Kubo formula for Floquet states and photoconductivity oscillations in a 2D electron gas

0.35 MB·English

by  Manuel Torres

Kubo formula for Floquet states and photoconductivity oscillations in a 2D electron
gas.
Manuel Torres1, and Alejandro Kunold2,
∗ †
1Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico,
5 Apartado Postal 20-364, M´exico Distrito Federal 01000, M´exico
0 2Departamento de Ciencias B´asicas, Universidad Aut´onoma Metropolitana-Azcapotzalco,
0 Av. San Pablo 180, M´exico D. F. 02200, M´exico
2 (Dated: February 2, 2008)

n The recent discovery of the microwave induced vanishing resistance states in a two dimensional
a electron system (2DES) is an unexpected and surprising phenomena. In these experiments the
J
magnetoresistanceofahighmobility2DESundertheinfluenceofmicrowaveradiation offrequency

3 ω at moderate values of the magnetic field, exhibits strong oscillations with zero-resistance states
1 (ZRS) governed by the ratio ω/ωc, where ωc is the cyclotron frequency. In this work we present
a model for the photoconductivity of a two dimensional electron system (2DES) subjected to a
] magnetic field. Themodel includesthemicrowave and Landau contributionsin a non-perturbative
l
l exactway,impurityscatteringeffectsaretreatedperturbatively. Inourmodel,theLandau-Floquet
a
states act coherently with respect to the oscillating field of the impurities, that in turn induces
h
- transitions between these levels. Based on this formalism, we provide a Kubo-like formula that
s takes into account the oscillatory Floquet structure of the problem. We study the effects of both
e
short-rangeand long-range disorder on thephotoconductivity. Ourcalculation yieldsa magnetore-
m
sistanceoscillatory behaviorwith thecorrect period andphase. Itisfoundthat,inagreement with
. experiment,negativedissipationcanonlybeinducedinveryhighmobilitysamples. Weanalyzethe
t
a dependence of the results on the microwave power and polarization. For high-intensity radiation
m multi-photonprocessestakeplacepredictingnewnegative-resistancestatescenteredatω/ωc =1/2,
- and ω/ωc =3/2.
d
n
o I. INTRODUCTION.
c
[
Two-dimensional electron systems (2DES) in a perpendicular strong magnetic field have been extensively studied
3 inrelationwiththe quantumHalleffect. Recently, twoexperimentalgroups1,2,3,4,reportedthe observationofanovel
v
phenomenon: the existence of zero-resistance states (ZRS) in an ultraclean GaAs/Al Ga As sample subjected
8 x 1 x
to microwave radiation and moderate magnetic fields. The magnetoresistance exhibits stron−g oscillations with ZRS
6
4 governed by the ratio ǫ = ω/ωc, where ωc is the cyclotron frequency. According to Zudov etal., the oscillation
7 amplitudes reach maxima at ǫ = ω/ω = j and minima at ǫ = j +1/2, for j an integer. On the other hand Mani
c
0 etal. reported also a periodic oscillatory behavior, but with maxima at ǫ = j 1/4 and minima at ω/ω = j+1/4.
c
4 Additionally experimental work appeared recently5,6,7,8,9. −
0 Inspiteofalargenumberoftheoreticalworks10,11,12,13,14,15,16,17,18,19,20,acompleteunderstandingoftheneweffects
/
t in 2DES induced by microwaves has not yet been achieved. A pioneering work put forward by Ryzhii10,11 predicted
a
m the existence of negative-resistance states (NRS). Durst and collaborators12 found also NRS in a a diagrammatic
calculationofthephotoexcitedelectronscatteredbyadisorderpotential. Apossibleconnectionbetweenthecalculated
-
d NRS and the observedvanishing resistance was put forwardin reference13, noting that a generalanalysis of Maxwell
n equations shows that negative resistance induces an instability that drives the system into a ZRS. Whereas some of
o the models10,11,12,17 are based on an impurity assisted mechanisms, there are alternative explanations5,19,20 in which
c the leading contribution arises from the modifications of the electrondistribution function induced by the microwave
:
v radiation. These models, as well as some of their predictions remain to be tested experimentally.
