Anisotropic effects and phonon induced spin relaxation in gate-controlled semiconductor quantum dots

0.47 MB·English

by  Sanjay Prabhakar

Anisotropic effects and phonon induced spin relaxation in gate-controlled
semiconductor quantum dots
Sanjay Prabhakar,1 Shohini Ghose,2 Roderick Melnik1,3 and Luis L. Bonilla3
1M2NeT Laboratory, Wilfrid Laurier University, Waterloo, ON, N2L 3C5 Canada
2Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, ON, N2L 3C5 Canada
3Gregorio Millan Institute, Universidad Carlos III de Madrid, 28911, Leganes, Spain
(Dated: January 26, 2012)
2 Inthispaper,adetailedanalysisofanisotropiceffectsonthephononinducedspinrelaxationrate
1 inIII-Vsemiconductorquantumdots(QDs)iscarriedout. Weshowthattheaccidentaldegeneracy
0 due to level crossing between the first and second excited states of opposite electron spin states in
2 both isotropic and anisotropic QDs can be manipulated with the application of externally applied
gate potentials. In particular, anisotropic gate potentials enhance the phonon mediated spin-flip
n
rate and reduce the cusp-likestructure to lower magnetic fields, in addition to thelower QDs radii
a
in III-V semiconductor QDs. In InAs QDs, only the Rashba spin-orbit coupling contributes to the
J
phonon induced spin relaxation rate. However, for GaAs QDs, the Rashba spin-orbit coupling has
6
a contribution near the accidental degeneracy point and the Dresselhaus spin-orbit coupling has
2
a contribution below and above the accidental degeneracy point in the manipulation of phonon

inducedspin relaxation rates in QDs.
]
l
l
a
h I. INTRODUCTION strong Rashba spin-orbit coupling in InAs QDs29 and
- trough strong Dresselhaus spin-orbit coupling in GaAs
s
e The study of electron spin states in zero dimensional QDs.25 Largeanisotropyeffects ofthe spin-orbitinterac-
m tioninself-assembledInAsQDshavebeenrecentlystud-
semiconductor nanostructures such as QDs is important
ied experimentally in Ref. 12. In this paper, we study
. forthedevelopmentofnextgenerationelectronicdevices
t the phonon induced spin-flip rate of electron spin states
a suchasspintransistors,spinfilters,spinmemorydevices
m and quantum logic gates.1–6 The electron spin states in inbothisotropicandanisotropicQDs. Ourstudies show
that the Rashba spin-orbit coupling has an appreciable
- QDs are brought in resonance or out of resonance by
d contribution to the spin-flip rate in InAs QDs. How-
applyingsuitablegatepotentialsinordertoreadoutthe
n spin states.7,8 Progress in nanotechnology has made it ever, the Rashba spin-orbit coupling has a contribution
o
to the spin-flip rate in GaAs QDs near the level crossing
possible to fabricate gated quantum dots with desirable
c
[ optoelectronic and spin properties.8–11 Very recently, it point and the Dresselhaus spin-orbitcoupling elsewhere.
Anisotropicgatepotentials,playinganimportantrolein
wasshownthattheelectronspinstatesingatedquantum
1 the spin-flip rate, can be used to manipulate the acci-
dots can be measured in the presence of magnetic fields
v
along arbitrary directions.12–16 dental degeneracy due to level crossing and avoided an-
9
ticrossingbetween the electronspin states 0,0, > and
4 A critical ingredient for the design of robust spin- | −
0,1,+ >. In this paper, we show that the anisotropic
5 tronic devices is the accurate estimation of the spin re- |
5 laxation rate. Recent studies by authors in Refs. 4 and gatepotentialscausealsoaquenchingeffectintheorbital
. angular momentum that enhances the phonon mediated
1 5 have measured long spin relaxation times of 0.85 ms
spin-flip rate and reduces its cusp-like structure to lower
0 in GaAs QDs by pulsed relaxation rate measurements
2 and 20 ms in InGaAs QDs by optical orientation mea- magnetic fields, in addition to lower QDs radii.
1 surements. These experimental studies in QDs confirm
:
v that the manipulationof spin-fliprate by spin-orbitcou-
i pling with respect to the environment is important for
X
the design of robust spintronics logic devices.17–19 The
ar spin-orbitcouplingismainlydominatedbytheRashba20 The paper is organized as follows: In section II, we
and the linear Dresselhaus21 terms in solid state QDs. develop a theoretical model and find an analytical ex-
TheRashbaspin-orbitcouplingarisesfromstructuralin- pression for the energy spectrum of electron spin states
version asymmetry along the growth direction and the for anisotropic QDs. In section III, we find the spin re-
Dresselhaus spin-orbit coupling arises from the bulk in- laxation rate of electron spin states for anisotropic and
version asymmetry of the crystal lattice.22–24 Recently, isotropic QDs. In section IV, we show that the cusp-
electricandmagneticfieldstunabilityoftheelectronspin like structure in spin-flip rate due to accidental degener-
states in gated III-V semiconductor QDs was manipu- acypointsinQDscanbe manipulatedtolowermagnetic
lated through Rashba and Dresselhaus spin-orbit cou- fields, in addition to lower QDs radii, with the applica-
plings.24–28 tionofanisotropicgatepotentials. Differentmechanisms
Anisotropic effects induced in the orbital angular mo- ofspin-orbitinteractionssuchasRashbavs. Dresselhaus
mentum in QDs suppresses the Lande´ g-factor towards couplings are also discussed in this section. Finally, in
bulk crystal.25,29 g-factor can be manipulated through section V, we summarize our results.
2
Also, the assymetric Gauge potential can be written as

