Radiation-induced resistance oscillations in 2D electron systems with strong Rashba coupling

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by  Jesus Inarrea

Radiation-induced resistance oscillations in 2D electron systems with strong Rashba
coupling
Jesu´s In˜arrea1,2
1Escuela Polit´ecnica Superior,Universidad Carlos III,Leganes,Madrid,Spain
and 2Unidad Asociada al Instituto de Ciencia de Materiales, CSIC Cantoblanco,Madrid,28049,Spain.
(Dated: April6, 2017)
We present a theoretical study on the effect of radiation on the mangetoresistance of two-
7 dimensional electron systems with strong Rashba spint-orbit coupling. We want to study the in-
1 terplay between two well-known effects in these electron systems: the radiation-induced resistance
0 oscillations andthetypicalbeatingpatternofsystemswithintenseRashbainteraction. Weanalyt-
2
ically derive an exact solution for the electron wave function corresponding to a total Hamiltonian

r with Rashba and radiation terms. We consider a perturbation treatment for elastic scattering due
p to charged impurities to finally obtain the magnetoresistance of the system. Without radiation we
A recoverabeatingpatternintheamplitudeoftheShubnikovdeHassoscillations: asetofnodesand
antinodes in the magnetoresistance. In the presence of radiation this beating pattern is strongly
5
modified following the profile of radiation-induced magnetoresistance oscillations. We study their

dependence on intensity and frequency of radiation, including the teraherzt regime. The obtained
]
l resultscouldbeofinterestfor magnetotransport ofnonidealDiracfermions in3D topological insu-
l
a lators subjected to radiation.
h
- PACSnumbers:
s
e
m
I. INTRODUCTION experimental advances have been made ruling out some
. of the previous existing theories. For instance a very re-
t
a cent experiment by T.Herrmann15 concludes that RIRO
m Radiation-induced resistance oscillations (RIRO) and
aremainlydependentonbulkeffects in2DES.Thenthis
zero resistance states (ZRS)1,2 are remarkable phenom-
- experiment rules out theories based on the effect of ra-
d ena in condensed matter physics that reveal a novel sce-
diation on edge states16 or others based on ponderomo-
n nario in radiation-matter coupling. Those effects rise
tive forces17 excited by contactillumination, as main re-
o up when a high-mobility, typically above 106 cm2/Vs,
c sponsibles of RIRO. Another example, the experiments
two dimensionalelectron systemunder a moderate mag-
[ by Mani et al.,18–21 on power dependence rule out the
netic field (B) is illuminated with microwave (MW) or
theories that claimed a linear dependence of oscillations
3 terahertz (TH) radiation. The mangetoresistance (R )
xx on P8. The current or future theoretical models deal-
v of such systems (2DES) shows oscillations with peaks
9 ing with RIRO or ZRS have to confront with the avail-
and valleys at a certain radiation power. When increas-
7 ableexperimentalresultstoprovehowgoodandaccurate
ing power, the R oscillations increase in turn and at
3 xx they are. Another approachto prove theories is to check
0 high enough intensity the valleys turn into ZRS. The out if they are able to predict results on novel scenar-
0 radiation-induced resistance oscillations show character-
ios where there are no experiments carried out yet. For
. istic traits such as periodicity in the inverse of B1,2, a
1 instance RIRO and ZRS obtained on different semicon-
1/4 cycle shift in the oscillations minima3, sensitivity to
0 ductor platforms other than GaAs, the most extensively
7 temperature4,5andradiationpower6. Forthelattercase, platformusedinthiskindofexperiments. Themainrea-
1 a sublinear law is obtained for the dependence of RIRO
sonforusingGaAsisthatthisplatformoffersthehighest
: on the radiation power, R Pα, where P is the ra-
v xx ∝ mobility22 to date, ( 3.0 107 cm2/Vs), among differ-
i diation power and, interestingly, the exponent is around ent semiconductor he∼terost×ructures.
X 0.5. This clearly indicates a squared root dependence.
r
a Agreatnumberofexperimentsandtheoreticalmodels In this article we present a theoretical study on the
have been presented to date to try to explain such strik- effect of radiation on the magnetotransport in samples
ing effects. From a theoretical standpoint, we can cite with strong Rashba23–25 spin-orbit interaction (RSOI),
forinstancethedisplacementmodel7 basedonradiation- such as InAs. The interest by heterostructures with
assistedinterLandaulevelscattering,theinelasticmodel InAs is increasing very fast in the last years, on the one
based on the effect of radiation on the nonequilibrium hand for their technological impact being part of new
electron distribution function8. Being these two models electronic devices26. On the other hand from basic re-
the most cited to date, other models are more success- search standpoint in fields such as spintronics transis-
ful explaining the basic features of RIRO, such as the tors and the realisation of Majorana fermions27. Usu-
one by Lei et al9, or the radiation-driven electron or- ally, the electron mobility in InAs has been always be-
bit model10–14. We have to admit that to date there is low 1.0 106 cm2/Vs and RIRO can hardly be seen.
×
no universally accepted theoretical approach among the Yet, there have been published very recently experimen-
people devoted to this field. On the other hand, some tal results demonstrating that improving MBE growth
2