i In this work we present a model which includes the Landau and radiation contribution (in the long-wavelength
X
limit) in a non-perturbative exact way. Impurity scattering effects are treated perturbatively. With respect to the
r
Landau-Floquet states, the impurities act as a coherent oscillating field which induces the transitions that proved to
a
beessentialinordertoreproducetheobservedoscillatorybehaviorofthemagnetoresistance. Basedonthisformalism
a Kubo-likeexpressionfor the conductance is provided. Our results displaya strongoscillatorybehaviorfor ρ with
xx
NRS. It is found that ρ vanishes at ǫ = ω/ω = j for j integer. The oscillations follow a pattern with minima at
xx c
ǫ=j+δ, and maxima at ǫ=j δ, adjusted with δ 1/5. The model is used to test chirality effects induced by the
− ≈
magnetic field, calculationsarecarriedoutfor variousE-fieldpolarization’s. Finally, weexplorethe nonlinearregime
in which multi-photon processes play an essential role.
Thepaperisorganizedasfollows. Inthenextsectionwepresentthe modelandthe methodthatallowusto obtain
the exact solution of the Landau-microwave system, as well as the perturbative corrections induced by the impurity
potential. In section III we develop the formulation of dc electrical linear response theory valid in arbitrary magnetic
2
field and microwave radiation. A discussion of relevant numerical calculations is presented in section (IV). The last
section contains a summary of our main results. Details of the calculations are summarized in the appendices.
II. THE MODEL.
We consider the motion of an electron in two dimensions subject to a uniform magnetic field B perpendicular to
the plane and driven by microwave radiation. In the long-wave limit the dynamics is governed by the Schr¨odinger
equation
∂Ψ
i~ =HΨ= H +V˜(r) Ψ, (1)
∂t {B,ω}
h i
here H is the Landau hamiltonian coupled to the radiation field (with λ ) and V˜(r) is any potential that
B,ω
cancan{be}decomposedinaFourierexpansion. Themethodappliesingeneralif→V˜∞(r)includesvariouspossibleeffects
such as: lattice periodic potential, finite wave-lengthcorrections,impurity scattering,etc; howeveras it will be lately
argued, the impurity scattering is the most likely explanation for the recent experimental results. One important
remarkwith relationto the impurity potentialin Eq. (1) is that to startwith, it shouldonly include the polarization
effects produced by the combined effects of the Landau-Floquet states and the impurity potentials, the broadening
effects produced by this potential are, as usually, included through the Kubo formula. Then we write V˜(r) as
V˜(r)=V(r) ∆V(r), ∆V(r)=W V(r)W , (2)
†
−
were W is the transformation that takes exactly into account the microwave-Landaudynamics, it is explicitly given
in Eq. (10). In the absence of the microwaveradiation, W 1 and V˜(r) vanishes. The impurity scattering potential
≡
V(r) is decomposed as
d2q
V(r)= V(q)exp iq (r r ) , (3)
(2π)2 { · − i }
i Z
X
here r is the positionofthe ithimpurity andthe explicitformofV(q) depends ofthe mechanismthat appliesunder
i
particular physical conditions, some examples will be consider in section (IV). H is then written as
B,ω
{ }
1
H = Π2, Π=p+eA, (4)
{B,ω} 2m∗
m is the effective electron mass and the vector potential A includes the external magnetic and radiation fields (in
∗
the λ limit) contributions
→∞
1 E
A= r B+Re 0 exp iωt . (5)
−2 × ω {− }
(cid:20) (cid:21)
We first consider the exact solution of the microwave driven Landau problem, the impurity scattering effects are
lately added perturbatively. This approximation is justified on the following conditions: (i): V /~ω <<1 and (ii):
c
| |
ωτ ω τ >> 1; τ is the transport relaxation time that is estimated using its relation to the electron mobility
tr c tr tr
∼
µ=eτ /m . As discussed in section (IV) both conditions are fully complied.
