Quantum transport in a cylindrical sub-0.1 μm silicon-based MOSFET

Quantum transport in a cylindrical sub-0.1 lm silicon-basedMOSFET

S.N. Balaban a,b, E.P. Pokatilov a, V.M. Fomin a,b, V.N. Gladilin a,b,J.T. Devreese b,c,d,*, W. Magnus e, W. Schoenmaker e, M. Van Rossum e, B. Sor�eee e

a Departamentul de Fizica Teoretic�aa, Universitatea de Stat din Moldova, MD-2009 Chis�in�aau, Republica Moldovab Theoretische Fysica van de Vaste Stof, Universiteit Antwerpen (UIA), B-2610 Antwerpen, Belgium

c Universiteit Antwerpen (RUCA), B-2020 Antwerpen, Belgiumd Technische Universiteit Eindhoven, 5600 MB Eindhoven, Netherlands

e IMEC, B-3001 Leuven, Belgium

Received 1 March 2000; received in revised form 5 December 2000; accepted 15 February 2001

Abstract

A model is developed for a detailed investigation of the current flowing through a cylindrical sub-0.1 lm MOSFET

with a closed gate electrode. The quantum mechanical features of the lateral charge transport are described by a Wigner

distribution function which is explicitly dealing with electron scattering due to acoustic phonons and acceptor impu-

rities. A numerical simulation is carried out to obtain a set of I–V characteristics for various channel lengths. It is

demonstrated that inclusion of the collision term in the numerical simulation is important for low values of the source–

drain voltage. The calculations have further shown that the scattering leads to an increase of the electron density in the

channel thereby smoothing out the threshold kink in the I–V characteristics. An analysis of the electron phase-space

distribution shows that scattering does not prevent electrons from flowing through the channel as a narrow stream, and

that features of both ballistic and diffusive transport may be observed simultaneously. � 2002 Elsevier Science Ltd. All

rights reserved.

1. Introduction

During the last decade significant progress has been

achieved in the scaling of the metal-oxide-semiconductor

field-effect transistor (MOSFET) down to devices with a

sub-0.1 lm size. In Refs. [1–3], silicon-based MOSFETs

with channel length shorter than 0.1 lm were fabricated

and investigated in a wide range of temperature. How-

ever, short channel effects together with random effects

in the silicon substrate are very well known to cause a

degradation of the threshold voltage and the appearance

of uncontrollable charge and current in regions far from

the gate electrode. Therefore in a sub-0.1 lm conven-

tional MOSFET, the controlling ability of the gate

electrode is substantially weakened. In order to achieve

an improved device performance, considerable attention

has been paid to Si-on-insulator (SOI) MOSFETs [4–6]

and double-gate MOSFETs [7,8]. As a result, a sub-

stantial reduction of the short-channel effect in the SOI

MOSFET was established as compared to MOSFETs

with a conventional geometry. In Ref. [9] we have in-

vestigated the thermal equilibrium state of a cylindrical

silicon-based MOSFET device with a closed gate elec-

trode (MOSFETCGE). An advantage of the latter is the

complete suppression of the floating body effect caused

by external influences. Moreover, the short-channel ef-

fect in these devices can be even weaker than that in SOI

and double-gate structures.

The main goal of the present work is the investigation

of quantum transport in a sub-0.1 lm MOSFETCGE

device. We have developed a flexible 2D model which

Solid-State Electronics 46 (2002) 435–444

*Corresponding author. Address: Theoretische Vaste Stof

Fysica, Departement Natuurkunde, Universiteit Antwerpen

(UIA), Universiteitsplein 1, B-2610 Antwerpen, Belgium. Tel.:

+32-3-820-24-59; fax: +32-3-820-22-45.

E-mail address: [email protected] (J.T. Devreese).

0038-1101/02/$ - see front matter � 2002 Elsevier Science Ltd. All rights reserved.

PII: S0038-1101 (01 )00117-4

optimally combines analytical and numerical methods

and describes the main features of the MOSFETCGE

device. The theoretical modeling of the quantum trans-

port features involves the use of the Wigner distribution

function formalism [10–12]. The paper is organized as

follows. In Section 2, a description of the system is

presented in terms of a one-electron Hamiltonian. In

Section 3, the quantum Liouville equation satisfied by

the electron density matrix is transformed into a set of

one-dimensional equations for partial Wigner distribu-

tion functions. A one-dimensional collision term is de-

rived in Section 4. In Section 5, we describe a numerical

model to solve the equations which have been derived

for the partial Wigner distribution function. In Section

6, the results of the numerical calculations are discussed.

Finally, in Section 7, we give a summary of our results

and conclusions about the influence of the scattering

processes in sub-0.1 lm MOSFETs.

