Coulomb interaction and valley-orbit coupling in Si quantum dots

Luyao Jiang,1 C. H. Yang,2 Zhaodi Pan,1 Alessandro Rossi,2 Andrew S. Dzurak,2 and Dimitrie Culcer3, 1

1ICQD, Hefei National Laboratory for Physical Sciences at the Microscale,University of Science and Technology of China, Hefei 230026, Anhui, China

2ARC Centre of Excellence for Quantum Computation and Communication Technology,School of Electrical Engineering & Telecommunications,

The University of New South Wales, Sydney 2052, Australia3School of Physics, The University of New South Wales, Sydney 2052, Australia

The valley-orbit coupling in a few-electron Si quantum dot is expected to be a function of itsoccupation number N . We study the spectrum of multivalley Si quantum dots for 2 ≤ N ≤ 4,showing that, counterintuitively, electron-electron interaction effects on the valley-orbit couplingare negligible. For N = 2 they are suppressed by valley interference, for N = 3 they vanish due tospinor overlaps, and for N = 4 they cancel between different pairs of electrons. To corroborate ourtheoretical findings, we examine the experimental energy spectrum of a few-electron metal-oxide-semiconductor quantum dot. The measured spin-valley state filling sequence in a magnetic fieldreveals that the valley-orbit coupling is definitively unaffected by the occupation number.

I. INTRODUCTION

The requirements of quantum bit (qubit) scalabilityand long coherence times have brought solid-state spinsystems to the fore of quantum computation1–5. Re-alizing spin qubits in Si quantum dots is particularlypromising thanks to the remarkable coherence proper-ties of Si6–11, and significant experimental progress haslately been reported12–15. The valley degree of freedomof Si16 is vital in quantum computing17–35: it enablesvalley-based information processing22,36 and resonance23,but hampers spin qubits24. A valley-orbit coupling(VOC) causes valley states to hybridize into valley eigen-states25–33, and the biggest issues at present are an un-derstanding of the magnitude of the VOC, its responseto applied fields, and its sensitivity to interactions withthe environment and among electrons.

The VOC of singly occupied quantum dots stems fromthe interface potential33, yet for few-electron quantumdots, interactions contribute to the VOC. Since Coulombrepulsion depends on the spatial separation of electrons,it is often appreciable in quantum dots, where electronsare strongly confined. The on-site interaction u can ex-ceed the confinement energy, and in Si double quantumdots repulsion contributes sizably to interdot tunneling24.The relative contribution of interactions to the VOC offew-electron quantum dots is an issue of substantial con-ceptual and practical significance.

This paper is a theoretical and experimental study ofinteraction effects on the VOC in a single Si quantumdot with 1 ≤ N ≤ 4. For N = 2, we show that theCoulomb interaction effect on the VOC is suppressed byvalley interference, while for N = 3 and N = 4 interac-tion terms vanish due to spinor overlaps. Experimentaldata supports this finding, showing no evidence of inter-actions in the valley-orbital spectrum of Si quantum dotswith 1 ≤ N ≤ 3. We argue that these observations applybeyond N = 4.

The outline of this paper is as follows. In Sec. II weintroduce the model of the quantum dot, and discuss the

spectrum for 1 ≤ N ≤ 4. In Sec. III we compare ourtheoretical findings with experimental data. Sec. IV isdevoted to a discussion of the underlying physics, and isfollowed by a summary and conclusions.

II. MODEL

We consider a dot located at the origin with Fock-Darwin radius a. The confinement potential

VD(x, y, z) =~

2m∗a2(x2 + y2

a2) + U0θ(−z) + eFz, (1)

with the electron charge −e, in-plane effective mass m∗,interface potential Vz = U0θ(−z), θ the Heaviside func-tion, and gate electric field F . The electron wave func-tions Dξ(x, y, z) = φD(x, y)ψ(z)uξ(r)eikξz, the valley in-dex ξ = {z,−z}, and kz,−z = ±k0 = ±0.85(2π/aSi),

with aSi = 5.43A the Si lattice constant. The en-velopes φD(x, y) = 1

a√πe−(x

2+y2)/2a2 , while ψ(z) =

Nz0ekbz/2θ(−z) +N(z + z0)e−kSiz/2θ(z) is a variational

envelope function, with kSi a variational parameter, kbfixed, and N the normalization22. Given the twofoldspin and valley degrees of freedom, the lowest single-particle levels can accommodate up to 4 electrons [seeFig. 1 (a)]. For N ≤ 4 the number of many-electronstates is the combination C4

