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Datta‑and‑Das spin transistor controlled by ahigh‑frequency electromagnetic field
Sheremet, A. S.; Kibis, O. V.; Kavokin, A. V.; Shelykh, I. A.
2016
Sheremet, A. S., Kibis, O. V., Kavokin, A. V., & Shelykh, I. A. (2016). Datta‑and‑Das spintransistor controlled by a high‑frequency electromagnetic field. Physical Review B, 93,165307‑.
https://hdl.handle.net/10356/83875
https://doi.org/10.1103/PhysRevB.93.165307
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PHYSICAL REVIEW B 93, 165307 (2016)
Datta-and-Das spin transistor controlled by a high-frequency electromagnetic field
A. S. Sheremet,1,2 O. V. Kibis,3,4,5,* A. V. Kavokin,2,6,7,8 and I. A. Shelykh1,4,5
1National Research University for Information Technology, Mechanics and Optics (ITMO), 197101 St.-Petersburg, Russia2Russian Quantum Center, Novaya 100, 143025 Skolkovo, Moscow Region, Russia
3Department of Applied and Theoretical Physics, Novosibirsk State Technical University, 630073 Novosibirsk, Russia4Science Institute, University of Iceland IS-107, Reykjavik, Iceland
5Division of Physics and Applied Physics, Nanyang Technological University, 637371 Singapore6School of Physics and Astronomy, University of Southampton, SO17 1BJ, Southampton, United Kingdom
7Spin Optics Laboratory, St.-Petersburg State University, 198504 Peterhof, St.-Petersburg, Russia8CNR-SPIN, Tor Vergata, viale del Politechnico 1, I-00133 Rome, Italy
(Received 4 March 2016; published 28 April 2016)
We developed the theory of spin dependent transport through a spin-modulator device (so-called Datta-and-Dasspin transistor) in the presence of a high-frequency electromagnetic field (dressing field). Solving the Schrodingerproblem for dressed electrons, we demonstrated that the field drastically modifies the spin transport. In particular,the dressing field leads to renormalization of spin-orbit coupling constants that varies conductivity of the spintransistor. The present effect paves the way for controlling the spin-polarized electron transport with light inprospective spin-optronic devices.
DOI: 10.1103/PhysRevB.93.165307
I. INTRODUCTION
One of most excited fields of modern condensed-matterphysics is the physics of spin-based electronic devices(spintronics), which is expected to play a crucial rolein the realization of high-performance information processing[1–4]. There are various kinds of logic devices based onspin-polarized electron transport in ferromagnets and semi-conductors [5–7]. In ferromagnetic materials, the spin transfercan be manipulated by an external magnetic field. As tosemiconductor structures, their spin properties are effectivelycontrolled with the spin-orbit interaction. Namely, the Rashbamechanism of spin-orbit coupling (based on the structureinversion asymmetry) and the Dresselhaus mechanism of spin-orbit coupling (originating from the bulk inversion asymmetry)allow tuning the spin states of electrons in semiconductorswithout external magnetic fields [8–10].
The concept of semiconductor spin-transfer device (spintransistor) was put forward by Datta and Das in theirpioneering work [11]. Constructively, Datta-and-Das spintransistor consists of two magnetized ferromagnetic electrodesand a semiconductor channel between them (see Fig. 1). Inthis design, the ferromagnetic electrodes 1 and 3 are usedto inject and collect spin-polarized electrons, whereas thesemiconductor area 2 serves to rotate electron spin via thespin-orbit coupling. As a result, the transmissivity of the spintransistor depends on the strength of the spin-orbit coupling inthe semiconductor channel [11–14]. Originally, the spin-orbitcoupling was proposed to be controlled with a gate voltageapplied to the semiconductor channel [11]. In the present study,we propose the alternative method of optical control.