8
- f=149GHz
(a) e jump NO MW 7
X(0)=X2(0)-X1(0) 6

) 5
X1(0) X3(0) X2(0) (
X
fixed orbits EF RX4

with MW
(b) - 3
Y e jump
dark
G
R 2
E A sin(w )
N E
F 1
E
0.0 0.1 0.2 0.3 0.4 0.5 0.6

X1(t) X3(t) X2(t) B(T)
FIG. 2: Calculated magnetoresistance, Rxx, vs magnetic
orbits move field,B,fordarkandradiation offrequencyf =149GHz,in
ahigh-mobility2DESwithstrongRSOIofRashbaparameter
α = 0.6×10−11eV ·m. For the dark curve we obtain, as
MW averaged time
(c) expected, a beating pattern profile made up of a system of
nodes and antinodes. The radiation curve exhibits similar
X( ) A sin(w )
beating pattern but modulated by the rise of the system of

X2(0) peaks and valleys of RIRO.(T=1K).
X1(0) X3(0)=X2(0)-A sin(w )
intensity,2DESsystemswithRSOIcangiverisetoZRS.
Thus, we start off based on the previous theory of the
DISTANCE radiation-driven electron orbit10–14. This theory stems
from the displacement model7 and shares with it the in-
terplay between charged impurity scattering and radia-
tion to be at the heart of RIRO. As a reminder, the dis-
FIG. 1: Schematic figure describing elastic scattering be- placementmodelincludes,inaperturbativeway,electron
tween Landau states or orbits. a) Dark scenario. The scat- promotionwiththecorrespondingphotonsabsorptionto
tering jumptakes place between fixedorbits. τ is thetimeit higher Landau states and subsequent electron advances
takes the electron to complete the jump and reach the final and recoils via scattering to explain R peaks and val-
xx
orbit. b) Under radiation the scattering jump is between os- leys. Yet, the radiation-drivenelectronorbittheorydoes
cillating orbits according to Asinwt. c) Scattering underra-
not consider any radiation-assisted electron promotion.
diation for an averaged time scenario, i.e., steady state. The
As a further evolution of the displacement model, our
stationary advanced distance turns out to be smaller than
theoryproceedsinanalternativeapproachstartingfrom
the dark one. This corresponds to a valley in the radiation-
the exact solution of the time-dependent Schrodinger
induced magnetoresistance oscillations.
equation for an electron under magnetic field and radia-
tion. TheobtainedexactwavefunctionrepresentsaLan-
daustatewheretheguidingcenterisharmonicallydriven
techniques in quantum wells of InAs electron mobilities backandforthbyradiationatthesamefrequency. Inter-
can be dramatically increased28. They claimed a mo- estingly, the Landau states guiding center follow a clas-
bility close to 3.0 106 cm2/Vs28. Therefore, samples sical trajectory given by the solution of the driven clas-
×
of InAs, with strong RSOI, can now become reasonable sical oscillator. According to this theory, the interaction
candidates to observe RIRO. Then, we could study the ofthedrivenLandaustateswithchargedimpuritiesends
interplay of Rashba interaction and radiation in these up giving rise to shorter and longer average advanced
kindofsystems. Wecouldalsopredictthatwithsamples distance by the scattered electrons. These distance are
withevenhighermobilitiesandathighenoughradiation reflected on irradiated R as valleys and peaks respec-
xx
3
tively. Aninterestingtraitofthismodelisthatitencom- orbitcouplingparameterandw thecyclotronfrequency.
c
passesquantumandclassicalconceptsunliketheinelastic TheSchrodingerequationcorrespondingtothe Hamilto-
andthe displacementmodels thatarefully quantum. At nianH canbeexactlysolvedandtheresultingstatesare
0
this point we canhighlighta recenttheoreticalcontribu- labeled by the quantum number N. For N = 0 there is
tion by Beltukov29 et al. that is fully classical. It should only one level of energy given by E =(~w gµ B)/2.
0 c B
−
notbesurprisingtotakeintoaccountandincludetotally For N 1 we obtain two branches of levels labelled by
≥
orpartiallyclassicalapproachesexplainingRIROconsid- + and - and with energies:
ering the very low magnetic fields where this effect rise
up. 1 8α2
E =~w N (~w gµ B)2+ N (3)
We have added to the same total Hamiltonian of the N± c ± 2r c− B R2
radiation-driven electron orbit theory the Rashba inter-
action,solvingexactlythecorrespondingtime-dependent ~
where R is the magnetic length, R = . The corre-
Schrodingerequation. Inotherwords,wedevelopasemi- eB
classical model based on the exact solution of the elec- sponding wave function for the + branqch is,
tronic wave function in the presence of a static B with
1 cosθφ
iamndpoartpaenrttuRrbaashtiboan tcroeuaptlminegntinfoterrealcatsitnigc swcaitthterriandgiafrtoiomn ψN+ = Lyeikyy(cid:18) sin2θ2φNN−1 (cid:19) (4)
charged impurities. We consider this type of scattering
p
