ee

√2√

3

E∗

f√2

c∗√

2 −d∗ − b∗√

3√2

b√2

0 Hso

, (4.2)

with

ss = 4tss(cxcz + cycz) + Vd,

Ee =4tsp[sy(cx + cz) + isx(cy + cz)]√

2

Ef =4√

2√3

itspsz(cx + cy) (4.3)

The Hund coupling between the localized spin and a hole in the d-orbitals is given by;

Jd = JsinθIe−iφId†

I↓dI↑ + JsinθIeiφId†

I↑dI↓ +

JcosθIe−iφId†

I↑dI↑ − JcosθIe−iφId†

I↑dI↓, (4.4)

while the hoppings between p and d orbitals can be expressed in terms of the free

parameters tsp:

tx,y

d↑, 32

= −tsp(1 + i)√

2,

tx,z

d↑, 32

= −tspi√2,

ty,z

d↑, 32

= −tsp1√2,

tx,y

d↑,− 1

2

= −tsp(1 − i)√

6,

57

tx,z

d↑,− 1

2

= tspi√6,

ty,z

d↑,− 1

2

= −tsp1√6,

tx,y

d↑, 12

= 0,

tx,z

d↑, 12

= ty,z

d↑, 12

= tspi

√

2

3,

tx,y

d↑,− 3

2

= tx,z

d↑,− 3

2

= ty,z

d↑,− 3

2

= 0,

tx,y

d↑, 12

= 0,

tx,z

d↑, 12

= ty,z

d↑, 12

= −tspi√3,

tx,y

d↑,− 1

2

= −tsp(1 + i)√

3,

tx,z

d↑,− 1

2

= −tspi√3,

ty,z

d↑,− 1

2

= tsp1√3,

tx,y

d↓, 32

= tx,z

d↓, 32

= ty,z

d↓, 32

= 0,

tx,y

d↓,− 1

2

= 0,

tx,z

d↓,− 1

2

= ty,z

d↓,− 1

2

= tspi

√

2

3,

tx,y

d↓, 12

= −tsp(1 + i)√

6,

tx,z

d↓, 12

= −tspi√6,

ty,z

d↓, 12

= −tsp1√6,

tx,y

d↓,− 3

2

= −tsp(1 − i)√

2,

tx,z

d↓,− 3

2

= tspi√2,

ty,z

d↓,− 3

2

= −tsp1√2,

tx,y

d↓, 12

= tsp(1 − i)√

3,

tx,z

d↓, 12

= −tspi√3,

58

ty,z

d↓, 12

= tsp1√3,

tx,y

d↓,− 1

2

= 0,

tx,z

d↓,− 1

2

= ty,z

d↓,− 1

2

= −tspi√3, (4.5)

where tsp is the hopping parameter between s and p orbitals following Slater-Koster[125].

4.2 Numerical Results

In the following, the results obtained by diagonalizing Eq. 4.1 using parameters for GaN will

be presented.

In Fig. 4.1, we present magnetization results obtained with the 6-orbital and the 8-orbital

models for GaAs and GaN, respectively. For (Ga,Mn)As the d-levels are placed deep into

the valence band and the results for 6 and 8-orbitals are similar as shown in Fig. 4.1.a. For

(Ga,Mn)N we show the results for different postions of the d-leveles: deep inside the valence

band (which reproduces 6-orbital results) and close to top of the valence band (as some

experiment indicates this is the case). if the holes go into the p-levels (VB) Tc of ≈ 450K

is obtained which is well above room temperature and in agreement with the Mean-Field

predictions,[3]. However, it can be seen that Tc is very much reduced, i.e., below room

temperature, if the holes are doped into the d-levels. This indicates that Mn doped GaN

may not be the appropriate material to be used in devices.

It is interesting to see that M vs T is linear (or even convex for large |Vd|) rather

than Curie-Weiss for (Ga,Mn)N. As mentioned in the previous chapter we believe that this

behavior may be a consequence of the small spin-orbit interaction in GaN due to the small

size of the N ions.

While in the 6-orbital model there are no free parameters, in the 8-orbital model the

position of the d-levels as well as the d-d and p-d hoppings are not well determined. Thus,

different values of these parameters were studied in order to explore the possible behavior

of (Ga,Mn)N if holes go into the d-levels of Mn. Very small values of td,d and tp,d, of the

order of 0.1 eV correspond to the case in which holes are not very mobile in the Mn d-level

(p-orbitals hoppings are of the order of 1.0 eV ). The left panel of Fig. 4.1.b shows the

temperature dependence of the magnetization for various values of Vd (that determines the

placement of the d-level with respect to the valence band) and the corresponding density of

59

0 50 100 150 200 250 300T(K)

0

0.2

0.4

0.6

0.8

M6-Orb8-Orbx=8.5 %

p=0.72

J=1.2 eV

tdd

=-0.1 eV

tpd

=0.1 eVGaAs

0 100 200 300 400 500 600T(K)

0

0.2

0.4

0.6

0.8

M

Vd=-3.8 eV

Vd=-4.2 eV

Vd=-4.5 eV

Vd=-4.8 eV

Vd=-5.2 eV

x=8.5%

p=0.72

J=2.49 eV

tdd

=-0.1 eV

tpd

=0.1 eV

a) b)

Figure 4.1: a) Temperature dependence of magnetization at x=8.5% for GaAs, obtainedwith 6-orbital (black curve) and 8-orbital (red curve) models. b) Same for GaN but withdifferent values of Vd

states (DOS) is presented in Fig.4.2 where the dotted lines show the position of the chemical

potential. As seen in Fig. 4.1 and Fig. 4.2, if the d-levels are deep inside the valence band the

maximum TC is obtained and the 6-orbital results (shown in Fig. 4.1.b) are reproduced and

no impurity band is observed. But when the d-level is moved towards the top of the valence

band, a decrease in Tc is observed and an IB starts to form (see Vd=-4.5 eV). When the

d-level moves further inside the semiconducting gap a dramatic decrease in magnetization

and Tc occurs and a well developed IB is observed (see Vd=-4.8 eV). Finally, for Vd=-5.2 eV

the IB is completely separated from the VB, the magnetization becomes very small, and Tc

is of the order of 100 K.

If we consider a strong hybridization between d and p orbitals both the magnetization

and Tc increase although an IB develops. This situation is presented in Fig. 4.3. More

experiments and theoretical studies are needed in this area.

4.3 Coulomb attraction

The results in the previous chapter showed that no impurity band is observed in Mn doped

GaAs for realistic values of J(1.2 eV ) and Mn doping (3%, 8.5%). However, it is known

60

-7 -6.5 -6 -5.5 -5 -4.5ω

2000

4000

6000

8000

10000

DO

S

Vd=-3.8 eV

Vd=-4.2 eV

Vd=-4.5 eV

Vd=-4.8 eV

Vd=-5.2 eV

x=8.5%

p=0.72

J=2.49

tss=-0.1 eVtsp=0.1 eV

Figure 4.2: Density of states (DOS) for different values of Vd considering weak p − dhybridization for Mn doped GaN.

0 100 200 300 400T(K)

0

0.2

0.4

0.6

0.8

1

M

tdd

= -0.1 eV, tpd

= 0.1 eV

x=8.5 %

p=0.72

J=2.49 eV

0 100 200 300 400T(K)

0

0.2

0.4

0.6

0.8

1

M

tdd

= -0.1 eV, tpd

= 0.1 eV

tdd

= -0.5 eV, tpd

= 0.5 eV

x=8.5 %

p=0.72

J=2.49 eV

-8 -7 -6 -50

DO

S

Figure 4.3: Magnetization versus temperature and Density od states (DOS) for strong p− dhybridization.

61

that if one Mn-ion is doped into GaAs, the hole gets bounded to the Mn-ion with a binding

energy of Eb=0.112 eV [6]. The hole is bounded by the Coulomb attraction to the more

negative Mn2+ centers. This Coulomb attraction is modeled in semiconductors by a central

cell potential (CCP).[150, 151] It is shown that the addition of a square-well like attractive

potential can generate an IB at small Mn doping x for values of J that are not strong

enough to generate IB by themselves. However, it is expected that as x increases and the

wave functions start to overlap, an IB will develop. This IB will become wider and it will

eventually merge with valence band. It is important to determine what is the value of xc

for which crossover from IB to VB occurs in the DMS. A DMFT analysis[152] indicates that

xc ≤ 0.1% in (Ga,Mn)As.

