Stroboscopic model of transport through a quantum dot with spin-orbit scattering

Stroboscopic model of transport through a quantum dot with spin-orbit

scattering

Bardarson, J.H.; Tworzydlo, J.; Beenakker, C.W.J.

Citation

Bardarson, J. H., Tworzydlo, J., & Beenakker, C. W. J. (2005). Stroboscopic model of transport

through a quantum dot with spin-orbit scattering. Retrieved from

https://hdl.handle.net/1887/4890

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Stroboscopic model of transport through a quantum dot with spin-orbit scattering

J. H. Bardarson,1J. Tworzydło,2and C. W. J. Beenakker1

1_{Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands} 2_{Institute of Theoretical Physics, Warsaw University, Hoża 69, 00-681 Warsaw, Poland}

共Received 23 August 2005; revised manuscript received 14 October 2005; published 5 December 2005兲

We present an open version of the symplectic kicked rotator as a stroboscopic model of electrical conduction through an open ballistic quantum dot with spin-orbit scattering. We demonstrate numerically and analytically that the model reproduces the universal weak localization and weak antilocalization peak in the magnetocon-ductance, as predicted by random-matrix theory共RMT兲. We also study the transition from weak localization to weak antilocalization with increasing strength of the spin-orbit scattering, and find agreement with RMT. DOI:10.1103/PhysRevB.72.235305 PACS number共s兲: 73.63.Kv, 05.45.Pq, 71.70.Ej, 73.20.Fz

I. INTRODUCTION

Electrical conduction in semiconductor heterostructures is affected by the spin degree of freedom through spin-orbit scattering. In quantum dots with chaotic scattering a statisti-cal approach is appropriate. The spin-orbit Hamiltonian共of either Rashba or Dresselhaus form兲 has a special structure, that of a non-Abelian vector potential. By a gauge transfor-mation Aleiner and Fal’ko identified all possible symmetry classes and described the crossovers between them by means of random-matrix theory 共RMT兲.1 _{This RMT has been} ex-tended by Brouwer et al. to the case that the spin-orbit scat-tering is nonuniform and thus the gauge transformation can-not be made.2,3_{Experiments are in good agreement with the} predictions of the theory.4,5 _{Recently a semiclassical theory} of quantum dots with spin-orbit scattering has been developed.6,7_{Exact quantum mechanical calculations of such} “Rashba billiards” have also been reported.8_{In this paper we} will focus on the regime of strong chaos, where RMT and semiclassics agree.

Here we present a fully quantum mechanical computer simulation to test the theory. In the case of spinless chaotic quantum dots, the stroboscopic model known as the quantum kicked rotator has been proven to be quite successful.9–15 This model exploits the fact that, although the phase space of the open quantum dot is four dimensional, the dynamics can be described, on time scales greater than the time of flight across the dot, as a mapping between points on a two-dimensional Poincaré surface of section. The kicked rotator gives a map on a two-dimensional phase space that has the same phenomenology as open quantum dots.

In this paper we extend the model of the open kicked rotator to include spin-orbit scattering in a perpendicular magnetic field. We begin by describing the known model for a closed chaotic quantum dot16 _{with spin-orbit scattering in} Sec. II A, before discussing the opening up of the model in Sec. II B. The relation of the model to RMT is given in Sec. III. This relation will give us a mapping between the model parameters and the microscopic parameters of a chaotic quantum dot. Numerical results for the weak 共anti兲localiza-tion peak and its dependence on magnetic field and spin-orbit scattering strength are presented in Sec. IV and compared with the analytical predictions from Sec. III.

II. DESCRIPTION OF THE MODEL A. Closed system

The symplectic kicked rotator has been introduced by Scharf16_{and studied extensively in Refs. 17–19. We} summa-rize this known model of the closed system before proceed-ing to the open system in the next subsection.

The symplectic kicked rotator describes an electron mov-ing along a circle with moment of inertia I0, kicked periodi-cally at time intervals ␶0 with a kicking strength that is a function of position and spin. We choose units such that

␶0⬅1 and ប⬅1. The Hamiltonian H is given by16,17

H =1 2共p + p0兲 2_{+ V共␪兲}

兺

n=−⬁ ⬁ ␦s共t − n兲, 共2.1a兲 V共␪兲 = K cos共␪+␪0兲 + KSO共␴1sin 2␪+␴3sin␪兲.