N . The case N = 1 has beenstudied at length, and the bare (single-particle) VOCis ∆0 ≡ |∆0| e−iφ0 = 〈Dz|U0 θ(−z) + eFz|D−z〉. Thevalley-orbit splitting is 2|∆0|, and the valley eigenstates

are D± = (1/√

2) (Dz ± eiφ0D−z)22.

A. Two-electron case

For N = 2 the Hamiltonian H2e = T (1) +V(1)D +T (2) +

V(2)D +Vee where superscripts i ∈ {1, 2} label electrons at

r1 and r2, T is the kinetic energy, and Vee = e2

4πε0εr|r1−r2|

arX

iv:1

308.

2728

v1 [

cond

-mat

.mes

-hal

l] 1

3 A

ug 2

013

2

D+

D-

EZ

2|Δ0|

2DEG

EF

ISET

ST

2DEG QD

SET

a)

b)

QD

Figure 1: (a) Energy diagram of the four lower spin-valleysingle-particle levels in the quantum dot. The Zeeman split-ting is assumed to be smaller than the valley-orbit coupling. Atunnel barrier separates the dot from a 2-dimensional electrongas (2DEG), acting as a reservoir. (b) Device architectureused for the experiments. A single-lead quantum dot (top)is capacitively coupled to a single-electron transistor chargesensor (bottom). Regions filled in orange represent electronlayers, non-filled regions are tunnel barriers. The readoutcurrent signal ISET is sensitive to the dot’s charge state viacapacitive effects.

is the Coulomb interaction, with ε0 the permittivity andεr the relative permittivity. The C4

2 = 6 two-particlestates are 3 spin singlets (φS) and 3 spin triplets (φT ),

φS±z,±z = D(1)±zD

(2)±zχS

φS,Tmix =1√2

(D(1)z D

(2)−z ±D(2)

z D(1)−z)χS,T ,

(2)

with χS and χT spin the singlet and triplet spin wavefunctions, given by

|χS〉 =1√2

(| ↑(1)↓(2)〉 − | ↓(1)↑(2)〉)

|χT,↑↑〉 = | ↑(1)↑(2)〉

|χT,↓↓〉 = | ↓(1)↓(2)〉

|χT,0〉 =1√2

(| ↑(1)↓(2)〉+ | ↓(1)↑(2)〉).

(3)

We use |χT 〉 generically for any of the three triplet wavefunctions.

In the basis {φSzz, φS−z−z, φSmix, φTmix}, the two-electronHamiltonian can be written as H2e = 2(ε0 + u)11 +

Hval, with ε0 = 〈D(i)ξ1|[T (i) + VD]|D(i)

ξ1〉, u =

〈D(i)ξ1D

(j)ξ2|V (i)(j)ee |D(j)

ξ2D

(i)ξ1〉, and

Hval =

0 0 ∆0

√2 0

0 0 ∆∗0√

2 0

∆∗0√

2 ∆0

√2 ∆ee 0

0 0 0 −∆ee

, (4)

with ∆ee = 〈D(1)z D

(2)−z |Vee|D

(2)z D

(1)−z〉. The eigenvalues of

Hval are 0, −∆ee and ∆± = ∆ee±√

∆2ee + 16|∆0|2, plot-

ted in Fig. 2 as functions of ∆ee. We note the crossoverof eigenvalues in Fig. 2. At large ∆0 � ∆ee, the groundstate is a spin singlet, yet when ∆0 is small enough, theground state corresponds to the eigenvalue −∆ee, and isa spin triplet. In such a case the ground state will movedown in energy in a non-zero magnetic field.

