Table Of ContentFast optical control of spin dynamics in a quantum wire
Z.-G. Zhu, C.-L. Jia, and J. Berakdar
Institut fu¨r Physik, Martin-Luther-Universita¨t Halle-Wittenberg, 06099 Halle, Germany.
The spin dynamicsin a quantumwire with aRashbaspin orbit interaction (SOI)is shown to be
controllableviasub-picosecondelectromagneticpulsesshapedappropriately. Ifthelightpolarization
vector is along the wire’s direction, the carriers experience a momentum boost while the phase
coherenceindifferentspinchannelsismaintained,afactexploitabletocontrolthespeedofaphoto-
1 driven spin field effect transistor. A photon pulse with a polarization vector perpendicular to the
1 wireresults in aspin precession whichis comparable tothatduetotheRashbaSOIandis tunable
0 bythe pulsefield parameters, an effect utilizable in optically controlled spintronics devices.
2
n
INTRODUCTION polarization axis [6, 12]. In case this axis is along the
a
J conductivechannelwefind thatboththe spinprecession
2 frequencyandthecarrierspeedincreaseuponirradiation
1 Optical semiconductor devices are indispensable but ∆θ0 remains unchanged. The operation speed is
components of nowadays technology [1] with applica- thus pulse-controllable. Ifthe opticalpolarizationvector
] tions ranging from optical fiber communication systems is perpendicular to the carriers propagation direction
l
al to consumer electronics. A new imputes is expected ∆θ0 and hence the SFET operation is determined by
h from spintronics devices [2, 3], i.e. from exploiting the the pulses parameters, as shown explicitly below.
- carriers spin in addition to their charge for efficient
s
e operation or to realize new functionalities. While
m the optical control and manipulation of charges in
. conventional semiconductors [4] have been the key MODEL SFET
t
a for the realization of ultra-fast electronic devices, an
m analogous optical control of the spins is however not
We follow Refs.[7 and 8] and focus on the central con-
- straightforward. The appropriateelectromagnetic pulses ductive region of the SFET that can be considered as a
d are U(1) fields; while spins belong to SU(2) fields. A
n one dimensional (1D) quantum wire (QW) of length L
laser pulse does not seem thus to couple to the spins
o with a spin orbit interaction (SOI). The ferromagnetic
directly, meaning a less efficient optical coupling to
c leads serve as spin injector and spin detector separated
[ the spins than to the charges. Indeed, the key to the
fromQWbyaninsulatingbarriertoachieveahigherspin
optical spin-manipulation are inherent interactions in
2 injection efficiency (Fig.1). Recently a SFET structure
v the system that couple the charge to the spin such as similartoFig.1hasbeenrealizedexperimentally[8]. The
9 SOI [5, 6]. Along this line we present here a way for an
experimental findings are in line with the predictions of
1 ultrafast control of the spin dynamics in a conventional
the stationary version of the present work. In the work
8
spin field effect transistor (SFET) [7, 8] driven by
4 presentedbelow, however,we consider 1D QW, for spin-
shaped electromagnetic pulses. The SFET relies on the
. flip transitions between the first and second subband is
9 Rashba SOI to perform a controlledrotation of a carrier
negligible for the system studied in this work.
0
spin that traverses a FET-type device [7, 8] with two
0
magnetic leads (cf. Fig.1). The conductance depends on
1
: the achieved rotation angle ∆θ0 at the drain lead. Here
v
we propose two ways to control the time-dependence of
i
X this rotation angle and hence the operation of SFET. yy
static
r The key ingredient is the use of asymmetrically shaped E(t) field
a linearly polarized electromagnetic pulses [9–12]. The gate εεεε0
pulse consists of a very short, strong half-cycle followed insulating layer
conductive channel
by a second long (compared to the ballistic transverse xx
zzzz
time) and a much weaker half-cycle of an opposite
polarity [9–11]. Hence such pulses are often called
half-cycle pulse (HCP). Experimentally the achieved
FIG. 1. Schematics of the optically driven spin field transis-
asymmetryratioof the positive andnegativeamplitudes tor. Ferromagnetic leads are separated from the conductive
can be 13:1, the peak fields can reach several hundreds channel by a tunneling barrier to enhance the spin injection
of kV/cm and have a duration t in the range between efficiency. AmetallicgateisusedtotunetheRashbaSOIvia
p
nano and subpicoseconds. The interaction of HCP with a static field ǫ0. E(t) is thetime-dependentelectric field.