共2.1b兲 We have introduced the symmetrized delta function

␦s共t兲=关␦共t+⑀兲+␦共t−⑀兲兴/2, with ⑀ an infinitesimal. The

di-mensionless angular momentum operator p = −iបeff⳵/⳵␪, with បeff=ប␶₀/ I₀the effective Planck constant, is canonically con-jugate to the angle ␪苸关0,2␲兲. The kicking potential V共␪兲 contains the Pauli spin matrices

␴1=

冉

0 1 1 0

冊

, ␴2=

冉

0 − i i 0

冊

, ␴3=

冉

1 0 0 − 1

冊

. 共2.2兲 Potential scattering is parameterized by the kicking strength

K and spin-orbit scattering by KSO. We choose smoothly varying functions of␪, corresponding to a smooth potential. Disorder can be added via a rapidly varying function of␪, see Ref. 20.

Spin rotation symmetry is broken if KSO⫽0. The gener-alized time-reversal symmetry16

T:␪哫 −␪, p哫 p, ␴i哫 −␴i, t哫 − t, 共2.3兲

is preserved if␪0= 0 and is broken if␪0苸共0,␲兲. A nonzero

Notice that the roles of p and ␪ are interchanged in T compared to the conventional time-reversal symmetry of the Rashba Hamiltonian and the spinless kicked rotator, which reads

T

⬘

:␪哫␪, p哫 − p, ␴i哫 −␴i, t哫 − t. 共2.4兲

For this reason time-reversal symmetry in the symplectic kicked rotator is broken by a displacement of␪, rather than by a displacement of p as in the spinless kicked rotator.21

The stroboscopic time evolution of a wave function is governed by the Floquet operator

F = T exp

冋

− i បeff

冕

t₀

t₀+1

H共t兲dt

册

, 共2.5兲 where T denotes time ordering of the exponential. In the range 关−1/2,1/2兲 only t0= 0 and t0= −1 / 2 preserve

T-symmetry for␪0= 0. We will find it convenient to choose

t0= −1 / 2 for numerical calculations and t0= 0 for analytical work.

For p0= 0 the reduction of the Floquet operator to a discrete finite form is obtained for special values of បeff, known as resonances.21 _{For បeff= 4␲/ M, with M an} integer, the Floquet operator is represented by an M⫻M matrix of quaternions. 共A quaternion is a 2⫻2 matrix

q = q01 + iq1␴1+ iq2␴2+ iq3␴3with qia complex number and1

the 2⫻2 unit matrix.兲 For this value of បeffthe momentum is restricted to p苸关0,4␲兲, i.e., one can think of the Floquet operator as describing a map on a torus. For t0= −1 / 2 the matrix elements in the p representation are given by

Fll⬘=共⌸UXU†⌸兲ll⬘, l,l

⬘

= 0,1, . . . , M − 1, 共2.6a兲 ⌸ll⬘=␦ll⬘e−i␲l 2 /M_{1, 共2.6b兲} Ull⬘= M−1/2e−i2␲ll⬘/M1, 共2.6c兲 Xll⬘=␦ll⬘e−i共M/4␲兲V共2␲l/M兲. 共2.6d兲

For t0= 0 one has instead

F = UX1/2_U†_⌸2_UX1/2_U†_{. 共2.7兲} These maps共2.6兲 and 共2.7兲 are equivalent to the Hamil-tonian共2.1兲 with p0= 0. A nonzero p₀may be introduced into the map by replacing⌸ with21

⌸ll⬘=␦ll⬘e−i␲共l + l0兲

2_/M

1, l0=

p0M

4␲ . 共2.8兲

This map is not rigorously equivalent to the Hamiltonian 共2.1兲, but it has the same classical limit 共for KSO= 0兲.13

The generalized time-reversal symmetry 共2.3兲 is ex-pressed by the identity

F = FR_{, if␪}

0= 0. 共2.9兲

The superscript R denotes the dual of a quaternionic matrix

FR_⬅␴

2FT␴2.

Here T denotes the transpose in the basis of eigenstates of p 共p representation兲. To verify Eq. 共2.9兲 note that

␴2␴i T_␴

2= −␴iand that the transpose in p representation takes

␪to −␪.

B. Open system

To describe electrical conduction we open up the kicked rotator, following the general scheme of Refs. 9–12. We model a pair of N-mode ballistic point contacts that couple the quantum dot to electron reservoirs, by imposing open boundary conditions in a subspace of Hilbert space repre-sented by the indices l_k共␮兲. The subscript k = 1 , 2 , . . . , N, with

N = N1+ N2, labels the modes 共both spin directions兲, and the superscript ␮= 1 , 2 labels the point contacts. The N⫻M quaternionic projection matrix P is given by

Pkk⬘=

再

1 if k

⬘

= lk共␮兲,

0 otherwise.

