The influence of anisotropic gate potentials on the phonon induced spin-flip rate in GaAs quantum dots

0.24 MB·English

by  Sanjay Prabhakar

The influence of anisotropic gate potentials on the phonon induced spin-flip
rate in GaAs quantum dots
Sanjay Prabhakar,1 Roderick V. N. Melnik,1,2 and Luis L. Bonilla2
1)M2NeT Laboratory,Wilfrid Laurier University, Waterloo, Ontario N2L 3C5,
Canada
2)Gregorio Millan Institute, Universidad Carlos III de Madrid, 28911, Leganes,
Spain
(Dated: December 16, 2011)
2
We study the anisotropic orbital effect in the electric field tunability of the phonon induced spin-flip rate
1
0 in quantum dots (QDs). Our study shows that anisotropic gate potential enhances the spin-flip rate and
2 reduces the level crossing point to a lower QDs radius due to the suppression of the Lande´g-factor towards
bulk crystal. In the range of 104 106 V/cm, the electric field tunability of the phonon induced spin-flip
n
−
a rate can be manipulated through strong Dresselhaus spin-orbit coupling. These results might assist the
J development of a spin based solid state quantum computer by manipulating spin-flip rate through spin-orbit

4 coupling in a regime where the g-factor changes its sign.
1

] Controlling the single electron spins in QDs through
l
l theapplicationofanisotropicgatepotentialisimportant
a for the design of solid state quantum computer.1–3 Tun- 105
h 1.0
s- atrboinlitgy-foafctthoreipnhIoInIo-Vn isnedmuicceodndsupcinto-flripQDrastecaanndbethmeaenleipc-- -1 s] 4 g/g0000...468
me ulatedthroughthe applicationofexternallyappliedgate T [11 0 00..02
at. pspoitne-notribailts.c4–o7upTlihneg isstrdeentgetrhmionfedthbeyRthaeshabsay-mDmreestsreilchatruis- ate, 1/103 -0.2 QD4s0 ra8d0iu s1,2 l00 (n16m0)
angular quantum well along z-direction which approxi- r
m p
nd- mjsutranutcectltiyounrea.s4lt,8iimnRvaeatresshsiobtnaheaspsdyienmn-somirtebytirtoyfcoaeulloepcnltgirnogtnhsaeragitsreotswheftrhohmedtiertrehoce-- Spin-fli1012 11700x145 0VV5// ccVmm/cm
o tionandDresselhausspin-orbitcouplingarisesfrombulk 10
c inversion asymmetry in the crystal lattice.9,10
[ 0
Recently, it has been made possible to measure the 10
2 electron spin states in gated QDs in presence of mag- 10 20 30 40 50 60 70 80 90
v netic fields along arbitrary direction.1 The physics be- QDs radius , l0 (nm)
8
hind this has been theoretically investigated by authors
5
5 in Refs. 11 and 12 which confirms that the spin-orbit FIG. 1. (Color online) Phonon induced spin-flip rate due to
0 coupling can be used as a control parameter in the elec- spin-orbit admixture mechanism as a function of QDs radius
. tric field and magnetic field tunability of the electron g- in symmetric QDs (a=b=1). We choose B=1 Tesla. Also,
1
insetplotshowstheg-factorvs. QDsradius. Thelevelcross-
factor tensor. The authors in Refs. 13 and 14 measured
1
ing point occurs at ℓ = 73 nm. The material constants for
1 the long spin relaxationtimes, approximately0.85 ms in 0
GaAs QDs have been chosen from Refs. 4 and 19 as follows:
1 GaAs QDs by pulsed relaxation rate measurements and
v: approximately 20 ms in InGaAs QDs by optical orienta- g20.3=4×−100.−454,ermg/=cm0,.0s6l7=, γ5R.14=×41.405˚Ac2m,/γsD,s=t =263.e0V3˚A×31,0e5hc1m4/=s
i tionmeasurements. Thesespin-flipratemeasurementsin and ρ = 5.3176 g/cm3. At E = 7×105 V/cm shown by
X QDs confirm the theoretical predictions of the suppres-
dashed-dotted lines, the admixture mechanism due to spin-
r sion of the phonon induced spin-flip rate by spin-orbit orbit coupling on the spin-flip rate is quite different because
a
couplingwithrespecttotheenvironment.8,15,16Ourwork theelectronspinstateschangetheirsign intheseregime(see
is along the lines of Refs. 4, 17, and 18 with severalnew inset plot).