as the most probable at the low temperatures normally and for the - branch,
used in experiments. Following the previous model and
aopbptalyininagnaexBporletszsmioannnfortrRanxsxpworitthmroaddeialtwioenaarnedaRblSeOtIo. ψN− = 1Lyeikyy(cid:18)−csions2θθ2φφNN−1 (cid:19) (5)
Inthe simulationswe obtain,firstwithout radiation,the
p
well-knownbeatingpatternwiththesystemofnodesand 2√2α√N
where θ is given by θ =arctan R and φ is the
antinodes of the Rashba magnetoresistance30–39. Then, (cid:20)gµBB−~wc(cid:21)
we switched on light obtaining Rxx that shows a strong normalized quantum harmonic oscillator wave function,
deformation of the previous beating pattern where the i.e., Landau state, N being the corresponding Landau
nodes and antinodes follow the peaks and valleys of level index. According to these results the Rashba spin-
RIRO.Westudythedependenceonpowerandfrequency orbit interaction mixes spin-down and spin-up states of
including the terahertz regime. 2DES with RSOI share adjacent Landau levels to give rise to two new energy
similarHamiltonianwith3Dtopologicalinsulators,then branches of eigenstates of the Hamiltonian H .
0
we consider that the results that we present in this arti- To analyze magnetotransport in 2DES with RSOI we
cle could be of application in the field of 3D topological calculate the longitudinal conductivity σ following the
xx
insulators under the influence of radiation. Boltzmann transporttheory40–42, where σ is givenby:
xx
II. THEORETICAL MODEL σxx =e2 ∞dEρi(E)[∆X(0)]2WI df(E) (6)
− dE
Z0 (cid:18) (cid:19)
We consider a 2DES in the x y plane with strong being E the energy, ρi(E) the density of initial states
−
Rashba coupling subjected to a static and perpendic- and f(E) the electron distribution function. ∆X(0)
ular B and a DC electric field parallel to the x di- is the shift of the guiding center coordinate for the
rection. Using Landau gauge for the potential vector, eigenstates involved in the scattering event,
A = (0,Bx,0), the hamiltonian of such a system, H
0
reads: H = (H + H ) where the different compo-
0 B SO
nents of H0 are: ∆X(0)=[X2(0)−X1(0)]≃2Rc (7)
H = p2x + 1m w2(x X )2 σ + 1gµ σ B+ X2(0)andX1(0)beingtheguidingcentercoordinatesfor
B (cid:20)2m∗ 2 ∗ c − 0 (cid:21) 0 2 B z final and initial states respectively and Rc the cyclotron
1 E2 radius. WI istheremotechargedimpurityscatteringrate
−eEdcX0+ 2m∗ Bd2c σ0 (1) because we consider that at very low temperatures (T)
(cid:20) (cid:21) this is the most likely source of scattering for electrons
α
H = [σ p σ eB(x X )] (2) in highmobility 2DES.According to the Fermi’s Golden
SO −~ y x− x − 0
Rule W is given by
I
X is the center of the orbit for the electron spiral mo-
0
tion: X0 =− ~ekBy − meE∗wdcc2 , Edc is the DC electric field WI = 2~πNI|<ψf±|Vs|ψi± >|2δ(Ef −Ei) (8)
parallelto the(cid:16)x direction, (cid:17)σ0 stands for the unit matrix,
σ = (σ ,σ ,σ ) are the Pauli spin matrices, g the Zee- where N is the impurity density and E and E are the
−→ x y z I i f
man factor, µ the Bhor magneton, α the Rashba spin- energies of the initial and final states respectively. V
B s
4
is the scattering potential for charged impurities41. The where α˜2 = α2 m∗ . To obtain the new expression for
matrix element inside W can be expressed as40–42: E we have n~e4wglcected the Zeeman term considering
I N
that±at the magnetic fields used in experiments and in
<ψ V ψ > 2 = V 2 I 2δ (9) simulations it is much smaller than the Rashba term37.
| f±| s| i± | q | q| | if| ky′,ky+qy Expressing the density of states in terms of Dirac δ-
X
function we can write:
whereV = e2 ,ǫthedielectricconstantandq isthe
q ǫ(q+qs) s eB
Thomas-Fermi screening constant41. The integral Iif is ρi(E) = δ(E E0)+
given by: h −
eB ∞
Iif = 12 ∞[±Φf−1,Φf] eiq0xx 0eiqxx ±ΦΦii−1 dx h NX=1[δ(E−EN+)+δ(E−EN−)]
Z−∞ (cid:18) (cid:19)(cid:20) (cid:21)(10) (12)
where we have considered that at low or moderate B
(used in experiments of magnetoresistance oscillations)
2√R2α√N and then θ π where E0 = ~w2c. To do the sum in the expression of ρi
(cid:20)gµBB−~wc(cid:21)→∞ ≃ 2 we use the Poisson sum rules,
Tocalculatethedensityofstatesρ (E)ofa2DESwith
i
perpendicular B and RSOI we proceed starting off with ∞ 1 ∞ ∞ ∞
the expression of the energy of the states, eq. (3) that f(n)= f(0)+ f(x)dx+2 cos(2πsx)f(x)dx
−2
can be rewritten in a more compact way: n=1 Z0 s=1Z0
X X
(13)
and after some lengthy algebra we get to an expression
1
E =~w N +~2Nα˜2 (11) that includes the state broadening and reads43,44:
N c
± " ±r4 #
ρi(E)= πm~∗2 1+X± 1± 14 + 2~Ewα˜αc˜22 +~2α˜4Xs∞=1e−~swπcΓ cos2πs~Ewc +~α˜2±s14 + 2Ewαc˜2 +~2α˜4
 q     (14)
 