matter is particular in that it delivers a definite amount
of momentum boost to the carriers along the optical Theinversion-asymmetryoftheconfiningpotentialre-
2
sultsinRashbaSOI[13]H . Foratwo-dimensionalelec- THE FIRST DYNAMIC CASE
R
tron gas H reads [13]
R
Having outlined the equilibrium case, we apply to the
H =α0(σ k σ k ), (1)
R R z x− x z quantum wire a linearly polarized HCP with the vec-
tor potential A = e A(t). The polarization vector e is
where σ ,i = x,y,z are Pauli matrixes, α0 = r ε is x x
i R R 0 along the x direction. Thus,
the static Rashba SOI coefficient which is proportional
to the perpendicular electric field ε resulting fromband
0 ∂A(t)
bending, rR is a material-specific prefactor [14]. Under E(t)=−ex ∂t =Fa(t)ex
equilibriumconditions,spintransportinsuchadevice is
investigated extensively [7, 14]; in brief, one chooses for where F is the peak amplitude of the electric field and
1D QW the z axis as the spin quantization axis (Fig1), a(t) describes the pulse profile. The single particle
meaning that Hamiltonian reads
HR =α0Rσzkx =µBσzB˜z, H = π2 +V(y,z) eΦ+ α0R(σ π) , (2)
2m∗ − ~ × y,kz=0
where µ is Bohr’s magneton, B˜ = α0k /µ is an ef-
fective mBagnetic field along z. Tzhis reRsulxts iBn the spin where π = p+eA(r,t), and p = i~▽ is the momen-
−
splitting2α0k betweencarriersinjectedwithspinpolar- tum operator. The second term in Eq. (2) is the QW
R x
izationparallelor antiparallelto z. The phase difference confinement potential, the third term is from the scalar
while passing through the length L is potential of the pulse field, and the fourth term in Eq.
(2) is the Rashba SOI. We write H =H0+Ht, with
2m∗α0L
△θ0 =(kx↑−kx↓)L= ~2R . H0 =p2/2m∗+V(y,z)+(α0R/~)(σ×p)y,kz=0,
The eigenenergies and the eigenstates are respectively and
Ekµ = ~22mk2 +µα0Rk Ht +=((eeα/20m/~∗))((σp·AA+)A. ·p)+(e/2m∗)A·A−eΦ(r)
R × y
and
H0isthepulse-freesingleparticleHamiltonian,andHtis
1 thetime-dependentpart. ChoosingagaugewhereΦ=0;
kµ = eikx η
| i √L | µi p A+A p=2Axpx and (σ A)y =σzAx results in
· · ×
where ηµ is the spin states. This is the original idea of e e2 eα0
| i Ht = A p + A2 + Rσ A . (3)
the Ref. [7]. m∗ x x 2m∗ x ~ z x
Considering the spins to be injected aligned along the x
or y directions, they precess around B˜ . In Heisenberg Definingthespinorfieldoperatorasψˆ(x)= c kµ ,
z kµ kµ| i
picture, the spin operators vary with time as wherec istheannihilationoperatorforthestates kµ ,
kµ
P | i
in the second quantization form, we obtain
σ˙ (t)= ω σ (t),
x(y) ∓ kx y(x) e~ e2 eα0
where Ht = m∗Axk+ 2m∗A2x+µ ~RAx c†kµckµ. (4)
kµ (cid:18) (cid:19)
2α0k X
ω = R x
kx ~ The total Hamiltonian reads H = εc (t)c† c ,
kµ kµ kµ kµ
with the time-dependent transient energy (measured
is the precession frequency. Therefore, P
with respect to the ground state of V(y,z))
σ±(t)=σx±iσy =σ±(0)e±iωkxt. 1 p p2
εc (t)= [(~k+eA (t)+µ so)2 so], (5)
Letusspecifytheinitialorientation,sayσ (0)=0,there- kµ 2m∗ x 2 − 4
y
fore
and
σx(t)=σx(0)cos(ωkxt) and σy(t)=σx(0)sin(ωkxt). p =~k = 2m∗α0R. (6)
so so ~
Aninitialspinalongthexdirectionrotatesanticlockwise
with the angular frequency ωkx. The accumulated angle To obtain the momentum distribution upon a short
throughthe length L is △θ0 = 2m∗~α20RL, which is exactly pulse application (say at t = 0) we may proceed as
equal to the phase shift for the spin along z. in [6, 12] and expand the single-particle excited state
3
Ψ (x,t) that starts from the state labeled by k µ tonian is
k0µ0 | 0 0i
in terms of the stationary eigenstates π2 α˜(t)
H = +V(r) eΦ+ (σ π) , (9)
2m∗ − ~ × y,kz=0
Ψ (x,t)= dkC (k ,µ ,t)kµ e−iEkµt/~. (7)
k0µ0 kµ 0 0 | i where
µ Z
X
α˜(t)=α0 +αt (t),
For a sudden-excitation, Ψ (x,t=0+) right after the R R
k0µ0
pulseevolvesfromthestatebeforethepulseΨk0µ0(x,t= and α0R is the static Rashba SOI.