importantfindings. Inparticular,inthispaper,westudy
theanisotropicorbitaleffectonthephononinducedspin-
fliprateforasystemwheretheareaofthesymmetricand
asymmetric QDs kept constant. Basedon both theoreti-
cal and finite element numerical simulation methods, we
find that the anisotropic potential enhances the spin flip
rateandreducesthelevelcrossingpointtoalowerquan-
tum dot radius due to the suppression of the g-factor
intheconductionbandinthepresenceofmagneticfieldB
towards bulk crystal.
alongz-direction. ThetotalHamiltonianH =H +H
xy so
Weconsider2DanisotropicsemiconductorQDsformed of an electron in the conduction band under the Kane
2
model5,20 can be written as 2 1/2
ω = 1 ω2+ω2 √a √b , ℓ = ~ , Ω =
Hxy = 2P~m2 + 12mωo2(ax2+by2)+ ~2σzωz,(1) ±ω02+ 412ω(cid:20)c2 cand ω0c(cid:16)= emB±is th(cid:17)e(cid:21)cyclotron frqequmeΩncy.
H = αR (σ P σ P )+ αD ( σ P +σ P ),(2) qWe nowturn to the calculationofthe phononinduced
so ~ x y − y x ~ − x x y y spin relaxation rate in between two lowest energy states
2me 2/3 in QDs. The interaction between electron and piezo-
αR =γReE, αD =0.78γD ~2 E2/3,(3) phonon can be written as5,16
(cid:18) (cid:19)
wmheenrteumP~ op=erat−oir~∇in+theeA~ aissymthmeet2riDc gealeucgteronA~mo=- uqphα(r,t)=s2ρV~ωqαei(q·r−ωqαt)eAqαb†qα +H.c. (4)
B ( y√b,x√a,0)andω =g µ B/~istheZeeman
√a+√b − z 0 B
frequency. Also, m is the effective mass, µ is the Bohr Here, ρ is the crystal mass density, V is the volume of
B
magneton, ~σ is the Pauli spin matrices, ω0 = m~ℓ20 is the ttohreqQDans,dbp†qoαlacrriezaatteisonaneˆa,cowuhsetricepαho=nol,ntw,tithawreavcehovseecn-
strength of the parabolic confining potential with quan- α 1 2
as one longitudinal and two transverse modes of the in-
tum dot radius ℓ . The externally appliedgate potential
0 duced phonon in the dots. Also, A = qˆqˆ eβ ej
(V =1/2mω2(ax2+by2)) in our theoretical model de- qα i k ijk qα
xy o is the amplitude of the electric field created by phonon
finesthelateralsizeoftheQDsalongx-andy-directions
strain, where qˆ = q/q and eβ = eh for i =
in the plane of 2DEG. By chosing ℓ0 =10 100 nm, we ijk 14 6
− k,i = j,j = k. The polarization directions of the
mimictheexperimentallyreportedlateralsizeoftheQDs 6 6
induced phonon are eˆ = (sinθcosφ,sinθsinφ,cosθ),
in Refs.1 and 21andthe potential inducedin this range l
eˆ = (cosθcosφ,cosθsinφ, sinθ) and eˆ =
is muchlowerthan the break downvoltage(400kV/cm) t1 − t2
( sinφ,cosφ,0). Based on the Fermi Golden Rule, the
in GaAs heterojunctions. The strength of the Rashba −
phonon induced spin transition rate in the QDs is given
and Dresselhaus spin-orbit couplings is determined by
by4,5
the relationα /α =1.5 10 3E1/3 whichtellsus that
R D −
α =α at the electric fi×eld E =3 106 V/cm. In the
raRngeofD104 106 V/cm,onlytheDr×esselhausspin-orbit 1 = 2π d3q M(qα) 2δ(~s q ∆), (5)
coupling has−an appreciable contribution in the manip- T1 ~ Z (2π)3 α=l,t| | α −
X
ulation of spin-flip rate in QDs. The asymmetric trian-
gularquantumwellpotential(E = dV/dz)arisesalong where sl,st are the longitudinal and transverse acoustic
the growth direction and usually h−as a major contribu- phonon velocities in QDs. The matrix element M(qα)