Γ, being the sateswidth. This equationis essentialinthe presentarticle because it revealsthe presence oftwo cosine
terms that could interfere. On the other hand, it also important to highlight that it is obtained from an expression
for the states energy that depends at the same time on the ”Landau” levelindex, both linearly andthrough a square
root. With this expression of the states density we recover the previous one obtained by Ch. Amann43 including
the states broadening. This last condition makes the expression much more useful to be used in theories explaining
experimental results on 2DES with RSOI. Considering that only electrons around the Fermi level participate in the
magnetotransport and the usual electron density used in these experiments28, it turns out that the E term is much
bigger than the Rashba term. Therefore, we can rewrite the expression of the density of states as:
ρi(E)= πm~∗2 1+ ∞ e−~swπcΓ cos2πs~Ewc +s14 + 2Ewαc˜2+cos2πs~Ewc −s14 + 2Ewαc˜2 (15)
 Xs=1 
    
Finally and after somealgebra we can write an expression for σxx, 
σxx = eπ2m~2∗(∆X0)2WI1+ ∞ e−~swπcΓsinXhSXS cos2πs~EwFc +s14 + 2EwFcα˜2+cos2πs~EwFc −s14 + 2EwFcα˜2
 Xs=1 
    (16)
 
whereE standsfortheFermienergyandX = 2π2kBT, to an interference effect that will become apparent as a
F S ~wc
k beingtheBoltzmannconstant. ToobtainR weuse beatingpattern. Thus,the physicaloriginofthe beating
B xx
the relation R = σxx σxx, where σ nie pattern, that has been experimentally observed, can be
xx σx2x+σx2y ≃ σx2y xy ≃ B trace back to the slightly different energies of the two
and σ σ , n being the 2D electron density. The
xx xy i
≪ eigenstates branches.
sumofcosinetermsintheexpressionofσ willgiverise
xx
5
The Hamiltonian H is the same as the one of the
0
surface states of nonideal Dirac fermions in 3D topo-
logical insulators. The only difference is that for the
latter the quadratic term is small compared to linear
term that it is the dominant when it comes to topo- 9
logical insulators. In real samples the surface states 103.08GHz
(a)
of 3D topological insulators are no longer described by 8
massles Dirac fermions. Experiments demonstrate im- 6.2mW
portant band bending and broken electron-hole symme- 7
try with respect to the Dirac point in the band struc- 10mW
4mW
ture of real 3D topologicalinsulators45,46. Therefore the 6
)
results presented above, especially the ones concerning 2.2mW
dark
density of states and states energy, could be of interest (X 5
in the study of magnetotransport in real 3D topological X 1mW
R
insulators. 4 0.5mW
If now we switch on radiation, first of all we have to
add to the Hamiltonian H a radiation term H and 3
0 R
then: H =(H +H +H ), where
0 B SO R
2 node
H = (x X )ε coswt X ε coswt (17) antinode
R 0 0 0 0
− − −
1
ε being the radiation electric field and w the corre-
0
sponding radiation frequency. H0 can again be solved 0
exactly10,11,47,48,andthesolutionfortheelectronicwave
0.0 0.1 0.2 0.3 0.4 0.5
function is made up, as above, of two states branches.
B(T)
The wave function for the + branch is,
)
ΨN+ = 1Lyeikyy(cid:18)cossin2θθ2ΨΨNN−(1x(x,t,)t) (cid:19) (18) R(XX 68 (b)antinode: 1.7P0.53 0.5
and for the - branpch, node: 1.2P
4
1 sinθΨ (x,t)
ΨN− = Lyeikyy(cid:18)−cos22θΨNN−(x1,t) (cid:19) (19) 2
0 2 4 6 8 10
where, p
P(mW)
Ψ (x,t)=Φ (x X x (t),t)
N N cl
− −
ehim~∗dxcdlt(t)[x−xcl(t)]+~i R0tLdt′i (20)
×
FIG. 3: Dependence on radiation power P of the calculated
as above, Φn is the solution for the Schr¨odinger equa- magnetoresistivity under light in 2DES with Rashba cou-
tion of the unforced quantum harmonic oscillator where pling. In panel (a) we exhibit Rxx as a function of B, for
x (t) is the classical solution of a forced harmonic different radiation intensities starting from dark and for the
cl
oscillator10,47,48, same frequency f = 103.08 GHz. In panel (b) we exhibit
∆Rxx = Rxx −Rxx(dark) versus P for B corresponding to
eεo dashed vertical lines on panel (a). One line corresponds to
x (t) = cos(wt β)
cl m (w2 w2)2+γ4 − the B-position of a node and the other of an antinode. For
∗ c − both,nodeandantinode,weobtainasquareroot (sublinear)
= Acos(wt β) (21)
p − dependenceshowing the corresponding fits.(T=1K).
γ is a phenomenologicallyintroduceddamping factor for
the electronic interaction with acoustic phonons. β is
thephasedifferencebetweentheradiation-drivenguiding
the center is displaced by x (t). In the presence of radi-
cl
center and the driving radiation itself. L with RSOI is