2. The Hamiltonian of the system

We consider a cylindrical MOSFETCGE structure

[9] (Fig. 1) described by cylindrical coordinates ðr;/; zÞ,where the z-axis is chosen to be the symmetry axis. The

structure contains a semiconductor pillar of Si with

radius R and length L. The source and drain regions

with the lengths Ls and Ld, respectively, are n-doped

by phosphorus with concentration ND ¼ 1020 cm�3,

whereas the p-doped channel with the length Lch has

an acceptor concentration NA ¼ 1018 cm�3 (boron). The

lateral surface of the semiconductor pillar is covered by

the SiO2 oxide layer, while the aluminum gate overlays

the whole oxide layer. In the semiconductor pillar the

electron motion is determined by the following Hamil-

tonian

bHHj ¼ � �h2

2m?j

1

ro

orro

or

� ��þ o2

o/2

�� h2

2mkj

o2

oz2þ V ðrÞ;

ð1Þ

where V ðrÞ ¼ VbðrÞ þ VeðrÞ is the potential energy asso-

ciated with the energy barrier and the electrostatic field,

respectively; m?j and mk

j are the effective masses of the

transverse (in the (x; y)-plane) and longitudinal (along

the z-axis) motion of an electron of the jth valley.

The electrostatic potential energy VeðrÞ satisfies

Poisson’s equation

DVeðrÞ ¼e2

e0eið�nðrÞ þ NDðrÞ � NAðrÞÞ; i ¼ 1; 2; ð2Þ

where e1 and e2 are the dielectric constants of the semi-conductor and oxide layers, respectively; nðrÞ, NDðrÞ andNAðrÞ are, correspondingly, the concentrations of elec-

trons, donors and acceptors as a function of r � ðr;/; zÞ.The barrier potential VbðrÞ is non-zero in the oxide layer

only, wherein its value is constant Vb. In our calculationswe assume that the source electrode is grounded whereas

the potentials at the drain and gate electrodes equal Vdand VG, respectively.

The study of the charge distribution in the cylindrical

MOSFETCGE structure in the state of the thermody-

namical equilibrium (see Ref. [9]) has shown that the

concentration of holes is much lower than that of elec-

trons so that electron transport is found to provide the

main contribution to the current flowing through the

MOSFET. For that reason, holes are neglected in

the present transport calculations.

3. The Liouville equation

In this section we consider ballistic transport of

electrons. Neglecting scattering processes and inter-

valley transitions in the conduction band, the one-elec-

tron density matrix can be written as

qðr; r0Þ ¼Xj

qjðr; r0Þ; ð3Þ

where qjðr; r0Þ is the density matrix of electrons residingin the jth valley satisfying Liouville’s equation

i�hoqjot

¼ ½Hj; qj�: ð4Þ

In order to impose reasonable boundary conditions

for the density matrix in the electrodes, it is convenient

to describe the quantum transport along the z-axis in a

phase-space representation. In particular, we rewrite Eq.

(4) in terms of f ¼ ðzþ z0Þ=2 and g ¼ z� z0 coordinatesand express the density matrix qj asFig. 1. Scheme of the cylindrical MOSFETCGE.

436 S.N. Balaban et al. / Solid-State Electronics 46 (2002) 435–444

qjðr; r0Þ ¼Xms;m0s0

1

2p

Z þ1

�1dk eikgfjmsm0s0 ðf; kÞWjmsðr;/; zÞ

W�jm0s0 ðr0;/

0; z0Þ ð5Þ

with a complete set of orthonormal functions

Wjmsðr;/; zÞ. According to the cylindrical symmetry of

the system, these functions take the following form:

Wjmsðr;/; zÞ ¼1ffiffiffiffiffiffi2p

p wjmsðr; zÞeim/: ð6Þ

The functions wjmsðr; zÞ are chosen to satisfy the equa-

tion

� �h2

2m?j

1

ro

orro

or

� �� m2

r2

�wjmsðr; zÞ þ V ðr; zÞwjmsðr; zÞ

¼ EjmsðzÞwjmsðr; zÞ; ð7Þ

which describes the radial motion of an electron. Here

EjmsðzÞ are the eigenvalues of Eq. (7) for a given value ofthe z-coordinate which appears as a parameter. It will beshown, that EjmsðzÞ plays the role of an effective poten-

tial in the channel, and that Wjmsðr;/; zÞ is the corre-

sponding wavefunction of the transverse motion at a

fixed z. Substituting the expansion (5) into Eq. (4), and

using Eq. (7), we arrive at an equation for fjmsm0s0 ðf; kÞ :

ofjmsm0s0 ðf; kÞot

¼ � �hk

mkj

o

offjmsm0s0 ðf; kÞ

þ 1

�h

Z þ1

�1Wjmsm0s0 ðf; k � k0Þfjmsm0s0 ðf; k0Þdk0

�Xs1 ;s01

Z þ1

�1bMMs1s01jmsm0s0 ðf; k; k0Þfjms1m0s0

1ðf; k0Þdk0; ð8Þ

where

Wjmsm0s0 ðf; k � k0Þ ¼ 1

2pi

Z þ1

�1ðEjmsðf þ g=2Þ

� Ejm0s0 ðf � g=2ÞÞeiðk0�kÞg dg; ð9Þ

bMMs1s01jmsm0s0 ðf; k; k0Þ ¼

1

2p

Z þ1

�1ds0s0

1

bCCmss1ðfh

þ g=2; k0Þ

þ dss1 bCC�ms0s0

1ðf � g=2; k0Þ

ieiðk

0�kÞg dg;