1015 b1/4 a1/4
14 pure Rashba case D A = x B ,A =x B . (5)
s) 1013 InAs QDs pure Dresselhaus case R x − 2 √a+√b y 1 √a+√b
T (1/111001112 Mixed casesboth Rand Dpresent By substitutin(cid:18)g Eqs. (3,(cid:19)4, 5) into E(cid:18)q. (2), we g(cid:19)et the
1/ 1010 Hamiltonian in the form:
e, 109
Spin-flip rat 11111100000045678 Energy (meV)223333......680246 (i) g/g00000001.......4567890 (ii) hwh=erPe12h+x=21+2√mℑaH(cid:0)Px2y2,+ℑx=22(cid:1)+℘ab(,x1℘P2=−xω22+Pω12ω)2+(c√(b12a/+ga)o√1µ/b4B)2σz1(/B62).,
110023 0 1 23.0M3.a23gn3e.4tic 3F.i6e4ld3,. 8B (4T.)05 6 1M7a2gnet3ic8 Fie4ld, B95 (T)610 A(lωsco/ωw0)e2 uasendthωe =relaeBtioinsqtmheω0cγycl=otr1(cid:2)o,ncwfrhe0eqrueenγc2y.=(cid:3)1 +
Magnetic Fie ld, B (T) (√a+√b)2 c m
The energy spectrum ofHamiltonian (6) canbe found
as follows. First, we need to find the canonical trans-
FIG. 1. (Color online) Contributions of Rashba and Dres-
formation U of the four-dimensional phase space, Pt
selhaus spin-orbit couplings on the phonon induced spin-flip ≡
(P ,P ,x ,x ) which diagonalizes the quadratic form of
rate vs magnetic fields in InAs QDs. Inset plots shows the 1 2 1 2
energy difference vs. magnetic fields near the level crossing the Hamiltonian (6). To be more specific, Hamiltonian
point and the g-factor vs magnetic fields. Here we choose (6) without Zeeman spin splitting energy can be written
E = 105 V/cm, ℓ0 = 20 nm and a = b = 1. The material as
constants for InAs QDs are chosen from Refs. 24 and 30 as
1 0 0 ℘/2
g0 =−15,m=0.0239, γR =110˚A2,γD =130eV˚A3,eh14 = 0 ℘/2 −0
0.54×10−5 erg/cm,sl =4.2×105 cm/s,st =2.35×105 cm/s h=PtMP, M= 0 ℘ℑ/2 1 0 , (7)
and ρ=5.6670 g/cm3.
℘/2 0 0
 − ℑ 
 