ation, the electronic orbit center coordinates change and
now given by,
aregivenaccordingtoourmodelbyX(t)=X(0)+x (t).
cl
This means that due to the radiation field all the elec-
1 1 α
L= 2m∗x˙2cl−2m∗wc2x2cl−~ [σym∗x˙cl−σxeBxcl] (22) tronic orbit centers in the sample harmonically oscillate
atthe radiationfrequencyinthexdirectionthroughx .
cl
Apart from phase factors, the wave function for H now Applying initial conditions, at t = 0, X(t) = X(0) and
0
is the same as the standard harmonic oscillator where then β = π/2. As a result the expression for the time
6
dependent guiding center is now: = ∆X(0) Asinwτ (26)
−
X(t)=X(0)+Asinwt (23)
According to our model, this expressionis responsible of
RIRO including the maxima and minima positions. In
Inthepresenceofchargedimpuritiesscatteringandradi-
ordertoobtainanexpressionandaphysicalmeaningfor
ationtheaverageadvanceddistancebyelectronsisgoing
τ, we can compare the condition fulfilled by the minima
to be different than in the dark, ∆X(0) (see Fig. 1a).
positions obtainedfromthe theoreticalexpressionto the
Now the positions of the Landau states guiding centers
one obtained in experiments1. These minima positions
are time-dependent according to the last expression. If
represent one of the main traits describing RIRO and
the scatteringeventbeginsatacertaintimet,the initial
were first found by Mani et al1 being given by:
LS is given by, X (t) = X (0)+Asinwt. After a time
1 1
τ, that we call flight time, the electron ”lands” in a fi-
nal LS that is no longer X as in the dark scenario. All w 5 9 13 1
2 = , , ....= +n (27)
LS have been displaced in the same direction, the same wc 4 4 4 (cid:18)4 (cid:19)
distance given by, Asinwτ. Thus, due to the swinging
nature of irradiated LS, its former position is taken by
where n=1,2,3..... Accordingto theory, i.e., expression
another LS that we can call X (see Fig. 1b). This final
3
(26), the minima positions are obtained when,
LS is written as, X (t+τ)=X (0)+Asinw(t+τ), and
3 3
thescattering-inducedadvanceddistancebytheelectron
reads, π 2π 1
wτ = +2πn w = +n (28)
2 ⇒ τ 4
(cid:18) (cid:19)
∆X(t) = X (t+τ) X (t)
3 1
−
= X3(0)+Asinw(t+τ) X1(0) Asinwt Then,comparingbothexpressionswereadilyobtainthat
− −
(24) the flight time is,
In order to obtain the steady-state regime for the ad- 2π
τ = (29)
vanced distance we time-averageovera period of the ra-
w
c
diation field,
In other words, τ equals the cyclotron period T . This
<∆X(t)> = <X (t+τ) X (t)> c
3 − 1 important result admits a semiclassical approach in the
= X3(0)+<Asinw(t+τ)> sense that during the time it takes the electron to ”fly”
−
X (0) <Asinwt> from one LS to another due to scattering, the electrons
1
−
= X (0) X (0) (25) in their orbits perform one whole loop.
3 1
−
Finally,the advanceddistancedue to scatteringinthe
where obviously we have taken into account that < presence of radiation reads,
Asinw(t+τ)>=0 and < Asinwt >=0. Here, the an-
gularbracketsdescribe time-averageovera periodofthe
w
time-dependentfield. Next,wehavetorelate<∆X(t)> ∆X(τ)=∆X(0) Asin 2π (30)
− w
with the advanced distance in the dark ∆X(0). This is (cid:18) c(cid:19)
straightforward considering first, that during the time τ
all LS have been displaced in phase the same distance Applying these last results to a Boltzmann transport
Asinwτ, and secondly, that the initially position occu- model, similarly as the first part of this section, we can
pied by X (t) is now occupied after τ by X (t + τ). get to an expression for the longitudinal conductivity
2 3
In other words, X is the only orbit always located at of the magnetotransport of a high mobility 2DES with
3
a distance Asinwτ from X (t). Thus, we necessarily strong RSOI in the presence of radiation. In this ex-
2
conclude that the respective guiding centers of both LS pressionthree harmonicterms turnup, two cosineterms
X (0)andX (0)areseparatedbythedistance,Asinwτ, depending on the Fermi energy and α, that interfere to
3 2
i.e.,X (0)=X (0) Asinwτ (seeFig. 1c). Substituting giverise to the beating patternprofile in the magnetore-
3 2
this result in the ab−ove expression we obtain, sistance. And one sine term depending on radiation pa-
rameters, frequency and power. We expect the latter to
<∆X(t)>=∆X(τ) = X (0) Asinwτ X (0) interfere on the beating pattern profile.
2 1
− −
7
σxx ∝(cid:20)∆X0−Asin(cid:18)2πwwc(cid:19)(cid:21)21+e−~wπcΓsinXhSXS cos2π~EwFc +s14 + 2EwFcα˜2+cos2π~EwFc −s14 + 2EwFcα˜2
    (31) 
 