tr ∗
The system posed by H can be recast as a forced harmonic oscillator, a problem that was solved long time
B,ω
ago by Husimi21. Followin{g th}e formalism developed in references22,23, we introduce a canonical transformation to
new variables Q ,P ; µ=0,1,2, according to
µ µ
Q =t, P =i∂ +eφ+er E,
0 0 t
·
√eBQ =Π , √eBP =Π ,
1 y 1 x
√eBQ =p +eA +eBy, √eBP =p +eA eBx. (6)
2 x x 2 y y
−
It is easily verified that the transformation is indeed canonical, the new variables obey the commutation rules:
[Q ,P ]=[Q ,P ]=[Q ,P ]=i; all other commutators being zero. The inverse transformation gives
0 0 1 1 2 2
−
x=l (Q P ), y =l (Q P ), (7)
B 1 2 B 2 1
− −
3
~
where l = is the magnetic length. The operators(Q ,P ) can be identified with the generatorsof the electric-
B eB 2 2
magnetic traqnslationsymmetries24,25. Final results are independent of the selected gauge. Fromthe operatorsin Eq.
(6) we construct two pairs of harmonic oscillator-like ladder operators: (a1,a†1), and (a2,a†2) with:
1 1
a = (P iQ ), a = (P iQ ), (8)
1 1 1 2 2 2
√2 − √2 −
obeying: [a1,a†1]=[a2,a†2]=1, and [a1,a2]=[a1,a†2]=0.
It is now possible to find a unitary transformation that exactly diagonalizates H , it yields
B,ω
{ }
1
W†H{B,ω}W =ωc 2 +a†1a1 ≡H0, (9)
(cid:18) (cid:19)
with the cyclotron frequency ω =eB/m and the W(t) operator given by
c ∗
t
W(t)=exp iη Q exp iξ P exp iη Q exp iξ P exp i dt , (10)
1 1 1 1 2 2 2 2 ′
{ } { } { } { } {− L }
Z
where the functions η (t) and ξ (t) represent the solutions to the classical equations of motion that follow from the
i i
variation of the Lagrangian
ω
= c η2+ζ2 +ζ˙ η +ζ˙ η +el [E (ζ +η )+E (η +ζ )] . (11)
L 2 1 1 1 1 2 2 B x 1 2 y 1 2
(cid:0) (cid:1)
The explicit form of the solutions for η (t) and ξ (t) are given in the appendix (VIA).
i i
Let us now now consider the complete Hamiltonian including the contribution from the V˜(r) potential. When the
transformation induced by W(t) is applied the Schr¨odinger equation in (1) becomes
P Ψ(W) =H Ψ(W)+V (t)Ψ(W), (12)
0 0 W
where Ψ(W) =W(t)Ψ and
V (t)=W(t)V˜(r)W 1(t)=W(t)V(r)W 1(t) V(r). (13)
W − −
−
Notice that the impurity potential acquires a time dependence brought by the W(t) transformation. The problem is
now solved in the interaction representation using first order time dependent perturbation theory. In the interaction
representation Ψ(W) =exp iH t Ψ(W), and the Schr¨odinger equation becomes
I { 0 }
i∂ Ψ(W) = V (t) Ψ(W). (14)
t I { W }I I
The equationis solvedin terms ofthe evolutionoperatorU(t), in sucha waythat Ψ(W)(t)=U(t t )Ψ(W)(t ). The
I − 0 I 0
solution of the evolution operator in first order perturbation theory is given by the expression
t
U(t)=1 i dt W (t)V˜(r)W(t) , (15)
′ † ′ ′
− I
Z−∞ h i