ð10Þ

bCCmss1ðz; k0Þ ¼�h

2mkj ibjmss1ðzÞ þ

�h

2mkj

cjmss1ðzÞ�� i

o

ofþ 2k0

�;

ð11Þ

and

bjmss1ðzÞ ¼Z

w�jmsðr; zÞ

o2

oz2wjms1ðr; zÞrdr; ð12Þ

cjmss1ðzÞ ¼Z

w�jmsðr; zÞ

o

ozwjms1ðr; zÞrdr: ð13Þ

Note that Eq. (8) is similar to the Liouville equation

for the Wigner distribution function, which is derived

to model quantum transport in tunneling diodes (see

Ref. [10]). The first drift term in the right-hand side of

Eq. (8) is derived from the kinetic-energy operator of the

longitudinal motion. It is exactly the same as the cor-

responding term of the Boltzmann equation. The second

component plays the same role as the force term does in

the Boltzmann equation. The last term in the right-hand

side of Eq. (8) contains the operator bMMs1s01jmsm0s0 ðf; k; k0Þ,

which mixes the functions fjmsm0s0 with different indexes

s; s0. It appears because wjmsðr; zÞ are not eigenfunctionsof the Hamiltonian (1). The physical meaning of the

operator bMMs1s01jmsm0s0 ðf; k; k0Þ will be discussed below.

In order to solve Eq. (8), we need to specify boundary

conditions for the functions fjmsm0s0 ðf; kÞ. For a weak

current, electrons incoming from both the source and

the drain electrodes, are assumed to be maintained in

thermal equilibrium. Comparing Eq. (5) with the cor-

responding expansion of the density matrix in the

equilibrium state, one obtains the following boundary

conditions

fjmsm0s0 ð0; kÞ ¼ dss0dmm02½expðEjsmkb � EFSbÞ þ 1��1;k > 0;

fjmsm0s0 ðL; kÞ ¼ dss0dmm02½expðEjsmkb � EFDbÞ þ 1��1;k < 0;

ð14Þ

where the total energy is Ejsmk ¼ �h2k2=2mkj þ Ejsmð0Þ for

an electron entering from the source electrode ðk > 0Þand Ejsmk ¼ �h2k2=2mk

j þ EjsmðLÞ for an electron entering

from the drain electrode ðk < 0Þ. In Eq. (14) b ¼ 1=kBTis the inverse thermal energy, while EFS and EFD are the

Fermi energy levels in the source and in the drain, re-

spectively. Note, that Eq. (14) meets the requirement of

imposing only one boundary condition on the function

fjmsm0s0 ðf; kÞ at a fixed value of k as Eq. (8) is a first orderdifferential equation with respect to f. Generally

speaking, the solution of Eq. (8) with the conditions (14)

depends on the distance between the boundary position

and the active device region. Let us estimate how far the

boundary must be from the active device region in order

to avoid this dependence. It is easy to show that the

density matrix of the equilibrium state is a decaying

function of g ¼ z� z0. The decay length is of the order ofthe coherence length kT ¼ ð�h2=mk

j kBT Þ1=2

at high tem-

perature and of the inverse Fermi wavenumber k�1F ¼ð�h2=2mk

j EFÞ1=2

at low temperature. So, it is obvious, that

the distance between the boundary and the channel must

S.N. Balaban et al. / Solid-State Electronics 46 (2002) 435–444 437

exceed the coherence length or the inverse Fermi wave-

number, i. e. Ls; Ld � kT or Ls; Ld � k�1F . For example,

at T ¼ 300 K the coherence length kT � 3 nm is much

less than the source or drain lengths.

The functions fjmsm0s0 ðf; kÞ, which are introduced in

Eq. (5), are used in calculations of the current and the

electron density. The expression for the electron density

follows directly from the density matrix as nðrÞ ¼ qðr; rÞ.In terms of the functions fjmsm0s0 ðf; kÞ, the electron den-

sity can be written as follows:

nðrÞ ¼ 1

2p

Xjmsm0s0

Z þ1

�1fjmsm0s0 ðz; kÞdkWjmsðr;/; zÞW�

jm0s0 ðr;/; zÞ:

ð15Þ

It is well known [13], that the current density can be

expressed in terms of the density matrix

jðrÞ ¼Xj

e�h2mji

o

or

�� o

or0

�qjðr; r0Þ

r¼r0

: ð16Þ

The total current, which flows through the cross-section

of the structure at a point z, can be obtained by an in-

tegration over the transverse coordinates. Substituting

the expansion (5) into Eq. (16) and integrating over rand u, we find

J ¼ eXj;m;s

1

2p

Z þ1

�1dk

�hk

mkj

fjmsmsðz; kÞ �2e�h

mkj

Xj;m;s;s0s0>s

cjmss0 ðzÞZ þ1

�1dk Imfjmsms0 ðz; kÞ; ð17Þ

where Imf is the imaginary part of f . The first term in

the right-hand side of Eq. (17) is similar to the expres-

sion for a current of the classical theory [13]. The second

term, which depends on the non-diagonal functions

fjmsms0 only, takes into account the effects of intermixing

between different states of the transverse motion.