where we have considered only the term s = 1, the obtainfor both, accordingto the calculatedfits (see Fig.
mostimportant,inthe sum. We considerthatthe above (2.b)), an approximately square root dependence on P,
results can be of application and can predict the behav- concludingthatRashbacouplingdoesnotaffectthesub-
iorofmagnetotransportin3Dtopologicalinsulatorssub- linear law. We can theoretically explain these results
jectedtoradiation;oncethesesystemsreachenoughmo- according to our model. In the expression of σ and
xx
bility to make patent the rise of RIRO. This could hap- then in R , P only shows up in the numerator of the
xx
pen at the same time that, without radiation, the R amplitudeAas√P ε ,butnotinthephaseofthesine
xx 0
∝
beating patter begins to be visible in magnetotransport function. Thus, on the one hand, P does not affect the
experiments in 3D topological insulators. phase of RIRO that remains constant as P changes,and
ontheotherhandR √P,givingrisetothesublinear
xx
∝
(squareroot)powerlawforthe dependenceofR onP.
xx
Finally,inthephaseofcosinetermsthereisnoradiation
parameters concluding that radiation will not affect the
III. RESULTS
B-positions of nodes and antinodes.
Allcalculatedresultspresentedinthisarticlearebased In Fig.(3) we presentthe dependence on radiationfre-
on the next list of parameteres regarding experiments in quency of irradiatedR for 2DES with Rashba interac-
xx
InAs quantum wells28,32–35,37: Rashba parameter α = tion. In panel(a)we show the lowfrequency case andin
0.6 10−11eV m, electron density ni =2.0 1016m−2, panel(b)thehighfrequency,obtainingsimilarresultsfor
× · ×
electron effective mass m∗ = 0.045me where me is the both. Thus,thenodesB-positionturnsouttobeimmune
electron rest mass and temperature, T =1K. to radiation frequency keeping the same ones as in the
In Fig.(1) we present calculated Rxx vs B for dark and darksituation. Thedeformationoftheantinodeschanges
irradiatedscenariosinahigh-mobility2DESwithstrong with the frequency. The reason is that the deformation
RSOI. For the latter, the radiation frequency f = 149 or modulation depends on the RIRO position and the
GHz. For the dark curve we obtain a very clear beating latter does change with radiationfrequency. As a result,
pattern profile made up of a system of nodes and antin- the same initial antinode in the dark will deform differ-
odes. The radiationcurveexhibits asimilarbeating pat- ently according to f. We also observe that the strongest
ternbutthistimedramaticallydeformedandmodulated deformation corresponds to the RIRO’s peaks irrespec-
by the rise of the system of peaks and valleys of RIRO. tive of radiation frequency. The immunity of nodes with
In the new beating pattern the node B-positions are not f can be readily explained as in the previous figure, ac-
affectedbythepresenceofradiationbutyetthedifferent cording to Eq. (23). In this equation the nodes position
antinodes are, according to their B-position. This pecu- dependsonlyonthecosinetermswheretheRashbaterm
liar profile in Rxx shows up as result of the interference α showsupin the correspondingphases. In these phases
effect between the sine and cosine terms that is reflected f does not show up and then its variation will not affect
in equation (23). the positions of either the nodes or the antinodes.
InFig.(2)wepresentthedependenceofcalculatedR
xx
onP for2DESwithimportantRashbacouplingunderra- InFig.(4)wepresenttheobtainedresultsforirradiated
diation. In panel (a) we exhibit calculated R vs B for R vsBfor2DESwithRashbacouplingintheterahertz
xx xx
aradiationfrequencyf =103.08GHz,differentradiation regime showing two frequencies: 300 GHz in panel (a)
intensities from dark to 10 mW: 0.5, 1, 2.2, 4, 6.2 and and 400 GHz in panel (b). For both panels we exhibit
10 mW and T = 1K. We easily observe, as expected, the dark case and two radiation curves. For the latter,
thatRIROincreasetheiramplitudes asP increasesfrom one is obtained at low radiation intensity and the other
dark. At the same time the deformation of antinodes at high. Apart form RIRO, due to radiation, and the
gets strongertoo,keepingconstantthe B-positionofthe beating patter, due to Rashba, we have obtained zero
nodes. Inpanel(b)weexhibit,againforf =103.08GHz, resistancestatesforB 0.4Tinthe upperpanelandfor
≃
∆R =R R (dark) versus P for B corresponding B 0.55T in the lower panel. In the former case ZRS
xx xx xx
− ≃
to dashed vertical lines on panel (a). One line corre- are obtained increasing P from an antinode in the dark
sponds to the B-position of a node and the other of an scenario. ThisantinodeendsuptotallywipedoutasZRS
antinode. WewanttocheckoutifthepresenceofRashba rise up. Similar situation is presented in the lower panel
coupling affects the previously obtained sublinear power but this time ZRS is obtained from a node. Similarly as
law for the dependence of RIRO on P. In this way we before,the nodedisappearsimmersedintheZRSregion.
8