where t represents the transpose of a vector. The or-
II. THEORETICAL MODEL
thogonal unitary matrix U which exactly diagonalizes
the matrix M can be written as,
We consider 2D anisotropic III-V semiconductor QDs
1 1 s s
in the presence of a magnetic field along the growth 1 1 1 −s − −s+
daniriescottiroonp.ic QTDhes intoctlualdinHgamspilitno-noriabnit ionftearnacteiloenctcroann bine U= (s+−s−) ss− −ss+ 11+ −11− , (8)
written as18,22,24 − + − − 
where ℘s 1 d and
H =Hxy+Hso, (1) ± ≡ℑ− ±
1
where the Hamiltonian Hso is associated with the ω+ b ωc2 ab b 2 2
s = 1 + 1 ,(9)
Rashba-Dresselhaus spin-orbit couplings and Hxy is the ± ω b 41 ra − ± ωq+2 −ra!  
Hamiltonianofthe electroninanisotropicQDs. Hxy can c a  
be written as (cid:0) (cid:1)  1 21/2
Hxy = 2P~m2 + 21mωo2(ax2+by2)+ 12goµBσzB, (2) ω± = 2(cid:20)ωc2+ω02(cid:16)√a±√b(cid:17) (cid:21) ,(10)
1
where P~ = p~+eA~ is the kinetic momentum operator, 4ωc2 ab b 2 2
p~ = i¯h(∂ ,∂ ,0) is the canonical momentum operator, d= + 1 . (11)
A~ is−the vxectyor potential in the asymmetric gauge, m ω2+ω2 √qa+√b 2 −ra! 
c 0
 
is the effective mass of the electron in the conduction  (cid:16) (cid:17) 
In the form of rotated operators P =UP, the Hamilto-
band, µ is the Bohr magneton,~σ =(σ ,σ ,σ ) are the ′
B x y z
Pauli spin matrices, ω = h¯ is the parabolic confining nian (6) can be written as
0 mℓ2
0 1 1
poTteontfiinaldatnhdeℓe0neisrgtyhespraecdtiruusmofotfhHeaQmDislt.onian (2), it is h= 2(℘s−+2) p′x2+x′2 + 2(℘s++2) p′y2+y′2 .(12)
(cid:16) (cid:17) (cid:16) (cid:17)
convenient to introduce the canonical transformation of The above Hamiltonian represents the superposition of
position and momentum operator as31,32 two independent harmonic oscillators. The energy spec-
trum of H can be written as
1 xy
x =a1/4x, P = P , (3)
1 1 a1/4 x 1
ε =(n +n +1)h¯ω +(n n )¯hω + g µ σ B,
x2 =b1/4y, P2 = b11/4Py. (4) n+n− + − + +− − − 2 0 B(13z)
3

15
10
14
1013 InAs QDs (i) a=b=3 (ii) a=b=9 (iii) a=1,b=9
1/s) 110012 Symmetric QDs Symmetric QDs Asymmetric QDs
T (11011
1/ 1010
ate, 1089
Spin-flip r 111100004567 EEE===151XX0X11001044 4VV V//cc/mmcm a=b=3 and E=1X104 V/cm
10
3
10
2
10
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 100 1 2 3 4 5 6 7 8 9 10
Magnetic F ield, B (T) Magnetic F ield, B (T) Magneti c Field, B (T)
FIG. 2. (Color online) Spin relaxation rate (1/T1) vs magnetic fields between the states |00+ > and |00− > in InAs QDs.
Here we chose ℓ0 = 20nm (the QD radius). As a reference in Fig. 2(iii), dashed-dotted line represents the spin-flip rate for
symmetricQDswitha=b=3. Weseethattheanisotropicpotentialenhancesthespin-fliprateandreducesthelevelcrossing
point tolower magnetic field.