It is well known that light is an effective tool to manipulateelectronic properties of various quantum systems in theregime of strong light-matter coupling. Since the stronglycoupled system “electron + electromagnetic field” should be
considered as a whole, the bound electron-field object—“electron dressed by electromagnetic field” (dressedelectron)—became a commonly used model in modern physics[15,16]. It has been demonstrated that a dressing field cruciallychanges physical properties of conduction electrons in variouscondensed-matter structures, including bulk semiconductors[17–19], quantum wells [20–25], quantum rings [26–28],graphene [29–38], etc. Therefore, one can expect that spin-tronic devices are strongly affected by a dressing field aswell. However, a consistent theory describing spin transport inelectronic devices subjected to electromagnetic radiation wasnot, to the best of our knowledge, elaborated on up to now.
II. MODEL
Let us consider a semiconductor channel of the spintransistor irradiated by a linearly polarized electromagneticwave (dressing field) propagating perpendicularly to thechannel (see Fig. 1). In what follows, temperature T is assumedto be close to zero. As a consequence, only electrons situatedin the close vicinity of the Fermi energy εF contribute toconductivity of the channel.
The Hamiltonian of the electrons reads
He = 1
2ms
(�k − eA)2 + α[σ × (�k − eA)]z
+β[σx · (�k − eA)x − σy · (�k − eA)y], (1)
where ms is the effective electron mass in the semiconductor,σ = (σx,σy,σz) is the Pauli matrix vector, k = (kx,ky) is theelectron wave vector, A = (0,[E/ω] cos ωt) is the vector po-tential of the wave, ω is the wave frequency, E is the amplitudeof electric field of the wave, and α and β are the Rashbaand Dresselhaus spin-orbit coupling constants, respectively. Inorder to describe the spin transistor, we specifically considerelectrons propagating in the one-dimensional channel alongthe x axis. Therefore, the Hamiltonian (1) can be rewritten as
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SHEREMET, KIBIS, KAVOKIN, AND SHELYKH PHYSICAL REVIEW B 93, 165307 (2016)
FIG. 1. Sketch of the spin transistor irradiated with an elec-tromagnetic (EM) wave. Under the applied voltage V the sourceferromagnetic contact 1 injects the spin-polarized current j into thesemiconductor channel 2 along the x axis. The drain ferromagneticcontact 3 collects the current. The linearly polarized EM wavepropagates perpendicularly to the channel and its electric field E
is directed along the y axis.
He = H0 + Hk , where
H0 = e2E2
2msω2cos2 ωt + (βσy − ασx)
eE
ωcos ωt (2)
is the Hamiltonian describing the dressed electron state atk = 0 and
Hk = �2k2
x
2ms
+ (βσx − ασy)�kx (3)
is the Hamiltonian of “bare” electron in the channel. Solv-ing the nonstationary Schrodinger equation, i�∂�0/∂t =H0�0, one can easily obtain the exact eigenspinors of theHamiltonian (2),
�±0 = 1√
2
(1
±α−iβ
γ
)e−i e2E2
4ms �ω2 (t+ sin 2ωt2ω )
e±i
eEγ
�ω2 sin ωt, (4)
where γ =√
α2 + β2 is the effective spin-orbit couplingconstant which takes into account both the Rashba spin-orbitinteraction mechanism and the Dresselhaus one. Since theeigenspinors (4) form the complete basis of the consideredelectron system, we can seek eigenspinors of the full Hamil-tonian He = H0 + Hk as
�k(t) = a+(t)�+0 + a−(t)�−
0 . (5)