0−) as [6]
αt (t)=r E(t)
R R
Ψ (x,t=0+)=eixp¯Ψ (x,t=0−).
k0µ0 k0µ0 is proportional to the HCP electric field. The time de-
Thus pendent part of the Hamiltonian is
Ckµ(k0,µ0,t=0+)=Ck−k¯,µ(k0,µ0,t=0−), (8) Ht = 2em2∗A2y + αtR~(t)(σ×p)y. (10)
where
The vector potential in the canonical momentum in the
tp/2 second term in Eq. (10) does not couple to the electric
~k¯ =p¯= eF a(t′)dt′ fieldoftheHCP.ThetotalHamiltonianisH =H0+Ht.
−
Z−tp/2 The first term in Eq. (10) results in a phase shift for all
is the momentum boost delivered to the carrier by the states in all subbands, an effect which is unimportant
pulse (the second weak and long half cycle of the pulse for the following discussion and hence we ignore it and
acts as a weak DC off-set field). For t < 0, eq.(7) may consider the total Hamiltonian
stand for the injected electron state in terms of the sta- ~2k2 α˜(t)
tionary states. For Ck′µ′(k,µ,t<0)=δ(k−k′)δµµ′, the H = 2m∗x + ~ (σ×p)y. (11)
injectedelectronoccupiesasingleeigenstate. Inthiscase
We note, the pulse field does not change the quantum
the wave function after the pulse is
number k nor the spin states. The time evolution oc-
Ψk0µ0(x,t>0)=ei(~k0+p¯)x|ηµ0ie−iE(k0+k¯)µ0t/~/√L. cisur1sDonplayrianmtehteerpasrpaamcee.terWspitahcevaspryeicnifigedαtb(yt)α˜(tth)ewshpiicnh-
R
The energy after the pulse is
dependent potential is changed and so does the energy.
1 p p2 To be specific let us consider the GaSb/InAs/GaSb sys-
εc (t>0)= (~k+p¯+µ so)2 so . tem with the parameters given in Ref. [15]. For InAs
kµ 2m∗ 2 − 4
(cid:20) (cid:21) QW with a width of 6.28 nm the second energy sub-
If Ck′µ′(k,µ,t < 0) models a Gaussian wave packet cen- band is separated from the ground state by ≈ 700 meV
tered at k for the injected state we infer from Eq. (8) [15]. To inspect the effect of the pulse field we show
0
that rightafter the pulse the shape ofthe wavepacketis in Fig. 2 the instantaneous energy spectrum. As evi-
maintained while its central momentum is shifted by p¯. dent from this figure the highest achievedenergy level is
We conclude: well below the first excited subband and hence we only
i)Thepulsefielddeliversatransientmomentumtransfer needtoconsidertheintra-banddynamicsinthefirstsub-
(which is proportional to the momentary vector poten- band. The behavior of the instantaneous energy for a
tial) and a net momentum given by the field-amplitude state kµα˜(t) is such that for positive k the energy of
| i
time-integrated over the field duration, for harmonic the spin up carrier is raised while the spin-down energy
fields this quantity vanishes whereas for HCP it is finite is lowered in the first quarter of the mono-cycle pulse;
and is equal to p¯. the opposite happens in the second quarter cycle. In
ii) This momentum boost is experienced by all the elec- the second half of the mono cycle the spin-resolved en-
trons speeding up the device operation. ergies evolves in an opposite way to the first half. As
iii) The phase difference ∆θ is maintained as for the the shift is determined by the magnitude of the elec-
0
static case, i.e. the operation speed is changed by an tric field peak, the very weak and long (on the trans-
amount proportional to the field strength while the spin port time) tail of HCP has a minor effect. From Fig. 2
coherence is unchanged, a fact exploitable for the real- we infer that the pulse results in a time-varying poten-
ization of an ultra-fast SFET. tial and hence an oscillationof all electrons in the Fermi
sphere in energy space. No holes are generated. The
spin operators develop as σ˙ (t) = iω˜ (t)σ (t), where
THE SECOND DYNAMIC CASE ω˜ (t)=2α˜(t)k /~. Hence,±we find∓ kx ±
kx x