The last term in the right-hand side of Eq. (8) takes

into consideration the variation of the wavefunctions

wjmsðr; zÞ along the z-axis. In the source and drain re-

gions, the electrostatic potential is essentially constant

due to the high density of electrons. In these parts of the

structure, the wavefunctions of the transverse motion

are very weakly dependent on z, and consequently, the

operator bMMs1s01jmsm0s0 has negligible effect. Inside the channel,

electrons are strongly localized at the Si/SiO2 interface as

the positive gate voltage is applied. Earlier calculations,

which we made for the case of equilibrium [9], have

shown that in the channel the dependence of wjmsðr; zÞ onz is weak, too. Therefore, in the channel the effect of the

operator bMMs1s01jmsm0s0 is negligible. In the intermediate re-

gions (the source–channel and the drain–channel), an

increase of the contribution of the third term in the

right-hand side of Eq. (8) is expected due to a sharp

variation of wjmsðr; zÞ. Since bMMs1s01jmsm0s0 couples functions

fjmsm0s0 ðf; kÞ with different quantum numbers (jms), it canbe interpreted as a collision operator, which describes

transitions of electrons between different quantum states

of the transverse motion. Thus, the third term in the

right-hand side of Eq. (8) is significant only in the close

vicinity of the p–n junctions. Therefore, this term is as-

sumed to give a small contribution to the charge and

current densities. Under the above assumption, we have

treated the last term in the right-hand side of Eq. (8) as a

perturbation. Hereafter, we investigate the steady state

of the system in a zeroth order approximation with re-

spect to the operator bMMs1s01jmsm0s0 . Neglecting the latter, one

finds that, due to the boundary conditions (14), all non-

diagonal functions fjmsm0s0 ðf; kÞ (m 6¼ m0 or s 6¼ s0) need tobe zero.

In the channel, the energy of the transverse motion

can be approximately written in the form [9]

EjmsðzÞ ¼ EjsðzÞ þ�h2m2

2m?js�RR2js

; ð18Þ

where EjsðzÞ is the energy associated with the radial

size quantization and �h2m2=2m?js�RR2js is the energy of

the angular motion with averaged radius �RRjs. Hence, inEq. (9) for the diagonal functions fjmsmsðf; kÞ the differ-ence Ejmsðf þ g=2Þ � Ejmsðf � g=2Þ can be substituted

by Ejsðf þ g=2Þ � Ejsðf � g=2Þ. Furthermore, summa-

tion over m in Eq. (8) gives

�hk

mkj

o

offjsðf; kÞ �

1

�h

Z þ1

�1Wjsðf; k � k0Þfjsðf; k0Þdk0 ¼ 0

ð19Þ

with

fjsðf; kÞ ¼1

2p

Xm

fjmsmsðf; kÞ: ð20Þ

In Eq. (19) the following notation is used

Wjsðf; kÞ ¼ � 1

2p

Z þ1

�1Ejsðf�

þ g=2Þ

� Ejsðf � g=2Þ�sinðkgÞdg: ð21Þ

The effective potential EjsðzÞ can be interpreted as

the bottom of the subband ðj; sÞ in the channel. The

function fjsðf; kÞ is referred to as a partial Wigner

distribution function describing electrons which are

travelling through the channel in the inversion layer

subband ðj; sÞ.


4. Electron scattering

In this section we consider the electron scattering

from phonons and impurities. For this purpose we in-

troduce a Boltzmann-like single collision term [13],

which in the present case has the following form

Stfjsmk ¼Xj0s0m0k0

Pjsmk;j0s0m0k0fj0s0m0k0

�� Pjsmk;j0s0m0k0fjsmk

�:

ð22Þ

As was noted above, we have neglected all transitions

between quantum states with different sets of quantum

numbers j and s. In the source and drain contacts the

distribution of electrons over the quantum states of

the angular motion corresponds to equilibrium. Conse-

quently, due to the cylindrical symmetry of the system,

we may fairly assume that across the whole structure the

electron distribution is given by

fjsmkðzÞ ¼ fjsðz; kÞwjsm; ð23Þ

where

wjsm ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�h2b2m?

j�RR2jsp

sexp

0@� b�h2m2

2m?j�RR2js

1A ð24Þ

is the normalized Maxwellian distribution function with

respect to the angular momentum m. The integration of

the both sides of Eq. (22) over the angular momentum

gives the one-dimensional collision term

Stfjsðz; kÞ ¼Xk0

Pjsðk; k0Þfjsðz; k0Þ�

� Pjsðk0; kÞfjsðz; kÞ�;

ð25Þ

where

Pjsðk; k0Þ ¼Xmm0

Pjsmk;jsm0k0wjsm0 : ð26Þ

This collision term is directly incorporated into the one-

dimensional Liouville equation (19) as

~WWjsðz; k; k0Þ ¼ Wjsðz; k � k0Þ þ Pjsðz; k; k0Þ� dk;k0

Xk0Pjsðz; k0; kÞ; ð27Þ

where ~WWjsðz; k; k0Þ is the modified force term in Eq. (19).