Substituting the expansion (5) into the Schrodinger equationwith the full Hamiltonian He = H0 + Hk , we arrive at the twodifferential equations describing the quantum dynamics of theconsidered system:
i�a±(t) =[
�2k2
x
2ms
± 2αβ
γ�kx
]a±(t)
∓iα2 − β2
γ�kxe
∓i2eEγ
�ω2 sin ωta∓(t). (6)
It follows from the Floquet theory of periodically drivenquantum systems [39–41] that the wave function (5) can bewritten in the form �k(t) = e−iεt/�(t), where the function(t) = (t + 2π/ω) periodically depends on time and ε
is the quasienergy (energy of dressed electron). Since the
quasienergy plays the same role in periodically driven quantumsystems as the usual energy in stationary ones, the presentanalysis is aimed at finding the energy spectrum ε(kx) forelectrons propagating through the irradiated semiconductorchannel. The periodicity of the function (t) allows seekingthe coefficients a±(t) in Eqs. (6) as the Fourier expansion,
a±(t) = e−iεt/�
∞∑n=−∞
a±n einωt . (7)
As to the exponents on the right sides of Eqs. (6), they canbe transformed with the use of the Jacobi-Anger expansion,eiz sin φ = ∑∞
−∞ Jn(z)einφ , where Jn(z) is the Bessel functionof the first kind. If the dressing field is both high-frequencyand nonresonant, the rapidly oscillating terms, einωt , makea negligibly small contribution into the quantum dynamicsequations (6). Physically, this is a general rule for pe-riodically driven quantum systems (see, e.g., the qualita-tively similar analysis for various dressed nanostructures inRefs. [23–25,30,34]). Therefore, the high-frequency harmon-ics einωt with n �= 0 can be omitted in Eqs. (6). As aconsequence, Eqs. (6) can be reduced to the expression(
�2k2
x
2ms
± 2αβ
γ�kx − ε
)a±
0 ∓ i�kx
α2 − β2
γ
× J0
(2eEγ
�ω2
)a∓
0 = 0. (8)
Solving Eqs. (8), we obtain the energy spectrum of dressedelectrons,
ε(kx) = �2k2
x
2ms
± γ �kx, (9)
where γ =√
α2 + β2, α and β are the spin-orbit couplingconstants renormalized by a dressing field:
α = α
[J0
(2eEγ
�ω2
)+ 2β2
γ 2
(1 − J0
(2eEγ
�ω2
))],
β = β
[J0
(2eEγ
�ω2
)+ 2α2
γ 2
(1 − J0
(2eEγ
�ω2
))]. (10)
Physically, the quantum dynamics equations (8) exactlycorrespond to the stationary Schrodinger equation with theeffective Hamiltonian of dressed electrons in the irradiatedsemiconductor channel,
Hs = �2k2
x
2m+ (βσx − ασy)�kx, (11)
which results in the same energy spectrum (9). It should benoted that the Hamiltonian of dressed electrons (11) exactlycoincides with the well-known Hamiltonian for “bare” elec-trons (3) with the formal replacement α,β → α,β. Conse-quently, all physical quantities describing dressed electronscan be easily derived from the “bare” ones with the samereplacement. The energy spectrum of dressed electrons (9)turns into the well-known spin-split energy spectrum of “bare”electrons in the absence of the dressing field (E = 0). Itshould be stressed that the spin-orbit constants of dressedelectrons differ from the same constants for “bare” electronsby the Bessel-function factor depending on the dressing field
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DATTA-AND-DAS SPIN TRANSISTOR CONTROLLED BY A . . . PHYSICAL REVIEW B 93, 165307 (2016)
amplitude E and the frequency ω [see Eqs. (10)]. In turn, theinfluence of the dressing field on the renormalized constants α
and β depends on the “bare” constants α and β. Particularly,the influence vanishes in the case of |α| = |β|.