IftheelectricfieldpolarizationoftheHCPisalongthe
σ (t)=σ (0)exp i ω˜ (t)dt . (12)
y direction, i.e. perpendicular to the 2DEG the Hamil- ± ± (± kx )
Z
4
2α0k = 4meV. We find then ω 2πps−1. The
40 40 R f kx ≈
Fermi velocity is 0.4µm/ps. Thus in 0.2 µm 1D QW
30 t=0 t=0.2 tp 30 thereare20electronsdistributedandthe transporttime
t for the electrons at the Fermi velocity is about 500
20 20 tr
fs. In what follows we based our discussion on these re-
V) 10 10 alistic numbers. HCPs with peak field of up to several
e
m (a) (b) hundreds of kV/cm and duration in the picosecond and
ergy ( 400 040 subpicosecond regimes can be experimentally generated
En 30 t=0.5 tp t=0.8 tp 30 [p1u0l]s.esNaosvsehloprrtinacsip0l.e1sfsalalonwd twhiethgeinnteernastiitoinesoufputnoip1o0la16r
W/cm2 [11]. The pulse induced precession angle is
20 20
θ =k L (Fγ/ε ),
10 10 p so p 0
(c) (d)
0 0 where Lp is the length traveled by a particle with a mo-
-3 -2 -1 0 1 2 3-3 -2 -1 0 1 2 3 mentum ~kx within the pulse duration tp. The total
kx (unit in 1/10 nm) kx (unit in 1/10 nm) precessionangleaccumulatedwhiletraversingthelength
L is
FIG. 2. The instantaneous energy spectrum for
GaSb/InAs/GaSb system [15] at different time fraction △θ =△θ +θ ,
of thepulseduration tp. In (a) thestatic spectrum with SOI 0 p
splitting is shown. The parameters are taken from Ref. [15]:
mk ∗==20..00(5150mnm0,)α−0R1. =In0.(9b×-d)10t−h9eeVh.ocrmiz,on2tαa0Rlkdfas=h-d4o.0tmlienVes, wToheervealtuhaetefirtshtetaenrmglestθepmwsefrcoomnsitdheerstthaetircatRioashba SOI.
f
indicate the position of the Fermi level for equilibrium
θ L Fγ
system, and the vertical dash-dot lines mark the positions of λ= p = p . (14)
the Fermi wave vectors. The solid (dash) curves correspond △θ0 L ε0
to the spin up (down) state. The other parameters are
selected as εF0 =3, a(t)=sin(πt/tp) for 0≤t≤tp. haiv)eSγingle1H. CTPh.eFsotrattipc=fie2ld0sfsεwaerneottyeptipca≪llyttorf, athnedowre-
0
≈
derofseveralkV/cm,forexample,inGaAs/Al Ga As
0.3 0.7
Defining τ = (F/ε )t γ, where γ = a(ξ′)dξ′, and quantumwell[14]. F canbe generatedwithseveralhun-
0 p
ξ′ =t′/tp. we o−btain ω˜kx(t)dt=ωkxt+Rθp, where dasre1d0s0o(fwkitVh/ocumt.inTduhceirnegforinet,eFr-/sεu0bbcaanndbteratnusnietdionass).high
Rθ = ω τ (13)
p − kx L =~k t /m∗
p x p
For an injected carrier with a spin polarization vec-
is about 8 nm during t = 20fs. Therefore, λ 4. The
tor along z, the wave function right after the pulse p
≈
induced accumulatedangle θ is a swift processionangle
Ψ (x,t>0) evolves from the stationary state before p
kµ0µ0 transfer. If the initial injected spin is in x direction it
the pulse Φ (x,t<0) as
kµ0µ0 suddenlyrotatesanticlockwiseoveranangleθ uponap-
p
Ψ (x,t>0)=e−iµ0θp/2Φ (x,t<0), plyingthepulse. Thisconclusionisexploitabletorealize
kµ0µ0 kµ0µ0 ananosizeSFET.Spinswithhigherdriftvelocityexperi-
meaning that the pulse causes a phase splitting in encealargerangulartransferduringtheperiodt . Thus,
p
the spin up and down channels. The phase difference a wave packet of spins that is initially polarized in the
betweenupanddownspinsis equalto theinducedangle samedirectionbutcontaindifferentvelocitycomponents
rotation in Eq. (13). will spread over a range of angles after scattering from
the pulses. This can be compensated by operating the
device in the linear response regime (small bias) where
electrons at the Fermi surface dominate the transport,
OPTICAL SFET
i.e. k =k . Acontourplotofthe ratioλ,asintroduced
x f
in eq.(14), as function of the external field parameters is
In the static case a 0.67 µm long 1D quantum wire shown in Fig. 3(a). The ratio increases with increasing
is needed to reach the phase shift △ θ = π in 2D F and t .