In this section, we study the effect of the attractive potential on the model described by

Eq. 3.30. First we studied an on-site potential by adding the term

V∑

I

nI (4.6)

to the Hamiltonian. In Fig. 4.4 it can be seen that using the parameters, including J, for

(Ga,Mn)As an IB eventually appears in the DOS when V=-6 eV. These results indicate that

the effect of V is to renormalize the value of J in agreement with findings of Ref.[152].

However, the effect of V should depend on x. When the hole wave functions start to

overlap as x increases, the effect of V on J should become negligible. A reason why this

behavior was not observed in our simulations is because of the zero range of the potential.

Thus, we considered the case of a nearest-neighbor range potential by adding the term

V∑

I

[nI

1

NI

+∑

µ

1

NI,µ+ν

nI+µ+ν ] (4.7)

where µ + ν indicates the 12 nearest neighbors of site I, and NI indicates the number of

impurity sites that surrond the site I. Since we work on a small system, values of x ≈ 0.1%

cannot be studied. Thus, we selected the large values of J (J=7 eV for the hopping parameters

of GaAs) that is below but close to the Jc at which the VB to IB crossover occurs at x = 8.5%

with V=0.

In Fig. 4.5 we show that if an on site potential V=-3.5 eV is added, an IB develops and

the chemical potential is on it. However, if an extended range V=-3.5 eV potential is used

62

−14 −12 −10 −8 −6ω

0

2000

4000

6000

8000

10000

12000

DO

S

V=0V=-2 .0eVV=-4.0 eVV=-6.0 eVV=-8.0 eV

x=8.5%p=0.72

J=1.2 eV

Figure 4.4: The density of states (DOS) for J=1.2 eV and various values of Vd. The verticallines indicate the position of the chemical potential.

the IB goes away and the DOS is similar to the one for V=0. This occurs because, at this

large doping, the extended potential allows for a more uniform distribution of the holes.

This demonstrates that it is crucial to consider extended range Coulomb attraction in order

to capture x dependence of the effect.

63

-6 -5 -4 -3 -2 -1 0 1 2 3 4ω−µ

0

2000

4000

6000

8000

DO

S

V=0V=-3.5 eVV=Vnn=-3.5 eV

J=7eV

IB

x=8.5%

p=0.72

VB

VB

VB

Figure 4.5: The density of states (DOS) for J=7 eV and V=0 (black line); for an on-site Coulomb attraction V=-3.5 eV (blue line); and for a Coulomb attraction with on siteintensity V and next nearest neighbors intensity Vnn=V=-3.5 eV (red line). The verticallines indicate the position of the chemical potential.

64

CHAPTER 5

Effect of Adiabatic Phonons on Striped and

Homogeneous Ground States For High Tc Cuprates

In this chapter, we argue that a Hamiltonian similar to the one used to describe DMS can

be used to study a very different problem: high Tc superconductivity. We provide a short

introduction to superconductivity which will lead us to describe some current problems that

our work will address.

Superconductivity was discovered in 1911 in by H. K. Onnes[153], when he observed a

sudden (and unexpected) drop in the electrical resistivity of mercury below 4.15 K. Despite

many efforts, the quantum theory of superconductivity by Bardeen, Cooper and Schriffer

(BCS)[19] came 46 years after Onne’s discovery. This theory, which explains the fascinating

properties of superconductors from first principles, is widely considered (along with Landau’s

Fermi liquid theory) as one of the most successful theories in condensed matter physics. BCS

superconductors have very low Tc’s that can be achieved with liquid He. In the year 1986,

Bednorz and Muller[154, 155] discovered superconductivity in La2−xBaxCuO4, a copper-

oxide material, at a temperature of about 30K. Soon after that other cuprates with Tc’s

above 100 K were obtained[156] and the study of the high temperature superconductors

begun. The cuprates are poor conductors, and thus their superconductivity properties were

unexpected. Presently dozens of “high-Tc” compounds have been discovered. Among these

is Y Ba2Cu3O7−x , whose critical temperature Tc =92 K is well above the condensation

temperature of nitrogen (77 K). To date the highest critical temperature is Tc =133 K

observed in HgBa2Ca2Cu3O8+x by Schilling et al.[156]. All these materials contained copper

and oxygen atoms in their structure. Soon it appeared that the superconducting properties

of these materials could not be explained in the framework of the BCS theory based only on

electron phonon interactions. It is believed that magnetism should play an important role

65

Figure 5.1: View of the La2CuO4 structure. Cu atoms are represented by red, La atomsby green and O atoms by blue balls. The six coordinated copper and the CuO2 planes areshown.

but there is not yet an accepted theory for high temperature superconductivity.

Crystalline structure

The crystalline structure of high temperature superconductors is similar to the perovskite

structure which consists of a big atom A, some transition metal T and oxygen atoms. 6

oxygen atoms surround T and together they form a TO6 octahedron. Fig. 5.1 shows the

crystalline structure of La2CuO4 . The six coordinated copper, the CuO6 octahedron is

clearly recognizable. This octahedron is distorted in the c direction. This distortion is

related to the Jahn-Teller effect. Also recognizable in the figure are the CuO2 planes.

66

Figure 5.2: View of the Y Ba2Cu3O7 structure. Cu are represented by red, Y atoms bygreen, Ba atoms by gold and O atoms by blue balls. This structure has two CuO2 planes inthe elementary cell, separated by an Y atom.

A common characteristics of all known high Tc superconducting materials is that they

contain one or several CuO2 planes[15] in their crystalline structure. These planes are

believed to be the electronically active elements, responsible for superconductivity. Some

of the extensively studied compounds are Y Ba2Cu3O6+x[157, 158] and Y Ba2Cu4O8 , both

containing one double CuO2 plane in their elementary cell. In Fig. 5.2 we display the

structure of Y Ba2Cu3O7[159] , with its double CuO2 plane.[15]

Electronic structure

The parent compounds of all of the high temperature superconductors are antiferromagnetic

insulators. Let us consider an ionic model and a copper atom in the CuO2 plane. This

copper is in a 3d9 state. Consequently, compared with a completely filled 3d10 shell, one

electron is missing. The missing electron, or in the hole representation the added hole,

has spin 1/2. The crystalline field triggers an energy splitting of the atomic 3d orbitals.

In the O6 octahedron the distance between the copper atom and the apex oxygen atom

67

Figure 5.3: Schematic phase diagram of the high temperature superconductors.

is larger than the distance between copper and planar oxygen atoms. As a consequence

of this distortion, the antibonding 3dx2−y2 orbitals have the highest energy. Furthermore,

the unpaired electron (hole) is in this atomic orbital. There is superexchange interaction

between the magnetic moments in the Cu ions. At low temperature, this exchange interaction

promotes antiferromagnetic long-range order. By suitable doping electrons are removed from

the CuO2 planes.In the hole representation this means that additional holes are introduced

into the planes. Doping can be achieved, for example, by heating up the sample in an

oxygen atmosphere or by changing the composition of the planes. In La2−xSrxCuO4 some La

atoms are replaced by Sr, in Y Ba2Cu3O7−x the oxygen content is modified. The immediate

consequence of doping is that the antiferromagnetic ordering becomes weaker and it vanishes

completely at a doping level x ≈ 0.04 in La2−xSrxCuO4. At a further increased doping level,

above x ≈ 0.06 the materials become conducting. A generic phase diagram is presented in

Fig. 5.3.

Theory

68

The search for new materials has been mainly empirical since no predictive theory is currently

known for the high-Tc compounds. The fact is that no one knows why the cuprates behave as

they do. It is believed that magnetic interactions may play an important role in the pairing

mechanism and this is the reason why most theoretical studies are based on simple models

that incorporate the most basic charge and magnetic degrees of freedom in the CuO2 planes.

From the exact treatment of the charge and spin of the Cu and O in the planes a complicated

3-band Hubbard model is derived.[15] Simplifications of this model are described below:

t − J Model: This Hamiltonian was proposed by Zhang and Rice by studying the ground

state of one single hole doped into a Cu-O cluster. A singlet forms between the spin of the

hole (in O) and the spin of the Cu. So effectively, it can be assumed that adding a hole is

equivalent to removing a spin in the Cu lattice. The resulting model is the t − J, defined as:

H = J∑

〈ij〉Si.Sj − t

∑

〈ij〉σ[c†iσcjσ + h.c.],

where Si are spin-1/2 operators at the sites i of a two dimensional square lattice, and J is the

antiferromagnetic coupling between nearest neighbors sites 〈ij〉. The hopping term allows

the movement of holes without changing their spin.