To assess the possibility of a triplet ground state, we

evaluate ∆ee. We expand uz(r) =∑K

czKeiK·r, with K

reciprocal lattice vectors, switch to center of mass andrelative variables r = r1 − r2,R = r1 + r2, and employcylindrical polar coordinates. The integration over R istrivial, while over r = (r⊥, φ, z) we make the substitution1r = 2√

π

∫∞0dte−r

2t2 , reducing the problem to

∆ee =e2Σ

2ε0εr√π

∫ ∞0

du cos(2k0u)eu2

2a2 Erfc

(u

a√

2

)L(u),

(5)

with Σ =∑

K1K2Q1Q2

cz∗K1c−z∗K2

c−zK1−Q1czK2−Q2

, Erfc the

complementary error function and

3

L(u) =2

12πN4z40kba

e−kbu +2√

2πN4z20δ3ka

e−kbu[eδkuδ2k(u+ z0)2 − 2eδkuδk(u+ z0) + δkz0(2− δkz0)− 2]+

N4π√2k5Sia

e−kSiu[2k4Siz20(u+ z0)2 + 2k3Siz0(u+ z0)(u+ 2z0) + k2Si(u

2 + 6uz0 + 6z20) + 3kSi(u+ 2z0) + 3]

(6)

Figure 2: Eigenvalues of Hvalley as functions of ∆ee for aquantum dot with N = 2 electrons, with ∆0 = 0.05 meV. Atlarge ∆ee the ground state is a spin triplet (green line).

Using a = 10nm as the lowest realistic quantum dot ra-dius, the effective εr = 7.9 for a sample Si/SiO2 inter-face, the parameters of Ref. 22 for the variational wave

function and the coefficients cξK of Ref. 37, we obtainnumerically ∆ee = 0.0352µeV . The Coulomb interac-tion contribution to the VOC is negligible compared tothe single-particle term ∆0, which is of the order of 0.05meV28.

B. Three-electron case

We consider next N = 3. The quantum dot can bein one of C4

3 = 4 states. Using | ↑〉, | ↓〉 for up anddown spinors respectively, one sample wave function isthe Slater determinant

φ3e1 (z ↑,−z ↑, z ↓) =1√6

∣∣∣∣∣∣∣D

(1)z↑ D

(1)−z↑ D

(1)z↓

D(2)z↑ D

(2)−z↑ D

(2)z↓

D(3)z↑ D

(3)−z↑ D

(3)z↓

∣∣∣∣∣∣∣ , (7)

and analogously φ3e2 (z ↑,−z ↓, z ↓), φ3e3 (z ↓,−z ↑,−z ↓),and φ3e4 (z ↑,−z ↑,−z ↓). The 3-electron Hamiltonian

H3e =∑3i=1 T

(i) + V(i)D + 1

2

∑3i6=j V

(ij)ee .

Each basis state contains 6 terms, and each Hamilto-nian matrix element has 36 terms. Spinor overlaps an-nihilate most of them, since the Coulomb interaction isspin-independent. We calculate 〈φ3e1 |H3e|φ3e1 〉 as an ex-

ample. The six terms in φ3e1 are φ11 = D(1)z↑ D

(2)−z↑D

(3)z↓ ,

φ12 = −D(1)z↑ D

(2)z↓ D

(3)−z↑, φ13 = D

(1)z↓ D

(2)z↑ D

(3)−z↑, φ14 =

−D(1)z↓ D

(2)−z↑D

(3)z↑ , φ15 = D

(1)−z↑D

(2)z↓ D

(3)z↑ , and φ16 =

−D(1)−z↑D

(2)z↑ D

(3)z↓ . We obtain

〈φ11|H3e|φ11〉 = Σ3i=1〈D

(1)z↑ D

(2)−z↑D

(3)z↓ |T

(i)|D(1)z↑ D

(2)−z↑D

(3)z↓ 〉

+ 3〈D(1)z↑ D

(2)−z−z↑D

(3)z↓ |Vee|D

(1)z↑ D

(2)−z↑D

(3)z↓ 〉

= 3ε0 + 3u

〈φ11|H3e|φ16〉 = −〈D(1)z↑ D

(2)−z↑D

(3)z↓ |H3e|D(1)