25
300GHz
(a)
(a) low MW freq.
5 dark
20
90GHz
4 15
80GHz

10
3
ZRS
) 5
2
) (
X
(XX 58GHz R2X 50
R 1 (b) 400GHz

7 (b)high MW freq. 149GHz 20
128GHz
6
15

5
10
103GHz
ZRS
4
5
3
dark 0
2
0.0 0.2 0.4 0.6 0.8 1.0
1 B(T)
0.0 0.1 0.2 0.3 0.4 0.5 0.6
B(T)

FIG. 5: Terahertz irradiated Rxx versus B for 2DES with
Rashba interaction for different temperatures with constant
power excitation for f = 300GHz in panel (a) and f =
400GHzinpanel(b). WeobtainaZRSregionineachpanelas
FIG. 4: Dependence on radiation frequency, f of irradiated
indicatedbyarrows. Inpanel(a)anantinodeiswipedoutim-
RxxvsBfor2DESwithRashbacoupling. Inpanel(a)weex-
mersedintheZRSregionastheradiationintensityincreases.
hibitthelowfrequencyscenarioandinpanel(b)thehighfre-
The same in panel (b) butnow for a node. (T=1K).
quency, obtaining similar results for both. Nodes B-position
is immuneto frequency and antinodes shape dependson fre-
quencybecausetheformerdependsonRIRO’spositionsthat,
in turn,deeply depend on f. (T=1K).
We have developed a perturbation treatment for elastic
scattering due to charged impurities based on the
Boltzmann theory to finally obtain an expression for
IV. CONCLUSIONS
the magnetoresistance of the system. In the presence
of radiation we have obtained that the typical beating
In summary we have presented a theoretical analysis pattern of a 2DES with Rashba interaction is strongly
on the effect of radiation on mangetotransport of 2DES modified following the profile of radiation-induced
with strong Rashba spint-orbit coupling. We have magnetoresistance oscillations. We have studied also
studied the interaction between the radiation-induced the dependence on intensity and frequency of radiation,
resistance oscillations and the typical beating pattern including the teraherzt regime. For this regime we have
showing up in the mangetoresistance of 2DES with also obtained ZRS beginning from the dark in the case
Rashba interaction. We have deduced an exact solution of a node and in the case of antinode. We think that
for the electron wave function corresponding to a total the results presented in this paper, could be of interest
Hamiltonian with Rashba coupling and radiation terms. for magnetotransport of nonideal Dirac fermions in 3D
9
topological insulators considering that both systems MINECO (Spain) under grant MAT2014-58241-P and
sharesimilarHamiltonian. Theonlydifferenceisthatfor ITN Grant 234970 (EU). GRUPO DE MATEMATI-
3D topological insulators the quadratic term is smaller CAS APLICADAS A LA MATERIA CONDENSADA,
than the linear that in this case is dominant. When 3D (UC3M), Unidad Asociada al CSIC.
topologicalinsulators reachenoughelectronmobility, all
the effects related above, from the beating pattern to
its modulation under the presence of radiation will be
clearly apparent.
V. ACKNOWLEDGMENTS
VI. REFERENCES
We acknowledge V. Tribaldos and J.M. Reynolds for
useful discussions. This work is supported by the
1 R.G.Mani,J.H.Smet,K.vonKlitzing,V.Narayanamurti, (1984).
W.B. Johnson, V. Umansky,Nature 420 646 (2002). 25 Yu.A.BychkovandEI.Rashba,17,J.Phys.C,17,6039,
2 M.A. Zudov,R.R.Lu,N. Pfeiffer, K.W. West, Phys.Rev. (1984).
Lett. 90 046807 (2003). 26 N.-T. Yeh, P.-C. Chiu, J.-I. Chyi, F. Ren, and S. J.
3 R. G. Mani, J. H. Smet, K. von Klitzing, V. Narayana- Pearton, J. Mater. Chem. C 1, 4616 (2013).
murti,W.B.Johnson,andV.Umansky,Phys.Rev.Lett. 27 J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma,
92, 146801 (2004). Phys. Rev.Lett. 104, 040502 (2010).