Hubbard Model: The Hubbard model was introduced in 1963 [25] to describe strongly

correlated electron systems. In the context of high Tc’s it was introduced by Anderson[24] as

a simplification of the 3-band Hubbard model. It is described by the following Hamiltonian:

H = −t∑

〈ij〉,σ(c†iσcjσ + c†jσciσ) + U

∑

i

(ni↑ −1

2)(ni↓ −

1

2),

where, as usual, c†iσ is a fermionic operator that creates an electron at site i of a square lattice

with spin σ. U is the on-site repulsive (Coulomb) interaction, and t the hopping amplitude.

In the limit of U/t → ∞ the Hubbard model is equivalent to the t − J with J/t → 0.[15]

Spin-Fermion Model: Spin-fermion models (SFM), were first introduced in the context

of the Kondo problem[160]. Many autors[15, 161, 162] applied it to the cuprates to mimic

the interaction between the spin of the doped holes and Cu spins. Although formally it is

difficult to study the Hubbard and t − J model, the SFM has the advantage that many of its

quantum mechanical properties can be achieved taking the classical limit for the localized

spins which simplifies its study.

69

5.1 Electron-phonon interactions in high TC cuprates

The pairing mechanism responsible for high Tc superconductivity is still unknown. The

electron-phonon interactions that satisfactorily explain pairing for traditional supercon-

ductors within the BCS theory [19] would require phonon frequencies incompatible with

the material stability in order to produce the observed high critical temperatures in the

cuprates.[19] For this reason many researchers believe that magnetic interactions, which are

observed in all the cuprates, may play an important role in the pairing mechanism.[163]

As a result of this hypothesis, most of the Hamiltonians proposed to study the physics of

the cuprates, such as the Hubbard and t-J models, only incorporate electronic and magnetic

degrees of freedom.[15] However, experiments indicate that there are active phonon modes in

the cuprates.[164, 165, 166, 167, 168] In addition, experimentally a very rich phase diagram,

particularly in the underdoped regime, is started to be unveiled. Some materials appear to

have ground states with electronic stripes as observed in neutron scattering results [167],

while scanning tunneling microscopy indicates nanosize patches of superconducting and

non-superconducting phases. [169, 170] The emerging phase complexity is reminiscent of

the experimental data for manganites where competing electronic, magnetic and phononic

degrees of freedom are responsible for the rich phase structure.[120]

For these reasons it is important to include electron-phonon interactions in models for

the cuprates. This would allow to understand whether EPI stabilize or destabilize charge

stripes, what kind of inhomogeneous textures, if any, develop and, eventually, whether the

interplay of magnetic and phonon interactions with the electrons is responsible for the pairing

mechanism.[171]

The first step towards the goal of introducing EPI in models for the cuprates is to

propose a simple but physically realistic Hamiltonian that can be studied with unbiased

techniques. The proposals already in the literature include momentum dependent electron-

phonon couplings that lead to long-range interactions in coordinate space[172, 173] and/or

quantum phonons which are very difficult to treat numerically.[174, 175] Most studies have

been performed using mean-field, slave-boson, or LDA approximations.[176, 177] Numerical

simulations have been done using the t−J model in very small lattices, with a limited number

of phonon modes and diagonal couplings, [174, 175] or on the one-dimensional Hubbard

model.[178, 179]

70

In this chapter we will study numerically a spin-fermion (SF) Hamiltonian for the

cuprates[161, 162] with electron-phonon interactions. The SP model is obtained as a

simplification of the three band Hubbard model to a two band Hubbard model proposed by

Emery[180] and further simplification introduced by Loh et al.[181] This model reproduces

many properties of the cuprates and presents stripes in the ground state, due solely to

spin-charge interactions, in some regions of parameter space.[161, 162] Thus, it provides a

framework particularly suitable to study the effects of electron-phonon interactions on the

preformed stripes. However, charge homogeneous ground states are also found in other

regions of parameter space which allows to investigate charge inhomogeneity induced by

EPI. Several phononic modes will be studied and diagonal and non-diagonal couplings, i.e.,

the dependence of the hopping and other Hamiltonian parameters on the lattice distortions,

will be considered. The work will be performed in the adiabatic limit, i.e., at zero phononic

frequency.

The chapter is organized as follows: in Section 5.2 the Hamiltonian is introduced; the

effect of electron-phonon interactions on striped states are presented in Section 5.3, while

Section 5.4 is devoted to the effects of EPI on homogeneous states. Section 5.5 contains the

Conclusions.

5.2 The model

The SF-model is constructed as an interacting system of electrons and spins, mimicking

phenomenologically the coexistence of charge and spin degrees of freedom in the cuprates

[182, 183]. Its Hamiltonian is given by

H = −t∑

〈ij〉α(c†iαcjα + h.c.) + J

∑

i

si · Si + J′∑

〈ij〉Si · Sj, (5.1)

where c†iα creates an electron at site i = (ix, iy) with spin projection α, si=∑

αβ c†iασαβciβ is

the spin of the mobile electron, the Pauli matrices are denoted by σ, Si is the localized spin at

site i, 〈ij〉 denotes nearest-neighbor (NN) lattice sites, t is the NN-hopping amplitude for the

electrons, J > 0 is an antiferromagnetic (AF) coupling between the spins of the mobile and

localized degrees of freedom, and J′ > 0 is a direct AF coupling between the localized spins.

The density 〈n〉=1−x of itinerant electrons is controlled by a chemical potential µ. Hereafter

t = 1 will be used as the unit of energy. J′ and J are fixed to 0.05 and 2.0 respectively, values

71

shown to be realistic in previous investigations [161, 162]. The temperature will be fixed to

a low value: T=0.01, which was shown before to lead to the correct high-Tc phenomenology.

[161, 162, 184]

The diagonal electron-phonon part of the Hamiltonian being proposed here is given by

H(j)e−ph = −λ

∑

i

Q(j)i ni, (5.2)

where ni =∑

σ c†iσciσ is the electronic density on site i and Q(j)i is the phonon mode defined

in terms of the lattice distortions ui,α which measures the displacement along the directions

α = x or y of oxygen ions located at the center of the lattice’s links in the equilibrium

position, i.e., ui,α = 0. The index (j) identifies the phonon mode. In this work, the following

phonon modes will be considered:

(a) The breathing mode given by

Q(1)i =

∑

α

(ui,α − ui−α,α); (5.3)

(b) The shear mode, in which the oxygens along e.g., x move in counterphase with the

oxygens along y, given by

Q(2)i =

∑

α

(−1)σ(ui,α − ui−α,α), (5.4)

with σ = 1(−1) for α = x(y);

(c) The half-breathing mode along x given by

Q(3)i = (ui,x − ui−x,x); (5.5)

and (d) The half-breathing mode along y given by

Q(4)i = (ui,y − ui−y,y). (5.6)

Note that although the proposed interactions seem local in coordinate space, they

correspond to cooperative lattice distortions which, in turn, will produce strongly momentum

dependent effective electron-phonon couplings, in agreement with the experimental evidence

observed in the cuprates. [165]

72

A term to incorporate the stiffness of the Cu-O bonds is added. The term bounds the

amplitude of the lattice distortions induced by He−ph. Its explicit form is:

Hph = κ∑

i,α

(Qi)2, (5.7)

where κ is the stiffness parameter that will be set to 1 here. In addition, we will consider the

off-diagonal interactions induced by the lattice distortions. To obtain these terms we follow

the approach of Ishihara et al. [178, 177] As a result the hopping t in Eq. 5.1 now becomes

site and direction dependent and it is given by

ti,j = t + γ[u(i) + u(j)], (5.8)

where γ is a parameter and

u(i) = ui,x − ui−x,x + ui,y − ui−y,y. (5.9)

The Heisenberg coupling J ′ in Eq. 5.1 also is affected by the lattice distortions and it has to

be replaced by

J ′i,j = J ′ + gJγ[u(i) + u(j)], (5.10)

where gJ is another parameter.

As stated above, the spin-fermion model with electron-phonon interactions will be studied

with a Monte Carlo (MC) algorithm. To simplify the numerical calculations, avoiding

the sign problem, the localized spins are assumed to be classical (with |Si|=1). This

approximation is not drastic since most of the high Tc phenomenology is reproduced in

this limit, and it was already discussed in detail in Ref. [161]. Details of the MC method

can be found in Ref. [185]. Square lattices with 8× 8 and 12× 12 sites will be studied here.