13 12
10 10
e, 1/T (1/s)11111000011189012 InAs QDs e, 1/T (1/s)111111000001178901 GaAs QDs
Spin-flip rat 11111000004567 a=b=3 (Symmetric QDs) Spin-flip rat 1111000023456 aa==b1=,b8= (6S4y (mAmsyemtrmic eQtrDics Q)Ds)
3 a=1,b=9 (Asymmetric QDs) 10
10 1
2 10
10 0
10
10 15 20 25 30 35 40 45 50 55 60 65 70 0 2 4 6 8 10 12 14 16 18 20
QDs radius, l0 (nm) Magnetic Field, B (T)
FIG. 3. (Color online) Spin relaxation rate (1/T1) vs QDs FIG.4. (Coloronline)Spinrelaxationrate(1/T1)vsmagnetic
radius between the states |00+ > and |00− > in InAs QDs. fields between the states |00+ > and |00− > in GaAs QDs.
Here, we choose B = 1T and E = 104V/cm (magnetic and Here we chose, QDs radius, ℓ0 = 32nm and electric field,
electric fields, respectively). We see that the anisotropic po- E = 104V/cm. Again, we see that the level crossing point
tential reduces thelevel crossing point to lower QDs radius. reduces to the lower magnetic fields. For GaAs QDs, we use
the material constants from Refs. 24 and 30 as g0 = −0.44,
m = 0.067, γR = 4.4 ˚A2, γD = 26 eV˚A3, eh14 = 2.34×
10−5 erg/cm,sl =5.14×105 cm/s,st =3.03×105 cm/sand
where n = a† a are the number operators. Here, a ρ=5.3176 g/cm3
± ± ± ±
and a† are usual annihilation (“lowering”) and creation
(“raisi±ng”) operators. In Eq. 13, we included Zeeman
spin splitting energy. and α . They are given by
D
The Hamiltonian associated with the Rashba and
linear Dresselhaus spin-orbit couplings can be written 2me 2/3
α =γ eE, α =0.78γ E2/3, (15)
as20,21,24 R R D D ¯h2
(cid:18) (cid:19)
H = αR (σ P σ P )+ αD ( σ P +σ P ), (14) where γR andγD arethe RashbaandDresselhaus coeffi-
so x y y x x x y y
¯h − ¯h − cients.
The energy spectrum of the Hamiltonian associated
wherethe strengthsofthe RashbaandDresselhausspin- with the Rashba and Dresselhaus spin-orbit couplings
orbit couplings are characterized by the parameters α can be written as
R
4

9
10
s) 108
T (1/1 107 GaAs QDs
1/ 106
e, 5
at 10
r 4
p 10
n-fli 103 pure Rashba case D
pi 102 pure Dresselhaus case R
S 1 Mixed cases both Rand Dpresent
10
0
10
0 2 4 6 8 10 12 14 16 18 20
Magnetic Fie ld, B (T)
FIG. 5. (Color online) Contributions of Rashba and Dres-
selhaus spin-orbit couplings on the phonon induced spin-flip
rate as a function of magnetic fields in GaAs QDs. Here we
chooseE =105V/cm,ℓ0 =32nmanda=b=1. Weseethat
only Rashba spin-orbit coupling gives the cusp-like structure
in thespin-flip rate.
H =α (1+i) b1/4κ (s i)a +b1/4κ (s +i)a +a1/4η (i s )a +a1/4η (i+s )a
so R + + + + + +
− − − − − − − −
h i
+α (1+i) a1/4κ (i s )a +a1/4κ (i+s )a +b1/4η ( i+s )a +b1/4η (i+s )a +H.c., (16)
D + + + + + +
− − − − − − − −
h i
where, ς = 1 1 α2 2α β 1 β2 ,(22)
1 1 eBℓ 1 ± 2(s+−s−)(cid:26)±s± ∓− ∓ ∓∓ s∓ ∓(cid:27)
κ = σ i σ ,(17)
x y
± 2(s+−s−)(cid:26)ℓ ± ¯h (cid:18)√a+√b(cid:19) (cid:27) α =a1/4 1 eBℓ 1 ,(23)
1 1 eBℓ 1 ± ℓ ± ¯h √a+√b 
η = σy i σx ,(18)  
± 2(s+−s−)(cid:26)ℓ ± ¯h (cid:18)√a+√b(cid:19) (cid:27) (cid:16) (cid:17)
1 eBℓ 1 
where H.c. represents the Hermitian conjugate, ℓ = β =b1/4 .(24)
¯h/mΩ is the hybrid orbital length and Ω = ± ℓ ± ¯h √a+√b 
 