0 p
In Ga As [16]. While a shorter 1D quantum wire ii) A train of HCPs: We apply a pulse train with a
x 1−x
is sufficient to reach π phase shift with length 0.2 µm total duration t = n(t + t ), where n is the number
t p s
in GaSb/InAs/GaSb system [15], since the Rashba SOI of HCP peaks, t is the time interval between two con-
s
is larger, i.e. α0 0.9 10−9eVcm. According to secutive HCPs. m = t /t is the number of electrons
R ≈ × t tr
Ref. [15] the charge density is n = 1012cm−2, and passing through the device during t . We define for a
t
5
tuminterferencesin(the higher-energy)inter-band,one-
80 80
(a) 10 (b) 100 photon, two-photon absorption [18]. Our photo-induced
s) 60 9 ) 60 88
f 8 s 75 current for the first dynamic is sizable: For the above
t (p40 56 t (fp40 5603 device based on GaSb/InAs/GaSb the static current is
4 38
3 25 usually 1.07 µA. The ratio between the induced and
20 01 20 013 the stati≈c current is ∆ = 2p¯/~kf which varies in a large
range. For F=10 kV/cm and t = 1 ps we find ∆ = 7.6
20 40 60 80100 20 40 60 80100 p
F/ 0 F/ 0 which means an induced current of ≈8µA.
FIG. 3. A contour plot for the ratio λ, as given by eq.(14),
as a function of the magnitude and of the duration of the
external field for a single HCP (a) and for a train of HCPs CONCLUSIONS
(b). In(a),theFermivelocity is chosen as0.4 µm/ps, L=0.2
µm, γ is set at 0.5. In (b), ts=0.5 ps, ttr =0.5 ps, γ ≈1.
In summary we studied theoretically the photo-
inducedspindynamicsinaquantumwirewithaRashba
spin orbit interaction. For an efficient and a sub-
single electron an averaged γ¯ = nγ/m and L = L. For
p picosecond control of the spin dynamics the pulses have
t = 1 ps, t = 0.5 ps, t = 125 ps, n = 50, m = 250,
p s t tobeshapedappropriatelysuchthatineffectalinearmo-
γ 1, thus γ¯ 0.2 and λ 20. In this case, spins with
≈ ≈ ≈ mentum boost is transferred to the charge carrier. For
highervelocitieswillpassthroughthedevicefasterwhile
linear polarized photons we find that if the photon po-
rotating faster. The additional precession angle induced
larization axis is along the wire’s direction, the phase
by the pulse field is comparable to the static one or may
coherence in different spin channels is maintained, even
evenbelarger. Toshowthebehaviormoreclearly,acon-
though the charge carriers are speeded up. In the case
tourplotofλwiththe tuningparametersoftheexternal
that the photon pulse polarization is perpendicular to
field is given in Fig. 3(b). Since the transport time is
the wire we predicta spin precessioncomparableto that
fixed, increasing t decreases the ratio λ which is in a
p inducedbythe RashbaSOI.Thephoton-inducedpreces-
contrast to the single HCP case. However, a larger λ is
sion is tunable in a considerable range by scanning the
alsoacheivedviaincreasingtheelectricfieldstrength. As
parameters of the pulse electric field.
discussedforthestaticcase,aninjectedspinpolarizedin
This work was supported by the cluster of excellence
positiveornegativez directionexperiencesaphaseshift,
”NanostructuredMaterials” of the state Saxony-Anhalt,
while passingthroughthe lengthL,thatis exactlyequal
and the DFG, Germany.
to the precession angles for the case of a spin injected
polarized along x or y.
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