5.3 Influence of Phonons on Striped States

Neutron scattering experiments have shown that doping causes magnetic incommensuration

in the cuprates.[186, 187] The origin of this phenomenon is still being debated. One possible

scenario is the formation of charge stripes in the ground state upon doping.[188, 189]

Experiments on nickelates such as La2−xSrxNiO4 (LSNO) have shown the presence of

diagonal static stripes [167] and there is evidence of stripes also in LNSCO, i.e., Nd doped

73

La2−xSrxCuO4 (LSCO)[190, 191] and in La2−xBaxCuO4 (LBCO).[192] It is conjectured that

the magnetic incommensurability observed in other high Tc cuprates is due to the presence

of dynamical stripes and that the stripe dynamics may be related to the electron-phonon

couplings in the different materials. Despite these theoretical scenarios, it has been very

difficult to find striped ground states in models for the cuprates when studied with unbiased

techniques. Stripes have been observed in the t − J model [193, 194] but they are difficult

to stabilize and it is not clear whether the striped state is the actual ground state or a very

low lying excited one. Stripes, on the other hand, have been obtained without biases in the

SF model that will be studied in this work.[161] This characteristic will allow us to explore

the effect of electron-phonon interactions in striped ground states. The charge texture will

be monitored by measuring the density structure factor N(q).[161] The magnetic structure

factor S(q) will provide information on the magnetic properties. Lattice distortions will be

monitored by measuring correlations between the displacements, although the development

of long range correlations in the lattice degrees of freedom is not anticipated.

The dynamic properties of the system will be monitored by measuring the density of states

N(ω), the one-particle spectral functions A(q, ω) and the optical conductivity σ(ω).[161, 184]

5.3.1 Diagonal electron-phonon term:

In our investigations it has been observed that in general the diagonal electron-phonon

interaction plays an stabilizing role on stripe structures. This behavior was obtained for

the four phonon modes studied here. For 0 ≤ λ ≤ 2 the holes become more localized in

the stripes as λ increases. This can be seen in Fig. 5.4, where snapshots for 〈n〉 = 0.875

are displayed for λ = 0 (Fig.1a), λ = 1 (Fig. 5.4.b), and λ = 2 (Fig. 5.4.c). The lines in

the snapshots indicate the lattice distortions. If all the displacements ui,α were 0 then, the

lines would cross at the middle point of the links that join the lattice sites (as in Fig. 5.4.a).

Out of center crossings indicate ionic displacements. It is clear from the figure that, as λ

increases, lattice distortions in the direction perpendicular to the stripe develop along the

stripe with the mode Q(2), further localizing it. In Fig.1c large displacements along the

horizontal direction can be seen in the links next to the stripe. The stripes become thinner

and the density of holes per site inside the stripes increases.

We have observed some differences between the effects of the various phonon modes

74

(a)

(c)

(b)

ni =0.5

ni =1.0

Sxy

i=0.5

Si

xy=1.0

Figure 5.4: (a) MC snapshot of an 8 × 8 lattice at 〈n〉=0.875 for λ = 0 and γ = 0. The sizeof the circles is proportional to the electronic density; the shaded circles have charge densitylarger than the average, i.e., ni ≥ 〈n〉 = 0.875. The arrows represent the projection of thelocalized spins in the plane x − y; the lines indicate lattice distortions (see text); (b) sameas (a) but for λ = 1 and mode Q(2); (c) same as (b) but for λ = 2.

studied here as the strength of the diagonal electron-phonon coupling λ increases. In Fig. 5.5

snapshots for λ = 2 at 〈n〉 = 0.875 are presented for Q(1), Q(2), and Q(3).[195] A clear

tendency to form diagonal stripes is seen for the breathing mode Q(1) (Fig. 5.5.a). Since

in this case holes are localized by being surrounded by four elongated bonds they cannot

be accommodated in vertical or horizontal formations. Note that the single stripe that is

stable at λ = 0 for the electronic density shown in Fig. 5.5 gets destabilized due to the

Q(1) strong electron-phonon coupling. This result agrees with the fact that diagonal stripes

are observed in LSNO and experiments indicate that the breathing mode is the mode most

strongly coupled to the electrons.[196] According to our results, a robust diagonal coupling

of the electrons to the breathing mode should be expected in the nickelates.

In Fig. 5.5b it can be observed that the shear mode Q(2) tends to stabilize vertical (or

75

(b)(a)

(c)

Figure 5.5: (a) MC snapshot of an 8 × 8 lattice at 〈n〉=0.875, for mode Q(1), λ = 2 andγ = 0; (b) same as (a) but for mode Q(2); (c) same as (a) but for mode Q(3).

horizontal) stripes because a large horizontal (or vertical) distortion occurs, localizing the

holes along the stripe. Interestingly, the half-breathing mode also produces vertical (or

horizontal) stripes but the holes are less localized since the lattice can distort only along the

horizontal (or vertical) direction. As a result, more dynamical stripes are observed for the

half-breathing modes even for strong diagonal electron-phonon couplings (Fig. 5.4.c). Notice

that the experimental evidence indicates that in LNSCO, where vertical and horizontal

stripes are observed, the mode more strongly coupled to the electrons is the half-breathing

mode.[165]

Another indication of increasing localization with increasing λ is observed in the peak in

N(q), at q = (π/4, 0) for the stripes shown in Fig.1a-c, which becomes better developed as

λ increases (Fig. 5.6.a). In addition, the pseudogap in the density of states at the chemical

potential becomes a full gap indicating an increase in insulating behavior (Fig. 5.6.b).

Similar results are obtained at 〈n〉 = 0.75 for which two stripes are stabilized, even at

76

-1.8 -0.9 0 0.9 1.8ω−µ

0

1

2

3

4

5

Ν(ω)

λ=0λ=1λ=2

(0,0) (π,0) (π,π) (0,0)q

0.0

0.2

0.4

0.6

0.8

N(q)

λ=0λ=1λ=2

(b)(a)

Figure 5.6: (a) The charge structure factor N(q) for various values of λ for the sameparameters as in Fig. 5.4; (b) The density of states N(ω) for several values of the diagonalelectron-phonon coupling λ, for the same parameters as in (a).The phonon mode is Q(2).

λ = 0, in the 8 × 8 systems studied here.

The general trend, for different electronic densities and phonon modes, is that the ground

state becomes more insulating as the diagonal electron-phonon coupling increases. The shear

mode Q(2) will be used as an example but similar qualitative behavior is observed for the

other modes. In Fig. 5.7.a it can be seen that the spectral weight at ω = 0 in the density

of states decreases with increasing λ for different values of the electronic density 〈n〉. The

Drude weight, shown in Fig. 5.7.b, also decreases and insulating behavior is obtained for all

densities at λ = 2.

Although the effect of charge localization with increasing λ is more pronounced at

densities for which stripes are observed, we see in the snapshots shown in Fig. 5.8 for

〈n〉 = 0.8 and mode Q(2) that the charge becomes more localized as λ increases and the

lattice distortions localizing the holes develop large values for λ ≈ 2. Note that AF domains

separated by walls of holes are observed. These inhomogeneous structures appear to replace

the stripes when the density of holes is not commensurate with the lattice size. An important

characteristic of these inhomogeneous states is that, although no features are observed in

the charge structure factor N(q), incommensurate magnetic correlations are still present and

77

the peaks in S(q) occur at (π, π − δ) and (π − δ, π), i.e., in qualitative agreement with the

incommensurate peaks observed in the cuprates.

0 1 2λ

0.0

0.5

1.0

1.5

2.0

Ν(ω=0)

<n>=0.937<n>=0.89<n>=0.859<n>=0.828

0 1 2λ

0

1

2

3

4

5

Dru

de

wei

ght

<n>=0.937<n>=0.89<n>=0.859<n>=0.828

(a) (b)

Figure 5.7: (a) Spectral weight in the density of states N(ω) at ω = 0 as a function of thediagonal electron-phonon coupling λ for several values of the electronic density 〈n〉 and modeQ(2); (b) Drude weight as a function of the diagonal electron-phonon coupling λ for severalvalues of the electronic density 〈n〉 and mode Q(2).

In the configurations shown in Fig. 5.8, S(q) has a maximum at q = (π, 3π/4) for

snapshot (a), but there is also a less intense peak at q = (3π/4, π).