4 R. G. Mani, C. Gerl, S.Schmult, W. Wegscheider and V. 28 T.Tschirky,S.Mueller,Ch.A.Lehner,S.Falt,T.Ihn,K.
Umansky,Phys.Rev.B81, 125320, (2010). Ensslin, and W. Wegscheider, arXiv:1612.06782v1
5 J.In˜arreaandG.Platero,Phys.Rev.B72193414(2005). 29 YM. Beltukov, and MI. Dyakonov, Phys. Rev. Lett. 116,
6 JesusInarrea,R.G.ManiandW.Wegscheider,Phys.Rev. 176801, (2016).
,82 205321 (2010). 30 X.F.WangandP.Vasilopoulos,Phys.Rev.B.67,085313,
7 A.C.Durst,S.Sachdev,N.Read,S.M.Girvin,Phys.Rev. (2003).
Lett.91 086803 (2003). 31 X.F.WangandP.Vasilopoulos,Phys.Rev.B.72,085344,
8 M.G. Vavilov et. al., Phys. Rev.B, 70, 161306(2004). (2005).
9 X.L. Lei, S.Y. Liu, Phys.Rev. Lett.91, 226805 (2003). 32 JunsakuNitta,TatsushiAkazakiandHideakiTakayanagi,
10 J. In˜arrea and G. Platero, Phys. Rev. Lett. 94 016806, Phys. Rev.Lett. 78, 1335 (1997).
(2005) 33 B. Shojaei, P.J.J. O’Malley, J. Shabani,P. Roushan,B.D.
11 Jesus Inarrea Euro. PhysLett. 113 57004, (2016) Schultz,R.M. Lutchyn,C. Nayak,J.M. Martinis and C.J.
12 J.Inarrea and G.Platero, Appl. Phys. Lett. 89, 052109, Palmstron, Phys. Rev.B 93, 075302 (2016).
(2006). 34 J. Luo, H. Munekata, F.F. Fang and P. J. Stiles, Phys.
13 J.Inarrea and G.Platero, Nanotechnology, 21, 315401, Rev.B 41, 7685 (1990).
(2010). 35 J. Luo, H. Munekata, F.F. Fang and P. J. Stiles, Phys.
14 J. Inarrea and G. Platero, Phys. Rev. B, 84, 075313, Rev.B 38, 10142 (1988).
(2011). 36 B.Das,D.C.Miller,S.Datta,R.Reifenberger,W.P.Hong,
15 T.Herrmann, I.A.Dmitriev, D.A.Kozlov, M.Schneider, P.K. Bhattacharya, J. Singh and M. Jaffe, Phys. Rev. B
B.Jentzsch, Z.D.Kvon, P.Olbrich, V.V.Belkov, A.Bayer, 39, 1411 (1989).
C.Schuh, D.Bougeard, T.Kuczmik, M.Oltscher, D.Weiss 37 J.P. Heida, B.J. van Wees, J.J. Kuipers, T.M. Klapwijk
and S.D.Ganichev Phys. Rev.B. 94, 081301, (2016) and G. Borghs, Phys.Rev.B 57, 11911 (1998).
16 A.D.ChepelianskiiandD.L.Shepelyanslky,Phys.Rev.B, 38 A Fete, S Gariglio, C Berthod, D Li, D Stornaiuolo, M
80, 241398(R) (2009). Gabay and J-M Triscone, New Journal of Physics, 16,
17 S.A. Mikhailov, Phys. Rev.B, 83, 155303 (2011). 112002 (2014)
18 R.G. Mani, A.N. Ramanayaka and W. Wegscheider 2011 39 B.Rupprecht,S.Heedt,H.Hardtdegen,Th.Schapers,Ch.
Phys. Rev.B. 84 085308, (2011). Heyn, M. A. Wilde and D. Grundler Phys. Rev. B 87,
19 A.N.Ramanayaka,R.G.Mani,J.InarreaandW.Wegschei- 035307 (2013).
der Phys.Rev.B. 85, 205315, (2012). 40 B.K. Ridley. Quantum Processes in Semiconductors, 4th
20 Tianyu Ye, Jesus Iarrea, W. Wegscheider and R.G. Mani ed. Oxford University Press, (1993).
Phys. Rev.B. 94, 035305, (2016). 41 T. Ando, A. Fowler and F. Stern, Rev. Mod.
21 Jesus Inarrea, 100, 242103, (2012). Phys.,54,(1982).
22 G.C.Gardner,S.Fallahi,J.D.Watson,andM.J.Manfra, 42 B.M. Askerov.Electron trnasport phenomenain semicon-
Journal of Crystal Growth 441, 71 (2016). ductors. World Scientific,Singapore, (1994).
23 FT. Vasko,JETP Lett. 30, 541 (1979). 43 Ch. Amann and M. Brack, J. Phys. A: Math. Gen. 35,
24 Yu. A. Bychkov and EI. Rashba, JETP Lett. 39, 78, 6009, (2002)
10
44 SK. Firoz and Tarun Kanti Ghosh, J. Phys: Cond. Matt. 46 Z.Ren,A.A.Taskin,S.Sasaki,K.Segawa, andY.Ando,
24, 035302, (2012). Phys. Rev.B 82, 241306(R) (2010).
45 Y.L.Chen,J.G.Analytis,J.-H.Chu,Z.K.Liu,S.-K.Mo, 47 E.H. Kerner, Can. J. Phys.36, 371 (1958).
X. L. Qi, H. J. Zhang, D. H. Lu, X. Dai, Z. Fang, S. C. 48 K. Park, Phys.Rev.B 69 201301(R) (2004).
Zhang, I. R. Fisher, Z. Hussain, and Z.-X. Shen, Science
325, 178 (2009).