−z↑D(2)z↑ D

(3)z↓ 〉

= −∆ee,(8)

and 〈φ11|H3e|φ12〉 = 〈φ11|H3e|φ12〉 = 〈φ11|H3e|φ13〉 =〈φ11|H3e|φ14〉 = 〈φ11|H3e|φ15〉 = 0. Adding up theseterms gives 〈φ3e1 |H3e|φ3e1 〉 = 3ε0 + 3u − ∆ee, and sim-ilarly 〈φ3e1 |H3e|φ3e4 〉 = −∆0. The matrix elements〈φ3e1 |H3e|φ3e2 〉 and 〈φ3e1 |H3e|φ3e3 〉 vanish because no termsin the bra and ket states share the same spin arrange-ment. Analogous arguments apply to all the remainingmatrix elements, yielding

H3e = (3ε0 + 3u−∆ee)11 +

0 0 0 −∆0

0 0 ∆0 00 ∆∗0 0 0−∆∗0 0 0 0

(9)

The finite matrix elements of H3e are 〈φ3en |H3e|φ3en 〉 =3εD + 3u − ∆ee (for 1 < n < 4) and 〈φ3e2 |H3e|φ3e3 〉 =−〈φ3e1 |H3e|φ3e4 〉 = ∆0. The eigenvalues are 3ε0 + 3u −∆ee ± |∆0|. As a result of spinor overlaps, the Coulombterm ∆ee does not appear in the off-diagonal matrix el-ements and does not contribute to the valley splitting,giving only an identical offset to all the energy eigenval-ues.

C. Four-electron case

For N = 4, all four lowest-energy single-particle statesare occupied, and only C4

4 = 1 many-particle state exists.The Slater determinant wave function φ4e for N = 4takes the form

φ4e =1√24

∣∣∣∣∣∣∣∣∣∣D

(1)z↑ D

(1)−z↑ D

(1)z↓ D

(1)−z↑

D(2)z↑ D

(2)−z↑ D

(2)z↓ D

(2)−z↑

D(3)z↑ D

(3)−z↑ D

(3)z↓ D

(2)−z↑

D(4)z↑ D

(4)−z↑ D

(4)z↓ D

(4)−z↑

∣∣∣∣∣∣∣∣∣∣. (10)

4

0 2 4 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

B (T)

E(m

eV)

N=0

N=1

N=2

N=3

0 2 4 6B (T)

Energy

D-

D+

a)

c)

b)

D- D+

D-

D-D+

D-D+

D-D+

S T↓↓

2|Δ0|

Bk

Figure 3: (a) Energy diagrams of the dot when voltage pulsesare superimposed to the DC bias. When a single-particlelevel is near the reservoir’s Fermi level, an electron can beloaded into the dot during the upper phase of the pulse (solidlines) and unloaded during the lower phase (dashed lines). (b)Magnetic field dependence of the spin-valley states filling forN ≤ 3, where D± label single-electron valley eigenstates andS, T↓↓ label spin states, consistent with Eq. 2. Data points

show where the differential sensor’s signal dISETdE

is maxi-mized. Solid lines are guides for the eye with slopes + gµB

2in red and − gµB

2in green. Boxes of different colors indicate

different valley occupancies. Blue (black) arrows represent thespin state of the current (previous) electron addition(s). Setsof data for different charge transitions are arbitrarily shiftedin energy for clarity. (c) Energy diagram of the 1-electronspin-valley states evolution in magnetic field. At the fieldvalue B = Bk the two valley states cross, and Zeeman andvalley-orbit splittings coincide. Dashed lines indicate statesthat are not probed in the experiments.

The energy 〈φ4e|H4e|φ4e〉 = 4ε0 + 6u contains no inter-valley terms, stemming from either the bare VOC or theCoulomb interaction, since interaction terms cancel be-tween pairs of electrons with different spinors.

III. EXPERIMENTAL SPIN-VALLEYSPECTRUM

To experimentally probe the effect of Coulomb interac-tions on the VOC, we use quantum dots fabricated withmetal-oxide-semiconductor (MOS) technology in nearlyintrinsic Si38. In these devices a three-layer Al-Al2O3-Al gate stack39 allows one to electrostatically modulatethe conduction band profile near the Si/SiO2 interface.As a result, an electron accumulation layer is selectivelyformed in the semiconductor substrate and can functionas either a quantum dot or a 2DEG reservoir, depend-

ing on the extent of planar confinement provided bythe gates. The details of the fabrication process andtypical bias configurations are reported elsewhere28,40.A schematic diagram of the device used is shown inFig. 1(b). The system works as a single-lead quantum dotcapacitively coupled to a single-electron-transistor (SET)acting as a charge sensor. Independent gate electrodescontrol the occupancy of the quantum dot, the tunnellingrate between the dot and the reservoir, the Fermi energyof the reservoir, and the bias point of the SET detector.The dot is reliably operated in the few-electron regimedown to the last electron40.