In this work we consider scattering by acceptor im-

purities and acoustic phonons described by a defor-

mation potential. The scattering rates are evaluated

according to Fermi’s golden rule

Pjsmk;jsm0k0 ¼2p�h

jsm0k0� bHHint jsmkj i 2d Ejsm0k0

�� Ejsmk

�;

ð28Þ

where bHHint is the Hamiltonian of the electron–phonon or

the electron–impurity interaction. Hereafter, we model

the potential of an ionized acceptor as UðrÞ ¼ 4pe2R2s=e1dðrÞ, where Rs determines a cross-section for scatteringby an impurity. Consequently, the absolute value of the

matrix element is

jsm0k0� Uðr � riÞ jsmkj i

¼ 4pe2R2s=e1W2jsðri; ziÞ: ð29Þ

Averaging this over a uniform distribution of acceptors

results in the following scattering rate

P ijsmk;jsm0k0 ¼ Ci

Z R

0

w4jsðr; zÞd Ejsm0k0

�� Ejsmkk

�rdr; ð30Þ

where Ci ¼ NA 4pe2R2s=e1

� �2=�h and NA is the acceptor

concentration.

At room temperature the rate of the scattering by

acoustic phonons has the same form. Indeed, for T ¼300 K the thermal energy kBT � �hxq, therefore the

acoustic deformation potential scattering is approxi-

mately elastic, and the emission and absorption rates are

equal to each other. For low energies we can approxi-

mate the phonon number as Nq � kBT=�hxq � 1 and the

phonon frequency xq ¼ v0q, where v0 is the sound ve-

locity. Assuming equipartition of energy in the acoustic

modes, the scattering rate is

P phjsmk;jsm0k0 ¼

2pVCph

Xq

jsm0k0� eiq�r jsmk � 2

d Ejsm0k0�

� Ejsmk�; ð31Þ

where the parameter Cph ¼ 4R2ukBT=9pqv20�h (Ru and q

being the deformation potential and the mass density,

respectively). Integrating over q yields the scattering rate

P phjsmk;jsm0k0 in the form (30) with Cph instead of Ci. The fullscattering rate Pjsmk;jsm0k0 ¼ P ijsmk;jsm0k0 þ P ph

jsmk;jsm0k0 is then

inserted into Eq. (26) in order to obtain the one-

dimensional scattering rate

Pjsðz; k; k0Þ ¼ Ci�

þ Cph

�aðzÞF �h2k02

2mkj

� �h2k2

2mkj

!; ð32Þ

where

aðzÞ ¼

ffiffiffiffiffiffiffiffiffiffi2m?

j

�h2pb

s�RRjs

Z R

0

w4jsðr; zÞrdr; F ðxÞ ¼ e�x=2K0ðjxj=2Þ;

where K0ðxÞ is a McDonald function [14]. In calcula-

tions of the scattering by acoustic phonons the follow-

ing values of parameters for Si are used: Ru ¼ 9:2 eV,

q ¼ 2:3283 103 kg/m3, v0 ¼ 8:43 105 cm/s [15].

5. Numerical model

The system under consideration consists of regions

with high (the source and drain) and low (the channel)

concentrations of electrons. The corresponding electron

distribution difference would produce a considerable


inaccuracy if we would have attempted to directly con-

struct a finite-difference analog of Eq. (19). It is worth

mentioning that, in the quasi-classical limit, i.e. Ejsðfþg=2Þ � Ejsðf � g=2Þ � ðoEjsðfÞ=ofÞg, Eq. (19) leads to

the Boltzmann equation with an effective potential

which has the following exact solution in the equilibrium

state (EF ¼ EFS ¼ EFD):

f eqjs ðf; kÞ ¼1

p

Xm

exp�h2k2

2mkj

b

0@24 þ EjsðfÞb

þ �h2m2b

2m?js�RR2jsðfÞ

� EFb

1Aþ 1

35�1

: ð33Þ

For numerical calculations it is useful to write down the

partial Wigner distribution function as fjsðf; kÞ ¼ f eqjs ðf;kÞ þ f djsðf; kÞ. Inserting this into Eq. (19), one obtains thefollowing equation for f djsðf; kÞ:

�hk

mkj

o

off djsðf; kÞ �

1

�h

Z þ1

�1Wjsðf; k � k0Þf djsðf; k0Þdk0

¼ Bjsðf; kÞ; ð34Þ

where

Bjsðf; kÞ ¼1

2p

Z þ1

�1dk0Z þ1

�1dg Ejs f

��þ g2

�� Ejs f

�� g2

�� oEjsðfÞ

ofg

� sin½ðk � k0Þg�f eqjs ðf; k0Þ: ð35Þ

The unknown function f djsðf; kÞ takes values of the sameorder throughout the whole system, and therefore is

suitable for numerical computations. In the present

work, we have used the finite-difference model, which is

described in Ref. [10]. The position variable takes the set

of discrete values fi ¼ Dfi for fi ¼ 0; . . . ;Nfg. The valuesof k are also restricted to the discrete set kp ¼ ð2p�Nk � 1ÞDk=2 for fp ¼ 1; . . . ;Nkg. On a discrete mesh,

the first derivative ofjs=ofðfi; kpÞ is approximated by

the left-hand difference for kp > 0 and the right-hand

difference for kp < 0. It was shown in Ref. [10], that such

a choice of the finite-difference representation for the

derivatives leads to a stable discrete model. Projecting

the equation (34) onto the finite-difference basis gives a

matrix equation L � f ¼ b. In the matrix L, only the di-

agonal blocks and one upper and one lower codiagonal

blocks are non-zero:

L ¼

A1 �E 0 . . . 0�V A2 �E . . . 00 �V A3 . . . 0

..

. ... ..

. . .. ..

.

0 0 0 . . . ANf�1

0BBBBB@

1CCCCCA: ð36Þ

Here, the Nk Nk matrices Ai, E, and V are

½Ai�pp0 ¼ dpp0 �2mk

jDf

�h2ð2p � Nk � 1ÞDkWjsðfi; kp � kp0 Þ;

½E�pp0 ¼ dpp0hNk þ 1

2

�� p

;

½V �pp0 ¼ dpp0h p�

� Nk þ 1

2

; ð37Þ

and the vectors are

½fi�p ¼ fjsðfi; kpÞ and ½bi�p ¼ Bjsðfi; kpÞ;i ¼ 1;Nf � 1; p ¼ 1;Nk : ð38Þ

A recursive algorithm is used to solve the matrix equa-

tion L � f ¼ b. Invoking downward elimination, we are

dealing with Bi ¼ Ai � VBi�1ð Þ�1E and ci ¼ Ai�ðVBi�1Þ�1 bi þ Vci�1ð Þ ði ¼ 1; . . . ;NfÞ as relevant matricesand vectors. Then, upward elimination eventually yields

the solution fi ¼ Bifiþ1 þ ci (i ¼ Nf � 1; . . . ; 1). If an

index of a matrix or a vector is smaller than 1 or larger

than Nf � 1, the corresponding term is supposed to

vanish.

In the channel, the difference between effective po-

tentials EjsðfÞ with different (j; s) is of the order of or

larger than the thermal energy kBT . Therefore, in the

channel only a few lowest inversion subbands must be

taken into account. In the source and drain, however,

many quantum states ðj; sÞ of the radial motion are

strongly populated by electrons. Therefore, we should

account for all of them in order to calculate the charge

distribution. Here, we can use the fact that, according to

our approximation, the current flows only through the

lowest subbands in the channel. Hence, only for these

subbands the partial Wigner distribution function of

electrons is non-equilibrium. In other subbands elec-

trons are maintained in the state of equilibrium, even

when a bias is applied. So, in Eq. (15) for the electron

density, we can substitute functions fjmsmsðz; kÞ of highersubbands by corresponding equilibrium functions. For-

mally, adding and subtracting the equilibrium functions

for the lowest subbands in Eq. (15), we arrive at the

following equation for the electron density

nðrÞ ¼ neqðrÞ þ1

2p

Xjs

Z þ1

�1dk fjsðz; kÞ wjsðr; zÞ

2�� f eqjs ðz; kÞ weq

js ðr; zÞ 2�; ð39Þ

where neqðrÞ and weqjs ðr; zÞ are the electron density and the

wavefunction of the radial motion in the state of equi-

librium, respectively. The summation on the right-hand

side of Eq. (39) is performed only over the lowest sub-

bands. Since the electrostatic potential does not pene-

trate into the source and drain, we suppose that the


equilibrium electron density in these regions is well de-

scribed by the Thomas–Fermi approximation:

neqðrÞ ¼ NC

2ffiffiffip

p F1=2 b eV ðr; zÞðð þ EF � ECÞÞ; ð40Þ

where the Fermi integral is

F1=2ðxÞ ¼Z 1

0

ffiffiffiffiffiffiffitdt

p

expðt � xÞ þ 1: ð41Þ

Here NC is the effective density of states in the conduc-

tion band and EF is the Fermi level of the system in the

state of equilibrium.

6. Numerical results

During the device simulation three equations are

solved self-consistently: (i) the Poisson equation (2), (ii)

the equation for the wavefunction of the radial motion

(7), and (iii) the equation for the partial Wigner distri-

bution function (19). The methods of numerical solution

of Eqs. (2) and (7) are the same as for the equilibrium

state [9]. The numerical model for Eq. (19) was described

in Section 5. In the present calculations the four lowest

subbands ðj ¼ 1; 2 and s ¼ 1; 2Þ are taken into account.