that is explicitly evaluated in the appendix (VIB). The interaction is adiabatically turned off as t , in which
0
→−∞
(W)
case the asymptotic state is selected as one of the Landau-Floquet eigenvalues of H , i.e. Ψ (t ) µ,k . The
0 | I 0 i → | i
solutiontotheoriginalSchr¨odingerequationinEq. (1)hasbeenachievedbymeansosthreesuccessivetransformations,
which expressions have been explicitly obtained:
Ψµ,k(t) =W† exp iH0t U(t t0) µ,k . (16)
| i {− } − | i
As discussed on the appendix (VIA) the index k labels the degeneracy of the Landau-Floquet states. Selecting the
P-representationthe dependence of the wave function on k becomes very simple, see Eq. (47), for simplicity in what
followstheindexk willnotbeshown. TheexpressionoftheKuboformulathatwillbederivedinsection(III)requires
the knowledge of the matrix elements of the momentum operator Π:
Ψ Π Ψ = µ U (t t ) WΠ W U(t t ) ν . (17)
h µ| i| νi h | † − 0 i † I − 0 | i
(cid:2) (cid:3)
4
Let us first consider the term inside the square brackets,using the explicit form of the operators in Eqs. (4) (6), and
(10) it yields
√eB(P η ), i=x,
WΠiW† = √eB(Q1− ξ1), i=y. (18)
(cid:26) 1− 1
If we now utilize the result for the evolution operatorU given in the appendix (VIB), we can explicitly work out the
matrix elements of the momentum operator
eB
Ψµ Πi Ψν = aj√µeiωctδµ,ν+1+bj√νe−iωctδµ,ν 1
h | | i r 2 −
(cid:0) +√eB ei(Eµν+(cid:1)ωl−iηt) ∆(µlν)(j). (19)
l
X
Here the following definitions were introduced: = , a =b =1 if j =x and a = b = i if j =y, and
µν µ ν j j j j
E E −E − −
∆(l)(j) is given by
µν
1 a q˜ C(l) b q˜C(l)
∆(l)(j)=δ ρ δ +ρ δ j ∗ µν + j µν , (20)
µν µν j l,1 ∗j l,−1 − √2" µν ωc+ωl iη µν +ωc+ωl iη#
E − − E −
(cid:2) (cid:3)
where q˜=il (q iq )/√2, and the expressions for the functions ρ , and Cl are worked out as
B x− y i µ,ν
el E ( iωǫ +ω ǫ ) el E (ω ǫ +iωǫ )
B 0 x c y B 0 c x y
ρ = − , ρ = , (21)
1 ω2 ω2+iωΓ 2 ω2 ω2+iωΓ
− c rad − c rad
and
d2q ∆ l
Cµ(l,)ν = (2π)2V(q)e−iq·riDµν(q) i∆ J˜l(|∆|) , (22)
i Z (cid:18) | |(cid:19)
X
where J˜ = J δ , J being the Legendre polynomials, and D (q) is given in terms of the generalized Laguerre
l l l,0 l µν
−
polynomials in (53), and
ω l2eE
∆= c B 0 [ω(q ǫ +q ǫ )+iω (q ǫ q ǫ )] . (23)
ω(ω2 ω2+iωΓ ) x x y y c x y− y x
− c rad
It is important to notice that the subtracted term J˜ =J δ , has its origin in the fact that the impurity potential
l l l,0
−
Eq. (2) includes only the dynamical effects, with the corresponding zero field-term conveniently subtracted. This
procedureis justified because the broadeningeffects producedby V(r) areseparatelyincluded via the Kubo formula,
see appendix (VIE). The subtractionJ˜ =J δ becomes essential,otherwise the longitudinal resistance would be
l l l,0
−
dominated by the l=0 term, producing incorrect results.