Incommensurate peaks in S(q) at q = (π, 3π/4) and q = (3π/4, π) with almost equal

weight are observed for 〈n〉 = 0.8 and λ = 1 (see Fig. 5.8.d) although no peak in N(q) is

observed. In Fig. 5.8.b the snapshot of the final configuration of the corresponding Monte

Carlo run is shown. Other snapshots of configurations appearing during the measuring part

of our simulation are displayed in Fig. 5.9. It can be seen that the ground state is not frozen

and that there is a dynamical charge redistribution. Thus, the magnetic incommensurability

observed in some cuprates could be due to charge and spin configurations similar to those

presented in Fig. 5.9, which could be interpreted as “dynamic stripes”. In fact, these results

may indicate that the patches in Fig. 5.8 and 5.9 are not random islands since, if that

were the case, we would expect that the maxima in S(q) would form a ring in momentum

space.[197] It is tempting to associate the observed patches with “dynamical” stripes. Notice

that for the states with “static” stripes only one peak at q = (π, π − δ) is observed in S(q)

78

if the peak in N(q) is at q = (0, 2δ). On the other hand, the states with “dynamical”

stripes naturally reproduce the four peaks observed in neutron scattering experiments for the

cuprates[186, 187] and the “patch-like” shape of the clusters would explain, at the same time,

the apparently random inhomogeneous structures observed in STM experiments.[169, 170]

As the diagonal electron-phonon coupling λ increases beyond λ = 2, important quanti-

tative changes are observed, in particular, the tendency to bipolaron formation induced by

strong lattice distortions. This behavior will be discussed in more detail in subsection C.

(π,π) (0,π) (0,0) (π,0) (π,π) (0,0)q

0

2

4

6

8

10

S(q)

(c)

(a) (b)

(d)

Figure 5.8: (a) MC snapshot of an 8× 8 lattice at 〈n〉=0.8 λ = 0 and γ = 0; (b) same as (a)but for λ = 1 with mode Q(2); (c) same as (b) for λ = 2; (d) Magnetic structure factor forthe parameters in (b).

79

(d)

(a)

(c)

(b)

Figure 5.9: (a) MC snapshot of an 8× 8 lattice at 〈n〉=0.8, for mode Q(2), λ = 1 and γ = 0,after 2600 measuring sweeps; (b) same as (a) but after 3750 measuring sweeps; (c) same as(a) but after 4250 measuring sweeps; (d) same as (a) but after 5000 measuring sweeps.

5.3.2 Off-Diagonal electron-phonon term:

In this first exploratory study of the effects of the off-diagonal terms due to the EPI we

will allow the parameter γ in Eq. 5.8 and Eq. 5.10 to vary in the interval (0, 0.6), while gJ

(see Eq. 5.10) will be kept equal to 1. In general, we have observed that the effect of the

off-diagonal term is to destabilize the stripes, since they become more dynamic. Examples

of this effect can be seen in the snapshots presented in Fig. 5.10 for 〈n〉 = 0.75 and γ = 0,

0.1, 0.2 and 0.6. The stripes become distorted as γ increases and, eventually, AF domains

separated by irregularly shaped hole-rich regions start to develop.

In Fig. 5.11.a, it is shown how the sharp maximum in the charge structure factor loses

intensity as the stripes become more dynamic. However, notice that the magnetic structure

factor (Fig.8b) still shows incommensurability at q = (π, 3π/4) which is in agreement with

80

(d)

(b)

(c)

(a)

Figure 5.10: Study of the effect of off-diagonal couplings. (a) MC snapshot of an 8×8 latticeat 〈n〉=0.75, λ = 0 and γ = 0; (b) same as (a) but for γ = 0.1 and mode Q(2); (c) same as(b) for γ = 0.2; (d) same as (b) but for γ = 0.6 .

neutron scattering data for dynamic stripes. For γ ≥ 0.4 a peak in S(q) develops at

q = (0, 0). This seems to occur because the hole domains become ferromagnetic as it

can be seen in Fig. 5.10.c and d.

The system also becomes more metallic since the Drude weight increases with γ

(Fig. 5.11.c) and more spectral weight appears in the pseudogap in the DOS (Fig. 5.11.d).

5.3.3 Diagonal and Off-Diagonal terms:

When both diagonal and non-diagonal electron-phonon couplings are active simultaneously,

we have observed that for values of λ ≤ 2, there is a competition between the localizing effect

of the diagonal electron-phonon coupling λ and the disordering tendency of the off-diagonal

parameter γ. As a result, larger values of γ than in the case of λ = 0 are needed to destabilize

the stripes. The stripe states are replaced by AF “clusters” separated by hole-rich regions

81

(0,0) (0,π) (π,π) (0,0)q

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

N(q)γ=0.0γ=0.1γ=0.2γ=0.4γ=0.6

0.0 0.2 0.4 0.6γ

0

2

4

6

8

10

Dru

de

Wei

gh

t

(0,0) (0,π) (π,π) (0,0)q

0

5

10

15

20

25

S(q)

γ=0.0γ=0.1γ=0.2γ=0.4γ=0.6

-0.6 -0.3 0 0.3 0.6ω−µ

0.0

1.0

2.0

3.0

Ν(ω)

γ=0.0γ=0.1γ=0.2γ=0.4

(a)

(c)

(b)

(d)

Figure 5.11: (a) Charge structure factor for different values of γ (strength of the off-diagonalEPI) and for λ = 0 on a 8× 8 lattice at 〈n〉=0.75, for mode Q(2); (b) the magnetic structurefactor for the same parameters as in (a); (c) the Drude weight for the same parameters asin (a); (d) the density of states for the same parameters as in (a).

as shown in Fig. 5.12.

We also have observed that for λ ≥ 3 the modes Q(1) and Q(2) induce charge density

wave (CDW) states. CDW domains are formed in order to accommodate the extra holes

away from half-filling as shown in Fig. 5.13.a for 〈n〉 = 0.875 and mode Q(2). This is the

only case in which we have observed long range order developing in the lattice degrees of

freedom. The dashed lines in the figure indicate different CDW domains. In Fig. 5.13.b it

can be seen how the off-diagonal coupling destabilizes the CDW state and a disordered state

with bipolarons (two electrons trapped at the same site) is observed.

As γ increases the off-diagonal term opposes the trend towards bipolaron formation

caused by the diagonal electron-phonon coupling and stripe-like structures reappear for some

82

(a) (b)

(d)(c)

Figure 5.12: (a) MC snapshot of an 8 × 8 lattice at 〈n〉=0.875, for mode Q(2), λ = 1 andγ = 0; (b) same as (a) but for γ = 0.1; (c) same as (a) but for γ = 0.2; (d) same as (a) butfor γ = 0.4.

dopings.

The half-breathing modes, on the other hand, are not able to stabilize a CDW state

even for large values of diagonal couplings and they only produce disordered states with

bipolarons when the diagonal electron-phonons coupling is large. Since CDW and super-

conducting states normally compete with each other, the half-breathing mode may enhance

the development of superconductivity. Unfortunately, off-diagonal long-range order which

characterizes S or D wave superconductivity, cannot develop with adiabatic phonons.[198]

Thus, this possibility should be explored for finite values of the frequency of the lattice

vibrations in a full quantum calculation.

5.3.4 Spectral Functions:

In this subsection the properties of the spectral functions A(q, ω) will be discussed in detail.

83

(b)(a)

Figure 5.13: (a) MC snapshot of an 8 × 8 lattice at 〈n〉=0.875, for mode Q(2), λ = 6 andγ = 0. The dashed lines separate CDW domains; (b) same as (a) but for γ = 0.2.

Numerical studies of the spin-fermion model without phonons in Ref.[184] have shown

that the underdoped regime is characterized by a depletion of spectral weight along the

diagonal direction in momentum space, where a very weak Fermi surface (FS) may exist.

On the other hand, strong spectral weight, very flat bands, and a well defined Fermi surface

(FS) are observed close to q = (π, 0) and (0, π). These results are in agreement with ARPES

studies for LSCO,[199] material believed to have dynamic stripes, and also with numerical

studies of models in which stripes have been built via a configuration dependent “stripe”

potential in the t-J model[200] or with stripe-like mean-field states.[201] ARPES results for

LNSCO, where stripes have been observed,[190] indicate that the low-energy excitation near

the expected d-wave node region is strongly suppressed.[202]

In the absence of EPI the above mentioned characteristics are well reproduced by the

SF model. In Fig. 5.14.a the spectral function A(q, ω) along the path q = (0, 0) − (π, 0)−(π, π) for 〈n〉 = 0.80 on a 12 × 12 lattice in the absence of EPI is presented. A well defined

quasi-particle peak is observed at (0, 0), a flat band appears close to (π, 0) and the peak

crosses the Fermi energy at (π, 5π/6) defining a hole-like FS. The spectral function along the

diagonal of the Brillouin zone is presented in Fig. 5.14.b. It can be seen how the spectral

weight becomes very incoherent as the Fermi energy is reached.