冎

共2.10兲

The matrices P andF together determine the scattering matrix

S共␧兲 = P共e−i␧−FQT_Q兲−1_FPT_{, 共2.11兲}

where ␧苸关0,2␲兲 is the quasi-energy and QT_{Q = 1 − P}T_P.

One readily verifies that S is unitary.

We need to ensure that the introduction of the point con-tacts does not break theT-symmetry

S共␧兲 = SR共␧兲 if␪0= 0 共2.12兲 or for nonzero␪₀the more general duality relation

S共␪0兲 = SR_{共−␪0兲. 共2.13兲}

This is assured by choosing the absorbing boundary condi-tions in a strip parallel to the␪ axis, rather than parallel to the p axis as in the spinless kicked rotator共see Fig. 1兲. The difference is due to the exchange of the roles of p and␪ in the time-reversal symmetry operation, compare Eqs. 共2.3兲 and共2.4兲.

By grouping together the N_␮ indices belonging to the same point contact, the N⫻N quaternionic matrix S can be decomposed into 4 sub-blocks containing the quaternionic transmission and reflection matrices

FIG. 1. Location of the absorbing boundary conditions 共grey rectangles兲 in the classical phase space of the open kicked rotator. To ensure that the openings do not break the time reversal symme-try they are oriented parallel to the p axis in the spinless kicked rotator 共left panel兲 and parallel to the ␪ axis in the symplectic kicked rotator共right panel兲.

BARDARSON, TWORZYDŁO, AND BEENAKKER PHYSICAL REVIEW B 72, 235305共2005兲

S =

冉

r t

t

⬘

r

⬘

冊

. 共2.14兲

The value of␧ is arbitrary; we will take ␧=0 in the analytical calculations and average over␧ in the numerics. The T sym-metry 共2.12兲 requires that r=␴₂rT_␴

2, r

⬘

=␴2r

⬘

T␴2, and

t

⬘

=␴2tT␴2.

The conductance G follows from the Landauer formula

G =e

2

hTrtt

†_{, 共2.15兲}

where the trace Tr is over channel indices as well as spin indices. Unitarity of S ensures that Tr tt†= Tr t

⬘

t

⬘

†. For

␪0= 0 the eigenvalues of tt†are doubly degenerate due to the

T symmetry 共Kramers degeneracy兲. It will prove useful to

write the Landauer formula in the form2,3

G =2e 2 h N1N2 N − e2 hTrS⌳S †_{⌳ ⬅ G0+␦G, 共2.16兲} with⌳ a diagonal matrix having diagonal elements

⌳jj=

再

N2/N, j = 1, . . . ,N1, − N1/N, j = N1+ 1, . . . ,N.

冎

共2.17兲 The term G0=共2e2/ h兲N1N2/ N is the classical conductance and the term␦G, of order e2/ h, is the quantum correction from the weak localization effect.

III. RELATION TO RANDOM-MATRIX THEORY

Random-matrix theory共RMT兲 gives universal predictions for the quantum correction␦G in Eq. 共2.16兲. We calculate

this quantity for the symplectic kicked rotator and compare with RMT. This will give us the relation between the eters of the stroboscopic model and the microscopic param-eters of the quantum dot.

The three universality classes of RMT are labeled by

␤= 1 , 2 , 4, with22 ␦GRMT= ␤− 2 2␤ e2 h. 共3.1兲

In the absence ofT symmetry one has␤= 2. In the presence ofT symmetry one has␤= 1共4兲 in the presence 共absence兲 of spin rotation symmetry. We will investigate the three sym-metry breaking transitions ␤= 1→2, ␤= 1→4, and

␤= 4→2 in separate subsections.

A.␤=1\2 transition

The␤= 1→2 transition takes place in the absence of spin-orbit scattering共KSO= 0兲. This transition was studied in Ref. 13 for the case that the symmetryT

⬘

rather thanT is broken. To fully characterize the model we need to reconsider this transition for the case ofT-symmetry breaking.