III. KUBO FORMULA FOR FLOQUET STATES.
In this section we shall develop the Kubo formula that applies when the dynamics includes Landau-Floquet states
as those in Eq. (16). We take the perturbing electric field to have the form E = E cos(Ωt) exp( η t). The
ext 0
− | |
static limit is obtained with Ω 0 and η represents the rate at which the perturbation is turned on and off. The
→
perturbing electric field is included in the vector potential, as we are interested in the linear response the perturbing
potential has the form
1 E
V = Π A , A = 0 sin(Ωt) exp( η t). (24)
ext ext ext
m · ω − | |
Besides the original Hamiltonian in (1), the complete Hamiltonian should include V and the part of the disorder
ext
potential (∆V(r) = W V(r)W) that was previously subtracted (see Eq. (2)). Hence the total Hamiltonian H is
† T
written as
H =H +V +∆V(r). (25)
T ext
5
The disorder potential ∆V(r) will induce broadening effects, and it will be lately included. Then, the time-evolution
for the density matrix ρ(t) obeys the von Neumann equation
∂ρ
i~ =[H +V ,ρ] . (26)
ext
∂t
Within the linear regime ρ is split in the sum ρ=ρ +∆ρ. The zero order term ρ must satisfy
0 0
∂ρ
i~ 0 =[H,ρ ], (27)
0
∂t
the conditions required to solve this equation will be established below. The first order deviation ∆ρ then obeys
∂∆ρ
i~ =[H,∆ρ]+[V ,ρ ] . (28)
ext 0
∂t
We shall now apply to this equation the three transformation that were utilized in the previous section in order to
solve the Schr¨odinger equation, hence in agreement with Eq. (16), ∆˜ρ is defined as
∆˜ρ(t)=UI†(t−t0)exp{iH0t}W(t)∆ρ(t)W†(t)exp{−iH0t}UI(t−t0). (29)
In terms of the transformed density matrix ∆˜ρ(t), Eq. (28) becomes
∂∆˜ρ
i~ = V˜ ,ρ˜ , (30)
ext 0
∂t
h i
where V˜ and ρ˜ are the external potential and quasi-equilibrium density matrix transformed in the same manner
ext 0
as ∆˜ρ is transformed in Eq. (29). The transformed quasi-equilibrium density matrix is assumed to have the form
ρ˜ = µ f( ) µ, wheref( )is the usualFermifunctionand the Landau-Floquetlevels(see appendixVIA).
0 µ| i Eµ h | Eµ Eµ
Itisstraightforwardtoverifythatthisselectionguaranteethatthequasi-equilibriumconditionin(27)isverified. The
P
justification for selecting a Fermi-Dirac distribution in the quasi-energy states is presented in the appendix (VIC).
It is shown, that under experimental conditions (τ τ τ ), the elastic and inelastic relaxation processes can
ω tr in
≪ ≪
be neglected as compared to the external field effects. The solution of the Boltzmann equation yields, for a weak
microwave intensity, a Fermi-Dirac distribution in the quasi-energy states. The expectation value of Eq. (30) in the
µ base can now be easily calculated using Eqs. (16), (24) and (29), solving the resulting equation with the initial
| i
condition ∆ρ(t) 0 as t , yields for t<0
→ →−∞
µ ∆˜ρ(t) ν = Ψ ∆ρ(t) Ψ
µ ν
h | | i h | | i
eE t ei(Ω iη)t′
= 0 − f Ψ Π(t) Ψ +(Ω Ω) , (31)
µν µ ′ ν
2 · " Ω h | | i →− #
Z−∞
where the definition f = f( ) f( ) was used. Substituting the expectation value for the momentum operator
µν µ ν
E − E
given in (19), the integral in the the previous equation is easily performed. The current density to first order in
the external electric field can now be calculated from J(t,r) =Tr ∆˜ρ(t)J˜(t) , the resulting expression represents
h i
the local density current. Here we are concerned with the macroscophic conductiivity tensor that relates the spatially
and time averaged current density j = (τωV)−1 0τωdt d2xhJ(t,r)i to the averaged electric field; here τω = 2π/ω.