As a general rule we have observed that electron-phonon interactions increase the

decoherence of the spectral functions, particularly close to the Fermi energy. Below, we will

discuss the effects of the different phonon modes for the most interesting case of dynamic

84

-4 -2 0 2 4 6 8ω−µ

(0,0)

(π,0)

(π,π)

A(q,ω)

-4 -2 0 2 4 6 8ω−µ

(0,0)

(π,0)

(π,π)

A(q,ω)

-4 -2 0 2 4 6 8ω−µ

(0,0)

(π/2,π/2)

(π,π)A(q,ω)

-4 -2 0 2 4 6 8ω−µ

(0,0)

(π/2,π/2)

(π,π)A(q,ω)

(d)

(b)(a)

(c)

Figure 5.14: (a) Spectral functions along the path q = (0, 0)− (π, 0) − (π, π) on a 12 × 12lattice at 〈n〉=0.8, λ = 0 and γ = 0; (b) same as (a) but along the diagonal direction in theBrillouin zone; (c) same as (a) but for λ = 2 and mode Q(3); (d) same as (b) but for λ = 2.

stripes (see, for example, Fig. 5.9).

The diagonal electron-phonon coupling renders the system insulator, as discussed in

subsection A. In Fig. 5.14.c the spectral weight along the path q = (0, 0) − (π, 0) − (π, π)

is displayed for λ = 2 and the half-breathing mode Q(3) in a 12 × 12 lattice. It can be seen

that the quasiparticles are less well defined and that a clear gap has opened at the Fermi

energy. Along the diagonal of the Brillouin zone (Fig. 5.14.d) the spectral weight becomes

more incoherent and a gap also is observed.

The off-diagonal EPI restores spectral weight in the gap but there are no quasiparticles

close to the Fermi energy as it can be seen in Fig. 5.15.a and b where the spectral functions

85

-4 -2 0 2 4 6 8ω−µ

(0,0)

(π,0)

(π,π)

A(q,ω)

-4 -2 0 2 4 6 8ω−µ

(0,0)

(π/2,π/2)

(π,π)A(q,ω)

(a) (b)

Figure 5.15: (a) Spectral functions along the path q = (0, 0)− (π, 0) − (π, π) on a 12 × 12lattice at 〈n〉=0.8, λ = 2 and γ = 0.4 for the half-breathing mode; (b) same as (a) but alongthe diagonal direction in the Brillouin zone.

for λ = 2, γ = 0.4 and mode Q(3) are shown along q = (0, 0) − (π, 0) − (π, π) and along the

diagonal of the Brillouin zone. For γ finite and λ = 0 the only observed changes is that the

peaks in Figs.11a and b become broader.

For λ = 2 the breathing mode creates a larger insulating gap than the half-breathing

mode and produces a larger decoherence of the quasiparticle peaks. The results for the

shear mode are qualitatively very similar.

The off-diagonal EPI for the shear and breathing modes tends to close the gap and

increase the decoherence.

5.4 Influence of Phonons on Homogeneous States andgeneration of Stripes

Up to this point we have considered the effects of EPI on ground states that already

presented charge inhomogeneity. However, it is important to study whether the electron-

phonon interactions proposed in this work can themselves generate charge inhomogeneity, in

particular stripes, in a previously homogeneous ground state.

In order to address this issue, we studied the S-F model with J = 1.5 instead of J = 2,

value which was used in the previous sections (all the other parameters are kept the same).

For 〈n〉 = 0.75 the ground state has an homogeneous charge distribution as it can be observed

86

in the MC snapshot presented in Fig. 5.16.a. Note that despite the charge homogeneity this

state presents magnetic incommensurability due to a spiral spin arrangement in the vertical

direction.

One of the main results of this paper is our observation that a strong diagonal coupling

with the shear mode generates two horizontal or vertical stripes. An example can be seen

in the snapshot presented in Fig. 5.16.b for λ = 2. In this case the holes act as boundaries

between undoped antiferromagnetic states and the magnetic incommensurability arises from

the inhomogeneous charge distribution. A π-shift among the AF domains is observed as

well.[203]

The breathing mode also induces charge inhomogeneity for a diagonal coupling λ = 2.

From the discussion in Section III-A diagonal stripes would be expected but we have observed

two stripes with a zig-zag shape, i.e, the holes align diagonally at short distance scales but,

on the whole, the stripe is still horizontal or vertical (see Fig. 5.16.c).

This result indicates that diagonal EPI are able to induce stripe-like charge inhomo-

geneities in otherwise homogeneous states. However, the S-F model also shows that although

diagonal EPI stabilize stripes they are not necessary to induce them. Charge inhomogeneities

can result even in the absence of EPI, just from magnetic interactions.

5.5 Conclusions

Summarizing, we have studied the effects of diagonal and off-diagonal electron-phonon

interactions using non-biased numerical techniques on a model that has both striped and

homogeneous ground states in the absence of EPI. We found that diagonal EPI tend to

either generate or further stabilize the existing stripes and turn the system into an insulator.

Horizontal and vertical stripes are stabilized by half-breathing and shear modes, but the

stripes generated by shear modes are more localized. The breathing mode stabilizes static

diagonal stripes for intermediate diagonal couplings. Large diagonal couplings stabilize CDW

states for breathing and shear modes but this state is not observed with the half-breathing

mode.

On the other hand, off-diagonal electron-phonon couplings destabilize the stripes making

the ground state more metallic, although non homogeneous. Instead of static stripes other

kinds of inhomogeneous structures characterized by antiferromagnetic domains separated by

87

(c)

(a)

(b)

Figure 5.16: (a) MC snapshot of an 8× 8 lattice at 〈n〉=0.75, J = 1.5, λ = 0 and γ = 0; (b)same as (a) but for λ = 2 with mode Q(2); (c) same as (b) with mode Q(1) . The dashed lineindicates the stripes.

barriers of holes are observed.

The electron-phonon interactions undermine the quasiparticle peaks in the spectral

functions close to the Fermi energy producing incoherent weight. Breathing and shear modes

tend to open large gaps at the Fermi energy creating insulating behavior. The half-breathing

mode, on the other hand, opens smaller gaps that are closed by relatively modest off-diagonal

couplings, allowing for metallic behavior.

Our results indicate that the half-breathing mode is most likely to play a role in non-

insulating materials with vertical and/or horizontal “dynamic” stripes, such as LSCO, while

the breathing mode should dominate on insulators with diagonal stripes like the nickelates.

The most important result of this study is the fact that different phonon modes promote a

diverse array of charge structures and that the relative strength of diagonal and off-diagonal

couplings influences the transport properties. Diagonal electron-phonon couplings promote

88

insulating behavior, while off-diagonal interactions are crucial to achieve metalicity. It

appears that the half-breathing mode off-diagonally coupled to the electrons is the most

likely to produce non-insulating states with dynamical stripes as observed in the cuprates.

This is in agreement with the experimental data which indicate the prevalence of half-

breathing modes in the high Tc cuprates.[165] The electron-phonon interaction introduces

decoherence of the quasi-particle peak in the spectral function in agreement with ARPES

measurements in LSCO.

The crucial issue that remains to be explored is whether electron-phonon interactions are

needed, in addition to the magnetic exchange, in order to develop long range D-wave pairing

correlations. Since superconductivity arises from off-diagonal long range order, it cannot be

generated with adiabatic phonons. The next step will be to study off-diagonally coupled

half-breathing modes at finite frequencies.

89

CHAPTER 6

NUMERICAL TECHNIQUES

As described in the Introduction, numerical techniques have been developed as invaluable

unbiased tools to study the properties of models for strongly correlated electrons that cannot

be exactly solved. Computational results can contribute to the acceptance or rejection of

mean-field based theories, and can also indicate directions in which new approaches should

be developed. In the following sections, we describe the numerical methods used in the

previous chapters of this dissertation.