For small␪₀, cos共␪+␪0兲⬇cos␪−␪₀sin␪ and the Floquet matrix共2.7兲 takes the form

F共KSO= 0,␪0→ 0兲 = e␪0WF0e␪0W, 共3.2a兲

W = UYU†_{, Y}

ll⬘=␦ll⬘i共KM/8␲兲sin共2␲l/M兲. 共3.2b兲

Here F0=F共KSO= 0 ,␪0= 0兲 is unitary symmetric and W is real antisymmetric. The scattering matrix共2.11兲 共at ␧=0兲 be-comes

S = T共1 − F0R兲−1F0T

⬘

, 共3.3a兲

T = Pe␪0W_, 共3.3b兲

T

⬘

= e␪0W_PT_, 共3.3c兲

R = e␪0W_QT_Qe␪0W_. 共3.3d兲

Substitution of S into Eq.共2.16兲 gives the conductance G. To make contact with RMT we assume thatF₀is a ran-dom matrix from the circular orthogonal ensemble 共COE兲, expand the expression for G in powers ofF0and averageF0 over the COE. In the regime 1ⰆNⰆM, we can perform the average over the unitary group with the help of the diagram-matic technique of Ref. 23. Since Tr⌳=0 only the maxi-mally crossed diagrams contribute to leading order in N. The result for the average quantum correction becomes

具␦G典 = −2e

2

h trT

†_⌳T共T

_⬘

_⌳T

_⬘

†_兲T 1

M − trR†RT. 共3.4兲

The factor of 2 comes from the spin degeneracy and the trace tr is over channel indices only. The two remaining traces are evaluated in the limit N , M→⬁ at fixed N/M. We find

M−1trT†⌳T共T

⬘

⌳T

⬘

†兲T=N1N2 N2 N M, 共3.5兲 M−1_trR†_RT_{= 1 − N/M −␪} 0 2_{共KM/4␲兲}2_{共1 − N/M兲. 共3.6兲} Substitution into Eq.共3.4兲 gives the average quantum correc-tion 具␦G典 = −e 2 h 2N1N2 N2 1 1 +共␪0/␪c兲2 , 共3.7a兲 ␪c= 4␲

冑

N KM3/2. 共3.7b兲

The RMT result has the same Lorentzian profile22,24

␦G_RMT= −e 2 h 2N1N2 N2 1 1 +共B/Bc兲2 , 共3.8a兲 Bc= C h eL2

冉

NL⌬ បvF

冊

1/2 , 共3.8b兲

with C a numerical constant of order unity, L =

冑

A the size of

of the Kramers degenerate levels, andvF the Fermi velocity.

Comparison of Eqs.共3.7兲 and 共3.8兲 allows us to identify

␪0/␪c= B/Bc. 共3.9兲

B.␤=1\4 transition

The ␤= 1→4 transition is realized by turning on spin-orbit scattering 共KSO兲 in the absence of a magnetic field 共␪0= 0兲. In this transition the quaternionic structure of the Floquet matrix plays a role. The Floquet matrix共2.7兲 has the form

F共KSO,␪0= 0兲 = eKSOAF0eKSOA, 共3.10a兲

A = U共␴1Y1+␴3Y3兲U†, 共3.10b兲 共Y1兲ll⬘= −␦ll⬘i共M/8␲兲sin共4␲l/M兲, 共3.10c兲

共Y3兲ll⬘= −␦ll⬘i共M/8␲兲sin共2␲l/M兲. 共3.10d兲

The matrix A is real antisymmetric and thus A*_{= −A, where} the asterisk denotes quaternion complex conjugation. 共The complex conjugate of a quaternion q is defined as

q*_{= q} 0 *_{1 + iq}

1 *_␴

1+ iq2*␴2+ iq3*␴3.兲 The scattering matrix takes the same form共3.3a兲, but now with

T = PeKSOA_, 共3.11a兲

T

⬘

= eKSOA_PT_, 共3.11b兲

R = eKSOA_QT_QeKSOA_. 共3.11c兲

The average ofF0over the ensemble of unitary symmet-ric matsymmet-rices only involves the channel indices and not the spin indices. To keep the quaternions in the correct order we adopt the tensor product notation of Brouwer et al.2,3 _The average of␦G overF0gives, to leading order in N,

具␦G典 =e

2

h

兺

_␮␯

冋

␶

trE丢E

⬘

*

M1丢1 − trR丢R*␶

册

_{␮␯;␮␯}, 共3.12兲

where ␶= 1丢␴2, E = T†⌳T, and E

⬘

= T

⬘

⌳T

⬘

†. The tensor product has a backward multiplication in the second argu-ment

共a丢b兲共c丢d兲 ⬅ ac丢db, 共3.13兲

and the indices␮and␯are the spin indices.