ω2+ω2/4. It is clear that the spin-orbit Hamiltonian
p 0 c (cid:16) (cid:17)
and the Zeeman spin splitting energy in both isotropic  
p
and anisotropic QDs obey a selection rule in which the III. PHONON INDUCED SPIN RELAXATION
orbital angular momentum can change by one quantum.
At low electric fields and small QDs radii, we treat
We nowturn to the calculationofthe phononinduced
the Hamiltonian associated with the Rashba and linear
spin relaxation rate between two lowest energy states in
Dresselhaus spin-orbit couplings as a perturbation. Us-
QDs. FollowingRef.33,theinteractionbetweenelectron
ing second order perturbation theory, the energy spec- and piezo-phonon can be written as18,34
trum of the electron spin states in QDs is given by
ε0,0,+ =¯h̟+− α2R¯hξω+x+α∆2Dς+ − α2R¯hς−ωy+α∆2Dξ−,(19) uqphα(r,t)=s2ρV¯hωqαei(q·r−ωqαt)eAqαb†qα +H.c. (25)
− −
ε0,0, =h¯̟ α2Rς++α2Dξ+ α2Rξ−+α2Dς−,(20) Here, ρ is the crystal mass density, V is the volume of
− −− ¯hωx+∆ − ¯hωy+∆ the QDs, b†qα creates an acoustic phonon with wave vec-
tor q and polarization eˆ , where α = l,t ,t are chosen
where, ̟ = ω ω /2, ω = ∆/¯h is the Zeeman fre- α 1 2
quency, ∆±=g µ+ ±B, ωz =ωz +ω , and ω =ω ω . as one longitudinal and two transverse modes of the in-
Also, 0 B x + − y +− − duced phonon in the dots. Also, Aqα = qˆiqˆkeβijkejqα
is the amplitude of the electric field created by phonon
1 1 1 strain, where qˆ = q/q and eβ = eh for i =
ξ = α2 +2α β β2 ,(21) ijk 14 6
± 2(s+−s−)(cid:26)±s± ± ± ±∓ s∓ ±(cid:27) k,i 6= j,j 6= k. The polarization directions of the
5
induced phonon are eˆ = (sinθcosφ,sinθsinφ,cosθ), where s ,s are the longitudinal and transverse acoustic
l l t
eˆ = (cosθcosφ,cosθsinφ, sinθ) and eˆ = phonon velocities in QDs. The matrix element M(qα)
t1 − t2
( sinφ,cosφ,0). Based on the Fermi Golden Rule, the for the spin-flip rate between the Zeeman sublevels with
−
phonon induced spin transition rate in the QDs is given the emission of phonon qα has been calculated pertur-
by24,34 batively.34,35 As a result, we have:
1 = 2π d3q M(qα) 2δ(h¯s q ε +ε ), 1 =c Mx 2+ My 2 , (27)
T1 ¯h Z (2π)3 αX=l,t| | α − 0,0,− 0,0,+ T1 (cid:0)| | | | (cid:1)
(26) where,
(eh )2(gµ B)3 1 4 1
14 B
c= + , (28)
35π¯h4ρ s5 3s5
(cid:18) l t(cid:19)
(is +1)Ξ (¯hω +∆)+( is +1)Ξ (h¯ω ∆) ( is +1)Ξ (h¯ω +∆)+(is +1)Ξ (h¯ω ∆)
1 x 3 x + 2 y + 4 y
Mx = − − − − + − − ,(29)
a1/4 (¯hω )2 ∆2 a1/4 (h¯ω )2 ∆2
x y
− −
(is +1)Ξ (¯hω +∆h )+( is +i1)Ξ (¯hω ∆) (is 1)Ξ (h¯ω +∆h )+( is i1)Ξ (h¯ω ∆)
+ 1 x + 3 x 2 y 4 y
My = − − + −− − −− − ,(30)
b1/4 (¯hω )2 ∆2 b1/4 (h¯ω )2 ∆2
x y
− −
h ℓ i h i
Ξ = [α (s +i)β +(1 is )α +α ( s i)α +( 1+is )β ], (31)
1 2(s s )2 R{ + + − − +} D{ − −− − − + −}
+
− −
ℓ
Ξ = [α (s i)β +(1+is )α +α (s i)α +(1+is )β ], (32)
2 2(s s )2 R{ −− + + +} D{ +− − − −}
+
− −
ℓ
Ξ = [α (s i)β +( 1 is )α +α ( s +i)α +(1+is )β ], (33)
3 2(s s )2 R{ +− − − − − −} D{ − − + + +}
+
− −
ℓ
Ξ = [α (s +i)β +( 1+is )α +α (s +i)α +( 1+is )β ].(34)
4 2(s s )2 R{ − − − + −} D{ + + − − +}
+
− −
In the above expression, we use c =c I +2c I where (see 2nd term of Eq. 36) and is absent in the Dressel-
l xl t xt
c = q2e2 ε 2, ε 2 = q2h¯ andq = gµBB. Also, haus case. Similarly, the degeneracy only appears in the
α (2π)2h¯2sα| qα| | qα| 2ρωqα h¯sα g-factorfortheRashbaspin-orbitcoupling.29 Thedegen-
g = ε0,0,−µB−Bε0,0,+ is the Lande´ g-factor. Also, for longi- eracyin the Rashbacase induces the levelcrossingpoint
tudinal phonon modes Ixl = Iyl = 83π5h214 and Ixyl = 0. and cusp-like structure in the spin-flip rate in QDs. By
For transverse phonon modes, I = I = 16πh2 and considering second power of ∆, the spin relaxation rate
xt yt 105 14
I =0. for isotropic QDs is given by
xyt
For isotropic QDs (a = b = 1, s = 1 and s = 1),
+
the spin relaxation rate is given by − − 1 = (eh14)2(gµBB)3 1 + 4 1 2∆2 α2 +α2
1 (eh )2(gµ B)3 1 4 1 T1 35π¯h4ρ (cid:18)s5l 3s5t(cid:19)¯h4Ω4 R D
T1 = 1435π¯h4ρB (cid:18)s5l + 3s5t(cid:19) |MR|2+|MD|2 , 1+2 2ωΩc 2+3 2ωΩc 4+··· . (cid:0) (3(cid:1)8)
(cid:0) (3(cid:1)5) (cid:20) (cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
whereM andM arethecoefficientsofmatrixelement
R D It can be seen that the spin-flip rate is highly sensitive
associatedtotheRashbaandDresselhausspin-orbitcou-
to the effective g-factor of the electron, Zeeman energy,
pling in QDs and is given by
hybrid orbital frequency and cyclotron frequency of the
QDs.
α 1 1
R
M = , (36)
R √2¯hΩ1 ∆ − 1+ ∆ 
− h¯(Ω+ωc) h¯(Ω ωc)
2 − 2 IV. RESULTS AND DISCUSSIONS
 