The macroscopicconductivity can now be workedout, results for the dark and microwave-inducedconductivities are
R R
quoted:
e2ω2 f a µδ b νδ
σDxi = i4~c Ωµν i+µΩ,ν+1iη + i+µΩ,ν−1iη +(Ω→−Ω) , (32)
µν (cid:26) (cid:20)Eµν − Eµν − (cid:21) (cid:27)
X
σω = e2ωc2 fµν ∆(µlν)(i)∆ν(−µl)(x) +(Ω Ω) . (33)
xi i4~ ( Ω µν +ωl+Ω iη →− )
µν l E −
X X
In these expressions the external electric field points along the x-axis. Hence, setting i=x or i=y the longitudinal
andHallconductivitiescanbeselected. ThedenominatorsontheR.H.S.ofthepreviousequationscanberelatedtothe
advancedandretardedGreen’sfunctionsG ( )=1/( iη). Tomakefurtherprogresstherealandabsorptive
±µ E E −Eµ±
6
parts of the Green’s functions are separated taking the limit η 0 and using lim 1/( iη) = P1/ +iπδ( ),
η 0
where P indicates the principal-value integral. As usual the re→al and imaginary p→arts coEn−tribute to thEe Hall aEnd
longitudinal conductivities respectively. In what follows details of the calculations are presented for the longitudinal
microwave-induced conductivity, the corresponding dark conductivity expressions as well as the Hall microwave-
induced conductance are quoted in the appendix (VID). Implementing the previous considerations and inserting a δ
function, the longitudinal microwave-inducedconductivity takes the form
e2ω2 f( +ωl+Ω) f( )
σω = c d δ( )∆(l)(x)2 E − E ImG ( +ωl+Ω)
xx − 4~ E E −Eµ | µν | Ω ν E
µν l Z (cid:26)
XX
+(Ω Ω) , (34)
→−
(cid:27)
where ImG ( )= 1 [G+( ) G ( )]. The static limit with respect to the external field is obtained taking Ω 0.
ν E 2i ν E − −ν E →
Inthecaseoftheimpurityassistedcontributionanadditionalaverageovertheimpuritydistributionhastobecarried
out, it is assumed that the impurities are no correlated, utilizing the explicit expressions for the velocity matrix
elements in (20), the final result for the averagedmicrowaveinduced longitudinal conductance is worked out as
e2
σω = d ImG ( )B(l)( , ) ω ρ 2δ (δ +δ )
h xxi π~ E µ E E Eν c| 1| µν l,1 l,−1
Z µν l (cid:26)
XX
d2q 2
+ n l2 q2 J˜(∆)V(q)D (q˜) ,
imp B (2π)2 y l | | µν
Z (cid:12) (cid:12) (cid:27)
(cid:12) (cid:12) (35)
(cid:12) (cid:12)
(cid:12) (cid:12)
where n is the two dimensional impurity density and the following function has been defined
imp
d
B(l)( , )= [f( +lω+ ) f( )]ImG ( +lω+ ) . (36)
ν 0 ν 0
E E − d { E E − E E E }
(cid:20) E0 (cid:21) 0=0
E
The photoconductivity in (35) has a first contribution that depends on the ρ factor (independent of the impurity
1
concentration), it represents the direct cyclotron resonance heating, arising when the W(t) transformationis applied
tothemomentumoperator(seeEq. (18)). Theimpurityinducedcontribution(secondrowin(35))takesintoaccount
the dynamics produced by the magnetic and microwave fields, combined with the resonant effect of the impurities;
the information is contained in the complete wave function in Eq. (16).