III. SPIN TRANSPORT
In the original proposal for the Datta-and-Das spin tran-sistor [11], the semiclassical model was used. Within thisapproach, the spin-orbit interaction was treated as an effectivemagnetic field which depends on the electron wave vectoralong the semiconductor channel, kx , and is perpendicularto the magnetization of ferromagnetic contacts. The spin-polarized electrons injected to the semiconductor channel fromthe source ferromagnetic contact are affected by the effectivefield. As a consequence, the spin of the electrons undergoesthe rotation, whose angle at the drain ferromagnetic contact isθ = 2msγL/�, where L is the length of the channel. Assumingthe semiconductor channel to be in the ballistic regime, theamplitude of their probability to enter into the drain is equal tocos θ/2. Controlling the spin-orbit constant γ =
√α2 + β2
with external actions, one can control the conductance ofthe spin transistor. Particularly, it follows from Eq. (10) thatthe spin-orbit coupling can be controlled with an irradiation.Performing the formal replacement, γ → γ , in the knownexpressions describing the spin transistor [11], we obtain thesemiclassical conductance of the irradiated transistor,
G =(
e2
2h
)[1 + cos
(2msγL
�
)]. (12)
To carry out the full quantum treatment of the spin transport,let us divide the model structure into the regions 1, 2, and 3,where the regions 1 and 3 correspond to the ferromagneticcontacts and the region 2 corresponds to the semiconductorchannel (see Fig. 1). The electronic properties of the irradiatedsemiconductor region 2 are described by the Hamiltonian(11), whereas the Hamiltonian of ferromagnetic regions 1and 3 can be written as Hf = p2
x/2mf − σz�/2, where px
is the operator of electron momentum along the x axis, mf
is the electron effective mass in the ferromagnetic, and � is theZeeman spin splitting of electron states in the ferromagnetic.Then the spinors describing the spin transistor in the threeregions, ψ1,2,3(x), are given by
ψ1(x) |x<0 = (eiq+x + C1e
−iq+x)(1
0
)+ C2e
q−x
(01
),
ψ2(x) |0<x<L =(C3e
ik(1)+ x + C4e
ik(1)− x
)(1
β−iα
γ
),
+(C5e
ik(2)+ x + C6e
ik(2)− x
)(1
β+iα
γ
)
ψ3(x) |x>L = CT eiq+x
(10
)+ C7e
−q−x
(01
), (13)
where �k(1)± = ±√
2msε + m2s γ
2 − msγ and �k(2)± =
±√2msε + m2
s γ2 + msγ are the electron momenta
in the semiconductor channel at the energy ε,�q± = √
2mf (�/2 ± ε) are the electron momenta in
the ferromagnetic contacts at the same energy, and electronsin the contacts are assumed to be fully spin polarized(ε < �/2). In order to find the transmissivity of the spintransistor, CT , we have to use the conventional continuityconditions at the borders of the regions [42],
ψ1(0) = ψ2(0), ψ2(L) = ψ3(L),
v1ψ1(0) = v2ψ2(0), v2ψ2(L) = v3ψ3(L), (14)
where v = i[H,x]/� is the velocity operator in the differ-ent regions: v1,3 = px/mf and v2 = px/ms + (βσx − ασy).Substituting the transmission amplitude CT found fromEqs. (13)–(14) into the Landauer formula,
G = e2
h|CT |2, (15)
we can calculate the quantum conductance of the spintransistor G.
IV. DISCUSSION AND CONCLUSIONS
Discussing the limits of applicability of the developedtheory, we have to underline that the considered dressingfield must not be absorbable by electrons. This is importantsince the absorption of radiation could be detrimental when itgenerates intraband excitations (heating the Fermi sea). It iswell known that the intraband absorption of radiation by freeconduction electrons is forbidden by the energy conservationlow and the momentum conservation low. Therefore, theintraband absorption of THz radiation can only take place dueto the electron scattering (collisional absorption). However,the semiconductor channel of the Datta-Das spin transistoris assumed to be in the ballistic regime. Since scatterers areabsent in the ballistic channel, the radiation-induced intrabandexcitations can be neglected. As to the optical absorption ofthe field by electrons, it can be neglected if the field frequencyω lies far from resonant electron frequencies corresponding tointerband (intersubband) electron transitions. Therefore, thefrequency ω should be low compared to the characteristicoptical frequencies. On the other hand, the frequency ω
should be high enough to satisfy the condition ωτ � 1,where τ ∼ L/
√2ε/ms is the characteristic transit time of
electron in the channel. Physically, this condition means thatthe dressing field experiences multiple oscillations while anelectron flies ballistically from the source contact 1 to thedrain contact 3 (see Fig. 1). Moreover, the photon energy�ω must be larger than the characteristic spin-orbit couplingenergy, εso ∼ γ
√2msε, in which case it is safe to neglect the
rapidly oscillating terms in the quantum dynamics equations(6). Therefore, the dressing field must be both high frequencyand nonresonant.
At low temperatures, the electrons contributing to thespin transport in the channel are situated at the Fermi level.Therefore, the energy ε in all expressions above coincideswith the Fermi energy in the semiconductor channel εF . Thetemperature T should be low compared to both the Fermi en-ergy εF and the spin-orbit coupling energy, εso ∼ γ
√2msεF .