6.1 Conventional MC Algorithm Applied to SpinFermion Models

Monte Carlo (MC)[204, 205] methods are stochastic techniques. They are based on the use

of random numbers and probability statistics and used in many different ways e.g. as a

technique of integration of a function, as a way to model stochastic (random) processes, as a

tool to calculate properties of a state such as energy, temperature, pressure, and volume, and

as a model to simulate a system of interacting particles e.g. ferromagnetic materials. Monte

Carlo methods work by considering only a randomly chosen representative group of all the

possible configurations in a problem. A configuration of the variables is proposed and it is

accepted or rejected by using algorithms that include the physical properties of the system

with random factors. In many cases, the probability with which a configuration is selected

is determined by the properties of the system. Thus, the values of physical observables can

be obtained after sampling a relatively small number of configurations.

90

6.1.1 Monte Carlo Formalism

In this section, the details on the Monte Carlo simulations of Eq. 2.2 will be presented. We

focus on one-dimensional systems for simplicity but generalization to higher dimensions is

straightforward. Exact consideration of the ~S.~σ term includes a four-fermionic operators

interaction. This quartic term would have to be decoupled using a Hubbard-Stratonovich

transformation[206]. The decoupling allows to express the Hamiltonian in terms of bilinears

in the fermionic operators but at the expense of introducing the so-called Hubbard-

Stratonovich fields which lead to negative terms in the partition function and produce the

famous ”sign problem”[207, 208] that prevents the studies of the doped Hubbard and t-J

models at low temperatures. In spin-fermion models the ~S.~σ term can be made bilinear

in the fermionic operators by considering the classical limit of the localized spin ~S. This

approximation retains many properties of the quantum mechanical properties of the model

and this is the reason why spin-fermion models have been applied to so many different

problems. With classical localized spins the Hamiltonian contains terms that are quadratic

in the fermionic operators. This allows for the huge simplification of having to solve only

a one-electron problem even for large densities, since such high densities can be reached by

simply filling the energy spectrum from bottom up. Hence, in practice the problem at hand

only involves gathering information about the ”one-electron” sector, with the total density

regulated by the chemical potential µ. Therefore, for the case of a chain with L sites the

basis to be used can be considered as a†1,↑|0〉, . . . , a

†L,↑|0〉, a

†1,↓|0〉, . . . , a

†L,↓|0〉. Then, in this

one-electron sector, H can be represented by a 2L × 2L matrix for a fixed classical spin

background.

Formulas needed to carry out a Monte Carlo study are described in this section, following

Ref.[120]. The local spins are assumed to be classical which allows the parametrization of

each local spin in terms of spherical coordinates:

~S(θi, φi) = S(sin θi cos φi, sin θi sin φi, cos θi). (6.1)

The partition function in the grand canonical ensemble can be written as:

Z =L

∏

i

D∏

α

(

∫ π

0

dθi sin θi

∫ 2π

0

dφiZg({θi, φi}). (6.2)

Here, D is the dimension, g represents the fermions and Zg({θi, φi}) = Trg[exp−βK], where

β is the inverse temperature and K = H − µN , with N being the number operator. The

91

trace is taken with respect to the mobile fermions, which are created and destroyed by the

operators a† and a. Representing K by a hermitian matrix, one can diagonalize it by a

unitary matrix U as:

U †KU =

ǫ1 0 . . . 00 ǫ2 . . . 0...

.... . .

...0 0 . . . ǫ2L

. (6.3)

The basis that diagonalizes this matrix is u†1|0〉, . . . , u†

2L|0〉, where um =∑

jσ U †m,jσajσ, and m

runs from 1 to 2L. In this basis, the operator K becomes∑

m ǫmnm, where we have defined

nm = u†mum. Calling nm the eigenvalues of nm one can write the trace as:

Trg(e−βK) =

∑

n1,...,n2L

〈n1 . . . n2L|e−βK |n1 . . . n2L〉

=∑

n1,...,n2L

〈n1 . . . n2L|e−β∑

2Lλ=1

ǫλnλ |n1 . . . n2L〉. (6.4)

Exponential now becomes a number and we can write:

Zg =∑

n1

〈n1|e−βǫ1n1|n1〉 · · ·∑

n1

〈n2L|e−βǫ2Ln2L |n2L〉

=2L∏

λ=1

Trλ(e−βǫλnλ). (6.5)

Since the particles are fermions, the occupation numbers can either be 1 or 0, and the sum

is restricted to those values:

Zg =2L∏

λ=1

1∑

n=0

e−βǫλn =2L∏

λ=1

(1 + e−βǫλ). (6.6)

Hence combining Eq. 6.2 and Eq. 6.6 one arrives at:

Z =L

∏

i

D∏

α

(

∫ π

0

dθi sin θi

∫ 2π

0

dφi

2L∏

λ=1

(1 + e−βǫλ). (6.7)

The integral over the angular variables can be performed using a classical Monte Carlo

simulation where a new configuration (primed) is obtained from the previous one (non-

primed) according to x′i = xi + (r − 0.5)∆x, where x is one of θ, φ, r is a random number

between 0 and 1, and ∆x is a predefined parameter, chosen to be relatively small compared

to the full range of the variable. Quantities that depend on the spin degrees of freedom

(θi, φi) are calculated by averaging over the Monte Carlo configurations.

92

6.1.2 Monte Carlo Simulation

The integral over the angular variables can be performed using a Classical Monte Carlo

simulation [209]. The eigenvalues for the fermionic matrix must be obtained for each classical

spin configuration using library subroutines (LAPACK and its variants were used in this

work). Finding the eigenvalues is the most time consuming part of the numerical simulation.

It is important to remark that the integrand is positive and, therefore, “sign problems” in

which the integrand of the multiple integral under consideration can be non-positive, are not

present in our study.

Since the formalism is in the grand-canonical ensemble, the chemical potential is adjusted

to give the desired carrier density, n. To do so, the equation n(µ) − n = 0 is solved for µ at

every Monte Carlo step by using the Newton-Raphson method[210]. This technique proved

very efficient in adjusting with the precision the desired fermionic density.

Usually, between 2000 to 6000 Monte Carlo iterations were used to let the system

thermalize, and then 500 to 3000 additional steps were carried out to calculate observables,

measuring every 5 of these steps to reduce autocorrelations.

The final thermalized configuration is always saved which means that measuring runs can

be extended if needed. In the case of models in which there are less than 1 localized spin per

site the random location of impurities induces extra calculations since averages over several

disorder configurations need to be performed. Approximately 6 to 10 disorder configurations

were generally used for the small lattices (33) and 4 to 6 for larger lattices (43,53). The

number of configurations used is determined by the desired degree of accuracy in the results.

In Chapter 3 section 3.3 we reported a comprehensive numerical Monte Carlo study of

a realistic lattice model for Mn-doped GaAs, including spin-orbit coupling, as well as the

effects of random Mn doping. This large-scale computational effort was possible by using

the Cray XT3 supercomputer operated by the National Center for Computational Sciences.

Our simulations made use of up to 1,000 XT3 nodes. Parallelization was used in different

ways: (i) to study different regions of parameter space (densities, couplings), and (ii) to

average over different configurations of Mn locations. In all cases, the use of hundreds of

processors in a single parallel run poses several technical challenges that are best handled

by supercomputers with low latency and scalability, rather than by conventional clusters of

PC’s. In fact, this study would have taken several years without access to a supercomputer

93

with thousands of processors, such as the Cray XT3.

Observables

Quantities that depend on the spin degrees of freedom (θi and φi in the previous formalism)

are calculated by averaging over the Monte Carlo configurations. Obviously any observable

that does not have the continuous symmetry of the Hamiltonian will vanish after very long

runs. Thus, it is standard to calculate the absolute value of the magnetization of the spins

normalized to 1, namely, |M |= 1xN

√

∑

I,R〈~SI · ~SR〉 as opposed to the magnetization vector.

Another useful quantity is the spin-spin correlation, defined by:

C(x) =1

N(x)

∑

y

~Sy+x · ~Sy (6.8)

where N(x) is the number of non-zero terms in the sum. The correlation at a distance d is

averaged over all lattice points that are separated by that distance, but since the system is

diluted, the quantity must be normalized to the number of pairs of spins separated by d, to

compare the results for different distances.

The observables that directly depend on the electronic degrees of freedom can be

expressed in terms of the eigenvalues and eigenvectors of the Hamiltonian matrix K [120].

The density of states (DOS), N(ω), is simply given by∑

λ δ(ω − ǫλ).