The previous expression would present a singular behavior that is an artifact of the η 0 limit. This problem
→
is solved by including the disorder broadening effects. A simple phenomenological prescription is dictated by simply
retainingafinitevalueofηthatisrelatedtothequasiparticleslifetime(η =2π/τ )27,28. Accordingtothisprescription
s
the density of states (DOS) of the µ level would have a Lorentzian form ImG ( ) = η/2π . A more formal
procedurerequirestocalculatethebroadeningproducedby,thesofarneglected,µpEartof(tEh−eEdµ)is2o+rηd2e/4rpotential∆V(r)
(see Eq. (25)); fortunatelyasexplainedinthe appendix(VIE)the calculationbecomes equivalentto thatcarriedout
by Ando30 and Gerhardts31, so the density of states for the µ-Landau level can be represented by a Gaussian-type
form32
π 2β ~2ω
ImG ( )= exp ( )2/(2Γ2) , Γ2 = µ c. (37)
µ E 2Γ2 − E −Eµ µ µ (πτ )
r µ tr
(cid:2) (cid:3)
Theparameterβ inthelevelwidthtakesintoaccountthedifferenceofthetransportscatteringtimedeterminingthe
µ
mobility µ, from the single-particle lifetime. In the case of short-rangescatterers τ =τ and β =1. An expression
tr s µ
forβ ,suitablefornumericalevaluation,thatappliesforthelong-rangescreenedpotentialin(40)isgiveninappendix
µ
(VIE). β decreases for higher Landau levels. This property becomes essential to generate NRS, because they only
µ
appear for a narrow Γ , a condition that is satisfied around the Fermi level in the case of large filling factors.
µ
Eqs. (35,36) contain the main ingredients that explain the huge increase observed in the longitudinal conductance
(andresistance),whenthematerialisirradiatedbymicrowaves. InthestandardexpressionfortheKuboformulathere
are no Floquet replica contribution, hence ω can be set to zero in (36), if that is the case B(l) becomes proportional
to the energy derivative of the Fermi distribution, that in the T 0 limit becomes of the form δ( ), and the
F
→ E −E
conductivity is positive definite depending only on those states lying at the Fermi level. On the other hand, as a
result of the periodic structure induced by the microwaveradiation, B(l) contains a second contribution proportional
to the derivative of the density of states: d ImG ( +lω). Due to the oscillatory structure of the density of states,
d ν E
E
7
this extracontributiontakesbothpositiveandnegativevalues. AccordingtoEq. (37)this secondterm(ascompared
to the first one) is proportional to the electron mobility, hence for sufficiently high mobility the new contribution
dominates leading to negative resistance states. The former observationbecomes fundamental, because in agreement
with experiment, our calculations show that NRS can only be induced in very high mobility samples (see Fig. 3).
As it was mentioned in section (II) the present method applies in generalif V˜(r) can be decomposed in its Fourier
expansion (3) e.g.: finite wave-length corrections, lattice periodic potential, impurity scattering, etc. The microwave
radiation by itself only produces transitions between adjacent Landau levels (first term on the R.H.S of Eq. (35))
leading to the cyclotron peak. In the case of a periodic potential the resulting spectrum will be dominated by the
region q 2π/a, where a is the lattice parameter; for the experimental conditions λ a and the contribution is
B
≈ ≫
negligible. So we are led to analyze the impurity assisted mechanism as a plausible scenario to explain the strong
oscillatory structure of the magnetoresistance.
IV. RESULTS.
The single-particle and transport relaxation rates induced by disorder are given by29
1 m 2kF V(q)2 1
ττ1trs (cid:27)= nimp π~3k∗F Z0 dq 1−| (q/2|kF)2 × ( 2qk2F2 , (38)
p
where V(q) is the Fourier transform of the impurity potential (3). Remarkably, we have a consistent formalism in
which: (i)the photoconductivity (35), (ii)the relaxationrates(38),and(iii)the levelbroadeningEqs. (37)and(60)
can all be consistently calculated once V(q) has been specified.
For neutralimpurities the potentialcanbe representedby a shortrangedelta interaction,the coefficientin Eq. (3)
correspondsto a constant that can be selected as V(q)=2π~2α/m . The expressionin Eq. (38) is readily calculated
∗
to yield the same value for the single-particle and transport relaxation rates
1 1 4π2~
= = α2n(N), (39)
τt(rN) τs(N) m∗ imp
the upper index N labels the neutral impurity case. The evaluation of the photoconductivity (35) requires in general
a time consuming numericalintegration. However,for moderatevalues of the microwaveradiationthe transitionsare
dominated by single photon exchange, in the neutral impurity case a very precise analytical approximation can be
explicitly worked out; see appendix (VIF).