tion to the E-filedtunability ofthe g-factorandspin-flip for the spin-flip rate between the Zeeman sublevels with
rate due to the interplay between Rashba-Dresselhaus the emission of phonon qα has been calculated pertur-
spin-orbit couplings. The expression (2meE/~2) 1/3 es- batively.5,24
−
timatestheaveragethicknessofthe2DEGwhereonecan In Fig. 1, we quantify the influence of the Rashba-
estimate the vertical average height of the QDs.4,12 By Dresselhaus spin-orbit admixture mechanism on the
chosingE =(0.1 10)105V/cm,weestimatetheaverage phonon induced spin-flip rate as a function of quantum
−
height of the QDs from 2 nm to 10 nm which is in the dot radius for symmetric QDs. For the electric fields
range of experimentally reported values.21 E =104and105V/cm(shownbysolidanddashedlines),
The above Hamiltonian (1) can be exactly diagonal- we find that the transition time between spin up and
ized22,23 and spin-orbitHamiltonian canbe used pertur- downstatesincreaseswiththeincreaseinQDsradiusand
bativelytofindtheenergystatesoftheQDs. Theenergy givesthe levelcrossingpointatℓ0 =73nm. Forthecase
spectrum can be written as E =7 105V/cm(shownbydasheddottedline),thespin
×
relaxation rate in QDs starts decreasing at ℓ = 35 nm
~ 0
H =(n +n +1)~ω +(n n )~ω + σ ω , becauseZeemanspinsplittingenergyisverysmallwhich
xy + + + z z
− − − − 2 implies small phonon density of states. It becomes neg-
Hso =αR(1+i)[b1/4κ+(s+ i)a++b1/4κ+(s +i)a ligible at ℓ0 = 50 nm, and thus the spin relaxation rate
− − − turnstobezero. Thespinrelaxationtimestartsincreas-
+a1/4η (i s )a +a1/4η (i+s )a ]
− − − + − + − ing atℓ0 =65 nm. However,in this regime,the opposite
+αD(1+i)[a1/4κ (i s )a++a1/4κ (i+s+)a spin state is dominating because the g-factor of an elec-
− − − − −
+b1/4η ( i+s )a +b1/4η (i+s )a ]+H.c., tronspinstates changestheir sign(see insetplots). This
+ + + +
− − − is an important result for the design of spin based logic
1 1 eBℓ 1
κ = σ i σ , devices. Indeed, in GaAs/AlGaAs QDs, wavefunctions
± 2(s+−s−)(cid:26)ℓ x± ~ (cid:18)√a+√b(cid:19) y(cid:27) of electrons penetrate from GaAs QDs to AlGaAs bar-
1 1 eBℓ 1 rier with the application of gate controlled electric fields
η = σ i σ ,
± 2(s+−s−)(cid:26)ℓ y ± ~ (cid:18)√a+√b(cid:19) x(cid:27) Twhheerleevtehlecrgo-sfasicntgortaokfeasninetolecatcrcoonuncth,adnugeestoitms isxiginng.2,6t–h2e8
and H.c. represents the Hermitian conjugate. Also, Zeemanspinstates 0,0, > and 0,1,+>inQDs. The
| − |
3
crossing point is theoretically investigated by the condi-
tion17,18 ε =ε i.e., ~(ω ω )= g µ B (see
106 Eq. 4). Su0,b0,s−titutin0,g1,+B = 1 T+in−the−abov|e0|coBndition,
asymmetric QDs
5 gives the crossing point at ℓ = 73 nm. Theoretically
-1 s] 105 EE==1 70x1 V0/5c Vm/c m investigatedlevelcrossingpoi0ntis inagreementwith the
T [1 4 symmetric QDs numerically investigated values in the spin flip rate (see
1/ 10 E= 105 V/cm Fig. 1). It can be seen that enhancement in the spin-
e, E= 7x105 V/cm flip rate occurs with the increase in electric fields. The
rat 103 level crossing point in the spin-flip rate is not affected
p bytheelectricfieldswhichtellsusthatthelevelcrossing
n-fli 102 point found in the spin flip rate is a purely orbital effect
pi and is independent of the Rashba-Dresselhausspin-orbit
S 1
10 interaction.