It should be noted also that employing the concept of dressingfield we assume that �/τϕ < g0, where τϕ is the characteristicdephasing time, and g0 = eEγ/ω is the characteristic constantof light-matter coupling in the semiconductor channel. In the
165307-3
SHEREMET, KIBIS, KAVOKIN, AND SHELYKH PHYSICAL REVIEW B 93, 165307 (2016)
FIG. 2. The quantum conductance of the irradiated spin transistorG as a function of the irradiation intensity I and the length ofthe semiconductor channel (path length) L for the Rashba andDresselhaus spin-orbit coupling constants α = β/4 = 5 × 103 m/s.
considered case of ballistic channel, the dephasing time canbe estimated as τϕ ∼ τ ∼ L/
√2ε/ms . Therefore, the field
amplitude E should be large enough to satisfy the conditionE > �ω
√2ε/ms/(eγL).
Numerical calculations of the conductance of the irradiatedspin transistor G with use of the semiclassical Datta-and-Dasformula (12) and the quantum Landauer formula (15) arepresented in Figs. 2 and 3 for the field frequency ω = 1 THz,the Fermi energy εF = 10 meV, the Zeeman splitting inferromagnetic contacts � = 40 meV, the electron effectivemasses mf = me (in the ferromagnetic contacts) and ms =0.067me (in the GaAs semiconductor channel), where me is themass of free electron. The difference between the quantum andsemiclassical results is due to the multiple electron scatteringbetween the source and drain contacts, which is not accountedfor within the semiclassical approach. The multiple scatteringleads to formation of the standing electron waves between thecontacts, which have the same physical nature as eigenmodesof the Fabry-Perot resonator. As a consequence, the amplitudeof quantum conductance experiences additional modulation(see Fig. 3).
The plotted dependence of the conductance G on the irradi-ation intensity I originates from the field-induced modificationof the spin-orbit coupling constants (10). Conventionally,only the gate voltage was used before to tune the constants[14,43–45]. In particular, the gate-controlled scheme of thespin transistor was proposed by Datta and Das [11]. Thepresent theory indicates that the spin-orbit coupling can beeffectively controlled with an external optical field as well.
To summarize, the strong coupling between electron spinsand a high-frequency electromagnetic field is an efficientcontrol tool for the spin precession rate in spin transistors.
1.0
0.8
0.6
0.4
0.2
0.00 10 20 30 40 50 60 70
(а)
1.0
0.8
0.6
0.4
0.2
0.00 20 40 60 80 100 120
(b)
Irradiation Intensity, I (W/cm2)
Con
duct
ance
, G (e
2 /h)
Con
duct
ance
, G (e
2 /h)
α = β/4α = β/2α = 3β/4
FIG. 3. The conductance of spin transistor G as a functionof the irradiation intensity I for the length of the semiconductorchannel L = 2.3 μm: (a) semiclassical conductance (dashed line)and quantum conductance (solid line) for the Dresselhaus constantβ = 2 × 104 m/s and the Rashba constant α = 0; (b) quantumconductance for the Dresselhaus constant β = 2 × 104 m/s and thedifferent Rashba constants α.
Since the dressing field renormalizes spin-orbit coupling con-stants, the variation of the dressing field intensity drasticallymodifies the spin transport properties of the transistors. Thus,the predicted effect opens the way to realization of new spin-optronic devices. Since light-controlled electronic devices aretypically much faster than those of electrically controlled, theoptically-controlled Datta-and-Das spin transistor is expectedto be faster than the gate-controlled one.
ACKNOWLEDGMENTS
This work was partially supported by FP7 IRSES projectsPOLATER and QOCaN, the Rannis project BOFEHYSS,Singapore Ministry of Education under AcRF Tier 2 GrantNo. MOE2015-T2-1-055, the Russian Target Federal Program“Research and Development in Priority Areas of Develop-ment of the Russian Scientific and Technological Complexfor 2014–2020” (Project No. RFMEFI58715X0020). A.S.S.thanks the University of Iceland for hospitality.
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