The excitation spectrum of the quantum dot is mea-sured using a pulsed voltage technique41. A train ofsquare voltage pulses at a frequency of few hundreds Hzis applied to the gate that directly affects the quantumdot potential (in addition to its DC voltage). This shiftsthe energy levels of the dot in time, inducing charge tran-sitions whenever the dot’s single-particle levels come intoresonance with the lead’s Fermi energy. An electron canbe loaded from the reservoir into the dot during the upperphase of the pulse and unloaded during the lower phase(as shown in Fig. 3 (a)). The time-dependent gate volt-age modulates the sensor current, ISET , via capacitiveeffects, and a lock-in amplifier selects the spectral com-ponent which is uniquely associated to tunneling events(i.e. the one at the frequency of the pulse). This tech-nique allows one to probe both ground and excited statesfor each N , as long as the relaxation rate is slower thanthe pulse frequency and the pulse magnitude spans therelevant energy separation. The measured spectrum re-veals that the energy separation between the first excitedorbital state and the ground state varies between 1-8meVaccording to dot’s occupancy. A comprehensive descrip-tion of the measurement set-up and parameters is foundin Ref. 40.

In Fig. 3(b) measurements of the spin-valley states’evolution in a magnetic field B ‖ [110] (parallel to theSi/SiO2 interface) are shown for 1 ≤ N ≤ 3. Each datapoint indicates a maximum in the differential sensing cur-rent signal for varying DC gate voltage bias and a fixedpulse amplitude of 20 mV. The energy scale on the y-axisis obtained by converting the gate voltage into electro-chemical potential using the appropriate lever arm con-version factor (≈ 0.3 eV/V). The vertical shift of eachtrace is arbitrary and does not reflect the quantum dotcharging energy. A strong sensing signal is due to chargetransfers to/from the quantum dot. Hence, the observedmagnetic field dependence maps the filling of spin-valleystates for individual electrons. The slope of E(B) is givenby42: δE

δB = −gµBB∆Stot(N), where g is the electron gy-romagnetic ratio, µB is the Bohr magneton and ∆Stot(N)is the change in total spin when the N -th electron isadded to the dot. Hence, a slope of + gµB

2 indicates aspin-up addition, while a − gµB2 slope is due to a spin-down electron. Solid lines shown in Fig. 3 (b) are guidesfor the eye with slopes ± gµB2 , using g = 2 for bulk Si.

To understand the interplay between spin and valley

5

degrees of freedom in a magnetic field, we analyse indi-vidual charge additions. The first transition (N = 0→ 1)reveals that for low magnetic fields the slope is positive upto B = Bk ≈ 2T , where a change of sign occurs; hence, aspin-up (spin-down) electron is loaded into the dot for low(high) B-fields. For the first electron addition, one wouldexpect to fill the spin-down lower valley eigenstate (theground state) at all B-field values (see red dashed linein Fig. 3(c)). However, the measurements indicate thatthe excited (ground) state of the lower D− (upper D+)valley is probed for this transition for B < Bk (B > Bk).This occurs because the tunnelling rate between the dotand the lead in our measurement is faster than the relax-ation rate, so that the largest contribution to the sensingsignal comes from the loading/unloading of the excitedstates. Using Fig. 3(c) we can explain the presence ofthe kink (change of slope sign). For increasing magneticfields the states |D↑−〉 and |D↓+〉 cross, making it en-ergetically favourable for a spin-down electron to chargethe dot. The position of the kink reveals the valley-orbitsplitting, which coincides with the Zeeman splitting, asgµBBk = 0.23 meV, yielding |∆0| = 0.115 meV.