The electron density in the channel is obtained from Eq.

(15), whereas in the source and drain regions it is de-

termined from Eq. (39). The calculations are performed

for structures with a channel of radius R ¼ 50 nm and

for various values of the length: Lch ¼ 40, 60, 70 and

80 nm. The lengths of the source and drain are Ls ¼Ld ¼ 20 nm. The width of the oxide layer is taken to be

4 nm and the barrier potential is Vb ¼ 3 eV. All calcu-

lations are carried out with Nf ¼ 100 and Nk ¼ 100. The

partial Wigner distribution functions, which are ob-

tained as a result of the self-consistent procedure, are

then used to calculate the current according to Eq. (16).

We investigate the quantum transport with and

without the electron scattering. The latter is so-called

ballistic transport. The distribution of the electrostatic

potential is represented in Fig. 2 for Vd ¼ 0:3 V and

VG ¼ 1 V. This picture is typical for the MOSFET

structure, which is considered here. The cross-sections of

the electrostatic potentials for r ¼ 0; 30; 40; 45; 48; 50 nmare shown in Fig. 3. The main part of the applied gate

voltage falls in the insulator (50 nm < r < 54 nm). Along

the cylinder axis in the channel, the electrostatic poten-

tial barrier for the electron increases up to about 0.4 eV.

Since the potential along the cylinder axis is always high,

the current mainly flows in a thin layer near the semi-

conductor–oxide interface. Varying Vd and VG mainly

changes the shape of this narrow path, and, as a con-

sequence, influences the form of the effective potential

EjsðzÞ. As follows from Figs. 2 and 3, the radius of the

pillar can be taken shorter without causing barrier

degradation. At the p–n junctions (source–channel and

drain–channel) the electrons meet barriers across the

whole semiconductor. These barriers are found to persist

even for high values of the applied source–drain voltage

and prevent an electron flood from the side of the

strongly doped source. The pattern of the electrostatic

Fig. 2. Distribution of the electrostatic potential in the MOSFET with Lch ¼ 60 nm at VG ¼ 1 V and Vd ¼ 0:3 V.


potentials differs mainly near the semiconductor–oxide

interface, where the inversion layer is formed. In Fig. 4

the effective potential for the lowest inversion subband

ðj ¼ 1; s ¼ 1Þ is plotted as a function of z for differentapplied bias Vd ¼ 0; . . . ; 0:5 V, VG ¼ 1 V, Lch ¼ 60 nm.

It is seen that the effective potential reproduces the dis-

tribution of the electrostatic potential near the semicon-

ductor–oxide interface. In the case of ballistic transport

(dashed curve), the applied drain–source voltage sharply

drops near the drain–channel junction (Figs. 3 and 4).

The scattering of electrons (solid curve) smoothes out

the applied voltage, which is now varying linearly

along the whole channel. Note that the potential ob-

tained by taking into account scattering is always higher

than that of the ballistic case. The explanation is clear

from Fig. 5, where the linear electron density is plotted

for VG ¼ 1 V and Lch ¼ 60 nm. It is seen that, due to

scattering, the electron density in the channel (solid

curve) rises and smoothes out. Hence, the applied gate

voltage is screened more effectively, and as a result, the

potential exceeds that of the ballistic case. It should be

noted that at equilibrium (Vd ¼ 0) the linear density and

the effective potential for both cases (with and without

scattering) are equal to each other. This result follows

from the principle of detailed balance.

The current–voltage characteristics (the current den-

sity I ¼ J=2pR vs. the source–drain voltage Vd) are

shown in Figs. 6 and 7 for the structures with channel

lengths Lch ¼ 40; . . . ; 80 nm. At a threshold voltage

Vd � 0:2 V a kink in the I–V characteristics of the device

is seen. At subthreshold voltages Vd < 0:2 V the deriv-

ative dVd=dJ gives the resistance of the structure. It is

natural, that scattering enhances the resistance of the

structure (solid curve) compared to the ballistic trans-

port (dashed curve). Scattering is also found to smear

the kink in the I–V characteristic. At a voltage Vd >0:2 V a saturation regime is reached. In this part of

I–V characteristics, the current through the structure

increases more slowly than it does at a subthreshold

voltage. The slope of the I–V curve in the saturation

regime rises when the length of the channel decreases.

This effect is explained by a reduction of the p–n junc-

tion barrier potential as the length of the channel be-

comes shorter than the p–n junction width. In Fig. 7,

Fig. 3. Cross-sections of the electrostatic potentials calculated:

(- - -) without scattering and (—) with scattering from acceptor

impurities and from an acoustic deformation potential.

Fig. 4. Effective potential as a function of z for various Vd,Lch ¼ 60 nm.

Fig. 5. Linear electron density in the channel as a function of zfor various Vd, Lch ¼ 60 nm.


one can see that when the transistor is switched off

(VG < 0:5 V), the influence of the scattering on the cur-

rent is weak. This fact is due to a low concentration of

electrons, resulting in a low amplitude of the scattering

processes.