α 1 1
D
M = .(37)
D √2h¯Ω1+ ∆ − 1 ∆  In Fig. 1, we investigate the contributions of the
h¯(Ω+ω2c) − h¯(Ω−ω2c) Rashba and the Dresselhaus spin-orbit couplings on the
 
Since ∆=g µ B is negativeforGaAsandInAs QDs,it phononinducedspinrelaxationrateasafunctionofmag-
0 B
means the degeneracy only appears in the Rashba case netic fields in symmetric InAs QDs. Since the strength
6
of the Dresselhaus spin-orbit coupling is much smaller the anisotropicpotential reduces the level crossing point
than the Rashba spin-orbit coupling (αR = 3.2 at E = to lower magnetic fields if the areaof the symmetric and
αD
105V/cm (see Eq. 15)), only the Rashba spin-orbit cou- asymmetric quantum dots is held constant. However, if
pling has a major contribution to the phonon induced we increase the area, the level crossing point extends to
spin-fliprate. Thecusp-likestructureisabsent(seeFig.1 the larger magnetic field.
(dashedline))andthespin-fliprate(1/T )isamonotonic In Fig. 3, we study anisotropic effects on the phonon
1
function of magnetic field (B) for pure Dresselhaus case inducedspin-flipratevs. QDsradiiinInAsQDs. Similar
(α = 0). We solve the corresponding eigenvalue prob- toFig.2(iii),theanisotropicpotentialenhancesthespin-
R
lemwithHamiltonian(1)byapplyingtheexactdiagonal- flip rate and reduces the accidental degeneracy point to
ization procedure and the Finite Element Method,25,36 lower QDs radii at ℓ0 =43nm.
obtainingthe energylevels. The insetplots showthe en- Next, we investigate the phonon induced spin relax-
ergydifferencevs. magneticfield(Fig.1(i))andeffective ationrateinGaAsQDs. InFig.4,weplotthephononin-
Lande´g-factor vs. magnetic field (Fig. 1(ii)). It can be ducedspin-flipratevs. magneticfieldsforbothisotropic
seen that the level crossing point occurs at B = 3.5T (a = b = 8, (solid line)) and anisotropic (a = 1,b = 64
which is the exact location of the accidental degeneracy (dashed line)) GaAs QDs. It can be seen that the cusp-
point in the spin-flip rate either for pure Rashba case like structure due to the accidental degeneracy can be
(α = 0) or mixed cases (both α and α present). manipulated to lower magnetic fields in the phonon in-
D R D
Similar results have been discussed in Refs. 22 and 23 duced spin-flip rate with the application of anisotropic
andweconsidertheseresultsasabenchmarkforfurther gate potentials.
investigationofanisotropicorbitaleffectsonthespin-flip The contributions of Rashba and Dresselhaus spin-
rate in QDs. orbit couplings on the phonon induced spin-flip rate
vs. magnetic fields in GaAs QDs are shown in Fig. 5.
Fig. 2 explores the anisotropic effects on the spin-
TheRashbaandDresselhausspin-orbitcouplingsbecome
flip rate vs. magnetic fields for the electric fields E =
equal at very large electric field E = 3.02 106V/cm in
104,5 104,105V/cm. It can be seen that the enhance- ×
× GaAs QDs. Below this value of electric field, only the
mentinthespin-fliprateoccurswiththeincreaseinelec-
Dresselhausspin-orbitcouplinghasamajorcontribution
tric fields. The accidental degeneracy point in the spin-
onthephononinducedspin-fliprateinGaAsQDs. How-
fliprateisnotaffectedbytheelectricfieldswhichtellsus
ever, near the level crossing point (for example, B =6T