However, the majority and minority DOS, N↑(ω) and N↓(ω), were also calculated in this

study. N↑(ω) indicates the component that aligns with the local spin, i.e., N↑(ω) is the

Fourier transform of∑

i < c†i↑(t)ci↑(0) >, where ci↑ = cos(θi/2)ci↑ + sin(θi/2)e−iφici↓. Then:

N↑(ω) =2N∑

λ

δ(ω − ǫλ)[N

∑

i

U †i↑,λUλ,i↑ cos2(θi/2) +

U †i↓,λUλ,i↓ sin2(θi/2) +

(

U †i↑,λUλ,i↓ exp(−iφi) + U †

i↓,λUλ,i↑ exp(iφi))

×

cos(θi/2) sin(θi/2)], (6.9)

where for sites i without an impurity θi = φi = 0 is assumed.

A similar expression is valid for N↓(ω). The optical conductivity was calculated as:

σ(ω) =π(1 − e−βω)

ωN

∫ +∞

−∞

dt

2πeiωt < ~jx(t) ·~jx(0) >, (6.10)

94

where the current operator is:

~jx = ite∑

jσ

(c†j+x,σcj,σ − H.c.), (6.11)

with x the unit vector along the x-direction. For ω 6= 0, σ(ω) can be written as:

σ(ω) =∑

λ6=λ′

πt2e2(1 − e−βω)

ωN× (6.12)

|∑

jσ(U †j+xσ,λUjσ,λ′ − U †

jσ,λUj+xσ,λ′)|2(1 + eβ(ρλ−µ))(1 + e−β(ρλ′−µ))

×

δ(ω + ρλ − ρλ′). (6.13)

Both N(ω) and σ(ω) were broadened using a Lorentzian function as a substitute to the

δ-functions that appear in Eqs. (6.9) and (6.13). The width of the Lorentzian used was

ǫ = 0.05 in units of the hopping, t.

The optical conductivity in d dimensions obeys the sum rule:

D

2=

πe2 < −T >

2Nd−

∫ ∞

0+

σ(ω) dω, (6.14)

where D is the Drude weight and T is the kinetic energy:

−T = t∑

<ij>,σ

(c†iσcjσ + H.c.). (6.15)

95

CHAPTER 7

CONCLUSIONS

This dissertation contributes to the understanding of electronic properties in strongly cor-

related electron systems; (III,Mn)V DMSs and High TC cuprates, using unbiased numerical

techniques.

In Chapter 1, an introduction to the strongly correlated electron systems was given. In

Chapter 2, we have discussed general properties of III-V DMS including experimental results,

origin of FM, theoretical scenarios and models for DMS.

Chapter 3 was devoted to the study of a multiband model for DMS using a powerful

nonperturbative techniques, MC. It was found that the addition of one orbital solves the

disagreement with experimental results in the behavior of Tc versus p obtained in the previous

theoretical studies. It is predicted that materials with small ratios between the heavy and

light hole masses would have higher Curie temperatures.

We also presented the first full-scale study of a realistic model for the III-V Mn-doped

semiconductors, performed with unbiased numerical techniques. More specifically, we have

constructed a real-space Hamiltonian with the fcc lattice structure that reproduces the

valence bands of undoped GaAs. Our analysis considers (1) a realistic representation of the

top of the valence band of the parent compound (with the hopping parameters, involving

6 orbitals, known functions of the Luttinger parameters), (2) an exact treatment of the

random Mn doping, (3) a proper consideration of the spin-orbit interaction, and (4) the

exact treatment of the disordered magnetic interactions. There are no free or arbitrary

parameters. The studied finite-size structures contain hundreds of ions. Results performed

in systems with different sizes indicate that there are no strong finite-size effects. We have

studied realistic regimes of Mn doping and hole concentration, and performed comparisons

with experimental data and previous theoretical scenarios. For the first time, a Curie-

96

Weiss shape of the magnetization curves, observed experimentally in highly doped annealed

samples, have been obtained from unbiased numerical lattice simulations in the context

of these materials. The shape of the magnetization curves as well as the actual Curie

temperatures are in good quantitative agreement with available experimental data in all

the regimes of doping investigated. Although there were sizable differences with mean-field

predictions, the system is found to be closer to a hole-fluid regime than to localized carriers.

In Chapter 4, we have developed a new phenomenological model that takes into account

the Mn d-levels that may be important for some III-V DMS materials such as GaN and

GaP in which d-orbitals are close to the top of the VB or inside the semiconductor gap. We

found that if the d-level is deep inside the VB, we reproduce the 6-orbital model results.

But if the d-level was moved towards the top of the VB or above it, a sharp decrease in

TC and magnetization was observed and an IB was generated. However, when a strong p-d

hybridization was considered, Tc and magnetization increased in a presence of IB.

We believe that this unbiased study will contribute to guide future experimental and

theoretical research. Just by changing the values of the parameters in the Hamiltonian a

variety of DMS can be studied in the future, such as Mn-doped GaSb, GaP, GaN and so on.

Finally in Chapter 5, the effects of adiabatic phonons on a spin-fermion model for high

Tc cuprates were studied using numerical simulations. In the absence of electron-phonon

interactions (EPI), stripes in the ground state were observed [161] for certain dopings while

homogeneous states were stabilized in other regions of parameter space. Different modes

of adiabatic phonons were added to the Hamiltonian: breathing, shear and half-breathing

modes. Diagonal and off-diagonal electron-phonon couplings were considered. It was

observed that strong diagonal EPI generate stripes in previously homogeneous states, while

in striped ground states an increase in the diagonal couplings tends to stabilize the stripes,

inducing a gap in the density of states (DOS) and rendering the ground state insulating. The

off-diagonal terms, on the other hand, destabilize the stripes creating inhomogeneous ground

states with a pseudogap at the chemical potential in the DOS. The breathing mode stabilizes

static diagonal stripes; while the half-breathing (shear) modes stabilize dynamical (localized)

vertical and horizontal stripes. It was also observed that the EPI induces decoherence of the

quasi-particle peaks in the spectral functions.

Summarizing, unbiased numerical techniques were applied to the study of two of the

most important current problems in condensed matter physics. Our results for the cuprates

97

indicate that electron-phonon interactions are important to understand the formation of

inhomogeneous states observed experimentally. In the case of DMS we have provided the

first unbiased study of a realistic model for Mn doped GaAs and we found that “valence

bans” scenarios are well suited to describe the system in the relevant range of doping x ≥ 3%.

98

APPENDIX A

Appendix

Here, we provide information about the change of base transformations.

The matrix M described in Section. 3.3.3 is given by:

M =

− 1√2

− i√2

0 0 0 0

1√6

− i√6

0 0 0√

23

0 0√

23

−1√6

−i√6

0

0 0 0 1√2

− i√2

0

0 0 − 1√3

− 1√3

− i√3

0

− 1√3

i√3

0 0 0 1√3

(AI)

and its inverse M−1 is:

M−1 =

− 1√2

1√6

0 0 0 − 1√3

i√2

i√6

0 0 0 −i√3

0 0√

23

0 −1√3

0

0 0 −1√6

1√2

−1√3

0

0 0 i√6

i√2

i√3

0

0√

23

0 0 0 1√3

(AII)

99

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113

BIOGRAPHICAL SKETCH

Yucel YILDIRIM

Personal Information

Place of birthDate of birthNationalityLanguages

Turkey.November 11, 1977Turkish.Turkish (native tongue) and English (fluent).

Education

200720032001

PhD in Physics, Florida State University.MSc in Physics, Middle East Technical University.BSc in Physics, Middle East Technical University.

Publications

“Realistic Lattice Model for Ga1−xMnxAs and Other Lightly Magnetically Doped

Zinc-Blende-Type Semiconductors,”

Yucel Yildirim, Gonzalo Alvarez, and Adriana More,

to appear in Phys. Rev. B.

“Large-Scale Monte Carlo Study of a Realistic Lattice Model for Ga1−xMnxAs,”

Yucel Yildirim, Gonzalo Alvarez, Adriana Moreo and E. Dagotto,

to appear in Phys. Rev. Lett., cond-mat/0612002.

“Critical temperatures of the two-band model for diluted magnetic semiconductors,”

F. Popescu, Yucel Yildirim, Gonzalo Alvarez, Adriana Moreo and E. Da gotto,

Phys. Rev. B. 73, 075206 (2006).

114

“Effect of Adiabatic Phonons on Striped and Homogeneous Ground states,”

Yucel Yildirim and Adriana Moreo,

Phys. Rev. B 72, 134516 (2005).

115