In Fig. 8a and b the contour plots of the partial

Wigner distribution function (j ¼ 1; s ¼ 1) are given for

both cases ((a) without and (b) with scattering). The

lighter regions in these plots indicate the higher den-

sity of electrons. Far from the p–n junction, where the

effective potential varies almost linearly, the partial

Wigner distribution function can be interpreted as a

distribution of electrons in the phase space. When elec-

trons travel in the inversion layer without scattering,

their velocity increases monotonously along the whole

channel. Therefore, in the phase-space representation

the distribution of ballistic electrons looks as a narrow

stream in the channel (Fig. 8a). As it is expected, scat-

tering washes out the electron jet in the channel (see

Fig. 6. Current–voltage characteristics at VG ¼ 1 V for different

channel lengths.Fig. 7. Current–voltage characteristics for MOSFET with

Lch ¼ 40 nm.

Fig. 8. Contour plots of the partial Wigner distribution function fjsðz; kÞ for the lowest subband (j ¼ 1; s ¼ 1) at VG ¼ 1 V, Vd ¼ 0:3 V,

Lch ¼ 60 nm: (a) without scattering, (b) with scattering.


Fig. 8b). It is worth mentioning that the electron stream

in the channel does not disappear. It means that in this

case the electron transport through the channel com-

bines the features of both diffusive and ballistic motion.

7. Summary

We have developed a model for the detailed investi-

gation of quantum transport in MOSFET devices. The

model employs the Wigner distribution function for-

malism allowing us to account for electron scattering

by impurities and phonons. Numerical simulation of a

cylindrical sub-0.1 lm MOSFET structure was per-

formed. I–V characteristics for different values of the

channel length were obtained. It is shown that the slope

of the I–V characteristic in the saturation regime rises

as the channel length decreases. This is due to the de-

crease of the p–n junction barrier potential.

Finally, we have demonstrated that the inclusion of a

collision term in numerical simulation is important for

low source–drain voltages. The calculations have shown

that the scattering leads to an increase of the electron

density in the channel and smoothes out the applied

voltage along the entire channel. The analysis of the

electron phase-space distribution in the channel has

shown that, in spite of scattering, electrons are able to

flow through the channel as a narrow stream although,

to a certain extent, the scattering is seen to wash out this

jet. Accordingly, features of both ballistic and diffusive

transport are simultaneously encountered.

Acknowledgements

This work has been supported by the Interuniversi-

taire Attractiepolen, Belgische Staat, Diensten van de

Eerste Minister, Wetenschappelijke, technische en

culturele Aangelegenheden; PHANTOMS Research

Network; F.W.O.-V. projects nos. G.0287.95, 9.0193.97

and W.O.G. WO.025.99N (Belgium).

References

[1] Nakajima Y, Takahashi Y, Horiguchi S, Iwadate K,

Namatsu H, Kurihara K, Tabe M. Appl Phys Lett 1994;

65:2833.

[2] Ono M, Saito M, Yoshitomi T, Fiegna C, Ohguro T, Iwai

H. IEEE Trans Electron Dev 1995;42:1822.

[3] Guo L, Krauss PR, Chou SY. Appl Phys Lett 1997;71:

1881.

[4] Woo JCS, Terril KW, Vasudev PK. IEEE Trans Electron

Dev 1990;ED-37:1999.

[5] Joachim HO, Yamaguchi Y, Ishikawa K, Inoue Y,

Nashimura T. IEEE Trans Electron Dev 1993;40:1812.

[6] Pidin S, Koyanagi M. Jpn J Appl Phys 1998;37:1264.

[7] Sekigawa T, Hayashi Y. Solid State Electron 1984;27:827.

[8] Pikus FG, Liharev KK. Appl Phys Lett 1997;71:3661.

[9] Pokatilov EP, Fomin VM, Balaban SN, Gladilin VN,

Klimin SN, Devreese JT, Magnus W, Schoenmaker W,

Collaert N, VanRossum M, De Meyer K. J Appl Phys

1999;85:6625.

[10] Frensley W. Rev Mod Phys 1990;62:745.

[11] Tsuchiya H, Ogawa M, Miyoshi T. Jpn J Appl Phys 1991;

30:3853.

[12] Jensen KL, Ganguly AK. J Appl Phys 1993;73:4409.

[13] Isihara A. Statistical physics. State University of New York,

Buffalo, New York: Academic Press; 1971.

[14] Abramowitz A, Stegun LA, editors. Handbook of Mathe-

matical Functions with Formulas, Graphs, and Mathemat-

ical Tables, National Bureau of Standards, Washington,

DC, 1972.

[15] Hellwege KH, Landolt-B€oornstein, editors. Semiconduc-

tors. Physics of II–VI and I–VII Compounds, Semimag-

netic Semiconductors, vol. 17, New Series, Group III,

Berlin: Springer; 1982.


Quantum transport in a cylindrical sub-0.1 μm silicon-based MOSFET

Documents

Transcript of Quantum transport in a cylindrical sub-0.1 μm silicon-based MOSFET