The trend in Fig. 3 (b) for the N = 1 → 2 transitionis similar to that for N = 0 → 1. The ground state forN = 1 is spin-down for all B. The spin-filling measure-ments reveal that for low B a spin-up electron is addedto form a singlet which fully occupies the − valley eigen-state. However, at larger B, a kink occurs at the same Bkas for N = 1. This suggests that for B > Bk a spin-down

electron is added to form the triplet |φT↓↓mix〉. The antisym-metry of the 2-electron wave-function requires one elec-tron to occupy an excited state; here, this is the + valleyeigenstate, which is significantly lower than the first ex-cited orbital state40. The observed change of slope is dueto a singlet-triplet crossing and, once again, the Zeemanand valley-orbit splittings coincide at B = Bk.

For N = 2 → 3, we again observe a change of slopeat B = Bk, but unlike the previous cases, the electronadded at low (high) fields is spin-down (spin-up). Tounderstand this, we note that for B < Bk the 2-electronground state is a spin singlet that fully occupies the lowervalley eigenstate. The next available state would be aspin-down in the upper valley eigenstate. However, for

B > Bk the 2-electron ground state switches to |φT↓↓mix〉,which would allow one more electron in the lower valleyeigenstate. This can only be spin-up because of Pauli ex-clusion. The change of slope at the same field value as forthe previous transitions confirms that |∆0| = 0.115 meVfor N = 3 as well. These measurements confirm that theVOC is independent of N up to N = 3, corroboratingthe prediction that ∆ee is negligible.

IV. DISCUSSION

The underlying physics of this problem is contained inthe two-electron case. We have seen that the ground statefor N = 2 is dependent on the magnitude of the quan-

tity ∆ee, which can also be understood as representingintervalley exchange. It is worth noting that ∆ee is theonly exchange parameter appearing in this problem. Thecentral message of this work is that there is a negligiblegain in Coulomb energy for an electron to switch valleystates.

Fundamentally, ∆ee is small because the Coulomb in-teraction varies on longer spatial scales than that set bythe separation of the valleys in reciprocal space, whichcorresponds to ≈ 1A in real space. In other words, ∆ee

is suppressed by valley interference. Interaction effectsare not expected to be observable even if ∆0 = 0, since∆ee corresponds to a few mK, while typical temperaturesin a dilution refrigerator are ≈100mK. For any realistic Sitwo-dimensional electron gas (2DEG), the ground stateof a multivalley two-electron single quantum dot is a spinsinglet28,29.

For N = 3 and N = 4 we have no new matrix elements.We note, however, that the eigenvalues of the N = 3case are doubly degenerate. Operating such a quantumdot as a qubit presents the same scalability issues as inRef. 24. Briefly, although single qubit operations can beperformed regardless of the state in which the systemis initialized, two-qubit operations typically depend onexchange, which can be vastly different for different valleyconfigurations.

V. SUMMARY

In conclusion, we have demonstrated theoreticallyand experimentally that in few-electron quantum dotsthe VOC has no significant contribution from electron-electron interactions. We expect these findings to be gen-eralizable to N > 4, since valley interference and spinoroverlaps are equally relevant to higher orbital excitedstates. Conceptually, this proves that interactions arenot sharp enough in real space to couple valleys, whichhave a large separation in k-space. Together with previ-ous research33, it shows that the VOC is fully determinedby the properties of the interface: the size and shape ofthe interface potential step, electric field and roughnessprofile.

These findings will aid experimentalists working onquantum computing in Si and other materials with val-leys, such as C and Ge. The valley degree of freedom canbe a significant impediment to single-spin and singlet-triplet qubits24, and it is essential for their operation thatthe VOC be fully characterised, since valleys complicatethe spin state spectrum on a fundamental level. Thesize of the VOC and its sensitivity to interactions andoccupation number is also a critical ingredient of valleyqubits22, which may reduce sensitivity to noise.

6

Acknowledgments

We thank S. Das Sarma, Xuedong Hu and R. J. Joyntfor enlightening discussions. L.J., Z.P. and D.C. acknowl-edge support by the National Natural Science Foun-dation of China under grant number 91021019 and by

LPS-NSA-CMTC. C.H.Y., A.R. and A.S.D. acknowledgesupport from the Australian Research Council (projectCE11E0096) and the U.S. Army Research Office (con-tract W911NF-13-1-0024). Experimental devices for thisstudy were fabricated with support from the AustralianNational Fabrication Facility, UNSW.

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