thatitispurelyanorbitaleffectandisindependentofthe
inFig.5), theaccidentaldegeneracyappearsdueto only
Rashba-Dresselhaus spin-orbit interaction. In Fig. 2(ii),
theRashbaspin-orbitcoupling,whichgivesthecusp-like
the accidentaldegeneracypointis foundatthe magnetic
structure in the spin-flip rate.
field B = 3.5T. However, this point increases to the
larger magnetic field B = 6.2T in Fig. 2(ii). The exten-
sion in the B-field tunability of the spin-flip rate mainly
V. CONCLUSIONS
occurs due to an increase in the area of the symmetric
quantum dots. Note that the area of the quantum dots
in Fig. 2(ii) is 9 times larger than the dots in Fig. 2(i). In summary, we have analyzed in detail anisotropy ef-
Wequantifytheinfluenceoftheanisotropiceffectsonthe fects on the electron spin relaxation rate in InAs and
spin-fliprateinFig.2(iii). Herewefindthatthequench- GaAsQDs,usingrealisticparameters. InFig.1,wehave
ing in the orbital angular momentum26,27 enhances the shown that only the Rashba spin-orbit coupling has a
spin-fliprateandreducestheaccidentaldegeneracypoint major contribution on the phonon induced spin-flip rate
to lower magnetic fields (B = 5.85T) compared to the in InAs QDs. In Fig. 2, 3 and 4, we have shown that
symmetric quantum dots (B =6.2T). As a reference, in a cusp-like structure due to the accidental degeneracy
Fig.2(iii), wealsoplottedthe spin-flipratevs. magnetic point appears in the phonon induced spin-flip rate and
fields (shown by dashed-dotted line) for symmetric QDs can be manipulated to lower magnetic fields, in addition
(a=b=3) at E =104V/cm and ℓ =20 nm. Note that to lower QDs radii, with the application of anisotropic
0
theareaoftheisotropicandanisotropicquantumdotsin gate potentials in III-V semiconductor QDs. Also, we
Fig.2(ii)andFig.2(iii)areheldconstant. Theexpression have shown that the anisotropic gate potential causes a
for the level crossing point is given by the condition22,23 quenching effect in the orbital angular momentum that
ε0 = ε0 i.e., ¯h(ω ω ) = g µ B (see Eq. 13). enhances the phonon induced spin-flip rate. Finally, in
F0o,r0,−isotrop0,i1c,+QDs(a=+b=−1)−,the c|o0n|diBtionforthelevel Fig. 5 for GaAs quantum dots, we have shown that the
crossing point is Ω ω /2= g µ B/¯h. It means, when Dresselhausspin-orbitcouplinghasamajorcontribution
c 0 B
− |
thedifferencebetweenthehybridorbitalfrequencytothe on the spin-flip rate before and after the accidental de-
halfofthecyclotronfrequencybecomesequaltotheZee- generacypoint,andtheRashbaspin-orbitcouplinghasa
manfrequencythenthedegeneracyappearsintheenergy contribution near the cusp-like structure. These studies
spectrum which give the level crossing point and cusp- provideimportantinformationforthedesignandcontrol
like structure near the degeneracyin the spin-fliprate in of electron spin states in QDs for the purposes of build-
QDs. If we compare the condition of the level crossing ing robust electronic devices and developing solid state
pointfor the isotropicandanisotropicQDs,we find that quantum computers.
7
ACKNOWLEDGMENTS No. FIS2008- 04921-C02-01 and FIS2011-28838-C02-01
(Spain).
This work has been supported by NSERC and
CRC programs (Canada) and by MICINN Grants
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