The two traces are calculated in the limit KSO→0, N,M

→⬁ at fixed N/M, leading to M−1trE丢E

⬘

*=N1N2 N2 N M1丢1, 共3.14a兲 M−1trR丢R*=共1 − N/M兲关1 − 4K_SO2 共M/8␲兲2兴1丢1 + 2K_SO2 共M/8␲兲2共1 − N/M兲 ⫻共␴1丢␴1+␴3丢␴3兲. 共3.14b兲 After substitution into Eq. 共3.12兲 there remains a matrix structure that can be inverted, resulting in

具␦G典 =e 2 h N1N2 N2

冉

1 − 2 1 + 2a2− 1 1 + 4a2

冊

, 共3.15a兲 a = KSO/Kc, Kc= 4␲

冑

2N M3/2 . 共3.15b兲

The RMT result has the same functional form,2 _with

a =共2␲បN/␶SO⌬兲1/2. Here ␶SO is the spin-orbit scattering time. Thus we identify

KSO/Kc=共2␲បN/␶SO⌬兲1/2. 共3.16兲

C.␤=4\2 transition

In the presence of strong spin-orbit scattering共KSOⰇKc兲

the Floquet matrix takes for small␪₀the same form as in Eq. 共3.2a兲 for KSO= 0, but nowF0=F共KSOⰇKc,␪0= 0兲 is a uni-tary self-dual matrix rather than a uniuni-tary symmetric matrix. We can repeat exactly the same steps as we did for KSO= 0 but withF0 a random matrix in the circular symplectic en-semble共CSE兲. We then average F0over the CSE. This leads to 具␦G典 =e 2 h N₁N₂ N2 1 1 +共␪₀/␪c兲2 , 共3.17兲

with ␪c as in Eq. 共3.7b兲. The width of the Lorentzian is

therefore the same in the␤= 1→2 and␤= 4→2 transitions, in agreement with RMT.22

IV. NUMERICAL RESULTS

The numerical technique we use is the same as has been used before for the spinless kicked rotator.12,13_A combina-tion of an iterative procedure for matrix inversion and the fast-Fourier-transform algorithm allows for an efficient cal-culation of the scattering matrix from the Floquet matrix.

The average conductance 具G典 was calculated with the Landauer formula共2.15兲 by averaging over 60 different uni-formly distributed quasi-energies and 40 randomly chosen lead positions. The quantum correction具␦G典 is obtained by

subtracting the classical conductance G₀. The numerical data is shown in Figs. 2 and 3. The magnetic field parameter␪0is given in units of␪cfrom Eq.共3.7b兲 and the spin-orbit

scat-tering strength parameter KSO is given in units of Kc from

Eq. 共3.15b兲. The solid lines are the analytical predictions 共3.7兲, 共3.15兲, and 共3.17兲 without any fitting parameter.

The small difference between the data and the predictions can be attributed to an uncertainty in the value G0 of the classical conductance. A small vertical offset共corresponding to a change in G₀of about 0.1%兲 can correct for this 共dotted lines in Fig. 2兲. The strongly non-Lorentzian lineshape seen by Rahav and Brouwer14,15_{in the spinless kicked rotator is} not observed here.

BARDARSON, TWORZYDŁO, AND BEENAKKER PHYSICAL REVIEW B 72, 235305共2005兲

V. CONCLUSION

We have presented a numerically highly efficient model of transport through a chaotic ballistic quantum dot with orbit scattering, extending the earlier work on the spin-less kicked rotator. Through a simple assumption of a ran-dom Floquet matrix we have derived analytical predictions for the conductance of the model as a function of spin-orbit scattering strength and magnetic field. The functional form

of the conductance coincides with random-matrix theory 共RMT兲 and through this correspondence we obtain a map-ping from microscopic parameters to model parameters. Nu-merical calculations are in good agreement with the analyti-cal predictions.

In this paper we have applied the model in a parameter regime where the transport properties of the system are ana-lytically known through RMT, in order to test the validity of the model. In future work this model may provide a starting point for studies of transport properties in parameter regimes where RMT is known to break down. In certain cases, for example, in the study of the effect of a finite Ehrenfest time on weak antilocalization, very large system sizes are required 共see Refs. 14 and 15兲. An efficient dynamical model, as the one presented in this paper, is then a valuable tool.

ACKNOWLEDGMENTS

This work was supported by the Dutch Science Founda-tion NWO/FOM. We acknowledge support by the European Community’s Marie Curie Research Training Network under Contract No. MRTN-CT-2003-504574, Fundamentals of Na-noelectronics.

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FIG. 2. Average quantum correction具␦G典 to the conductance as

a function of the T-symmetry breaking parameter ␪₀. The data points are for the symplectic kicked rotator characterized by

K = 41, M = 500, N₁= N₂= 10, l₀= 0.2. The solid lines are the analyti-cal predictions共3.7兲 and 共3.17兲 in the absence and presence of spin-orbit scattering. The dotted lines are the solid lines with a vertical offset, to account for a difference between the predicted and actual value of the classical conductance G₀.

FIG. 3. Average quantum correction具␦G典 to the conductance as

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