Fig. 2 explores the influence of anisotropic effects on
0
10 the spin-flip rate vs. QD radius for the electric fields
10 20 30 40 50 60 70 80 90
E =105 and E =7 105 V/cm. It can be seen that the
QDs radiu s, l0 (nm) anisotropic potentia×l (a = 0.5,b = 2) enhances the spin-
flip rate by approximately one half order of magnitude
FIG. 2. (Color online) Phonon induced spin-flip rate due to compared to that of symmetric potentials (a = b = 1).
spin-orbit admixturemechanism as a function of QDsradius Note that we chose the above confining potentials in
in asymmetric QDs (solid and dotted lines). As a reference, such a way that the area of the symmetric and asym-
we also plotted spin-flip rate vs. QDs radius for symmetric metric QDs are held constant. The level crossing point
QDs(dashedanddasheddottedlines). Wechoosethepoten- determined by the condition ~(ω ω )= g µ B for
+ 0 B
tials characterized by a = 0.5 & b = 2 for asymmetric QDs anisotropic QDs is smaller than for−the−case|of|isotropic
and a=b=1 for symmetric QDs. Also we choose B = 1 T.
QDs if we held the area of the QDs constant. The cross-
As we see, spin-flip rate increases approximately by one half
ing point for symmetric QDs was first studied by the
order of magnitude in asymmetric QDs.
authors of Refs. 17 and 18. However, in this paper, we
presenttheconditionofthelevelcrossingpointforasym-
metricQDsandbyutilizingboththeoreticalandnumer-
ical methods, we report that the anisotropic potential
reduces the level crossing point to a smaller QD radius
aswellastosmallermagneticfields. SimilartoFig.1,at
1.4
1.00 the electric field E = 7 105 V/cm for anisotropic QDs
0.98 Suppression of g-factor 1.2 inFig.2(dottedlines),s×mallZeemanenergyimpliesneg-
towards bulk crystal
0.96 1.0 ligible phonondensity ofstateswhichgiveszero spinflip
g/g0 00..9924 aa==b0.=51, b=2 g/g 00.8 Stouwpaprrdess sbiuolnk ocfr ygs-tfaalctor (rsapteinatstℓa0te=s c4h6anngme athnedirthseiggn-)faicstoorbsweritvhedopaptoℓs0ite=sig6n3
0.90 0.6 nm. Note that these numerically estimated values (zero
00..8868 g 0,0, BB0,0, 0.4 PPMuuixrreee dRD careassshsebesal hcbaaoustshe cRaRsaendDDpresent saptisnm-flailplerraQteDarnadditihfeorga-fnaicstootrrowpiitchQoDpspothsiatne isnigins)otorcocpuicr
(a) 0.2 (b)
0.84 QDs. Thistellsusthatthe anisotropicpotentialleadsto
10 20 30 40 50 60 70 80 90 100 0 20 40 60 80 100
QDs radius, l0 (nm) Electric Field, E (104 ) (V/cm) the quenching effect in the orbital angular momentum7
that pushes the g-factor of an electron towards the bulk
crystal which causes the level crossing point to occur at
FIG. 3. (Color online) (a) The anisotropic effect on the g- a smaller QD radius for anisotropic QDs (see Fig. 3(a)).
factor vs. QDs radius at the potentials characterized by a= FromFig.3(b),weseethattheg-factorcanbetunedwith
b=1(solidline)forisotropicQDsanda=0.5,b=2(dashed- the spin-orbit interactions. Recall that the electric field
dotted line) for anisotropic QDs. We choose E = 105 V/cm controls the strength of Rashba and Dresselhaus spin-
and B = 1T. Anisotropic potential gives the suppression of orbit couplings (see Eq. 3). Here, we again see that the
theg-factor towards bulk crystal and hencereduces thelevel
suppression of g-factor towards bulk crystal induces the
crossing point to lower QDs radius. Accidential degeneracy
anisotropic effect due to the interplay between Rashba-
appearsintherangeof70−80nmQDsradiuswhichgivesthe
Dresselhaus spin-orbit couplings.29,30
cusp like structure in the spin-flip rate (see Refs. 17, 18, and
25). (b)TheinterplaybetweenRashbaandDresselhausspin- To conclude, we have shown that the electron spin
orbit couplings on the g-factor vs. the electric field in QDs states in the phonon induced spin-flip rate can be ma-
induces the anisotropic effect due to the suppression of the nipulated with the application of externally applied
g-factor towards bulk crystal. Here, we choose ℓ0 = 20 nm, anisotropic gate potentials in QDs. The anisotropic po-
B=1 T and a=b=1.
tentialcausesthesuppressionoftheg-factortowardsbulk
crystal that causes the enhancement of the spin-flip rate
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