For charged impurities the Coulomb potential is long-range modified by the screening effects. Although electron
motionis restrictedto2-dimensions,the electricfieldisthree dimensionalandthere arecontributionsfromthe impu-
rities localized within the doped layer of thickness d. The screened potential can then be represented in momentum
coordinates by the expression29
π~2 e qd e2m
− ∗
V(~q)= , q = , (40)
m 1+ q TF 2πǫ ǫ ~2
∗ qTF 0 b
where the Thomas-Fermi (TF) approximation is implemented in order to calculate the dielectric function. Here
ǫ represents the relative permittivity of the surrounding media. The expression in (40) corresponds to a screened
b
potential, that in realspace has a r 3 decay for large r. The rates in Eq. (38) can be evaluated numerically,however
−
an accurate analytical results is obtained observing that the decaying exponential in (40) causes the integral to die
off for q >> 1/d, and the upper limit in the integral can therefore be set to infinity. Additionally for the relevant
| |
parameters (see below) the following conditions are observed k >> 1/d and q >> 1/d; consequently, it is
F TF
| | | |
reasonabletodropthe factorq/q inthe denominatorof(40)andreplacethe squarerootinthe denominatorofEq.
TF
(38) by unity. These simplifications yields for the transport relaxation rate
1 π~
= n(C) , (41)
τt(rC) 8m∗(kFd)3 imp
the upper index C labels the charged impurity case. As expected, for charged remote impurities the single-particle
lifetime differs from the transport lifetime, the approximated relation reads τ(C) (2k d)2τ(C).
Theparametervalueshavebeenselectedcorrespondingtoreportedexperimetrnts2≈,3inuFltracsleanGaAs/Al Ga As
x 1 x
samples: effective electron mass m =0.067m , relative permittivity ǫ 13.18,fermi energy =10meV, ele−ctron
∗ e b F
≈ E
8
ρ (microwave)
60
xx
ρ (dark) 0.1ρ
xy
xx
40
)
Ω
( 20
ρ
0
0.1ρ
xy
-20
0.1 0.2 0.3
B (tesla)
FIG. 1: Longitudinal resistance, both total (continuos line) and dark (dotted line) as a function of the magnetic field. The
figurealso includes theHall resistance (dashed line, ρxy is rescaled by a factor 1/10). Results corresponds toneutral impurity
scattering obtained with the approximated solution (appendix VIF) and the selected parameters are: µ≈0.25×107cm2/Vs,
T ≈1K, f =100Ghz, |E~|≈2.5V/cm, α2n(N) =5×106cm−2. The values of the other parameters used in the calculations
imp
are discussed in thetext.
mobility µ 0.1 2.5 107cm2/Vs, electron density n = 3 1011cm 2, microwave frequencies f = 50 100Ghz,
−
≈ − × × −
magnetic fields in the range 0.05 0.4 Tesla and temperatures T 0.5 2.5K. The reported specimen is an
5mm 5mmsquare. Typicalmicro−wavepoweris10 40mW,however≈itises−timated1 thatthemicrowavepowerthat
× −
impinges onthe sample surfaceis of the order100 200µW,hence the microwaveelectric field intensity is estimated
as E~ 1 3V/cm. Using these values,it is verifi−edthat the weak-overlappingconditionholds: ω τ 100 1000.
c tr
| |≈ − ∼ −
Recalling that µ = eτ /m , one can use Eqs. (39) and (41) to determine the values of n corresponding to
tr ∗ imp
neutral or charged scatterers respectively. For example assuming µ 2.5 106cm2/Vs, one estimates for neutral
≈ ×
scattering α2n(N) 1 107cm 2. Although α and n(N) are not separately fixed, one notices that the condition for
imp ≈ × − imp
weak disorder potential as compared to the Landau energy can be be expressed as V(q)/(l2~ω ) = 2πα 1; e.g if
B c ≪
α 0.01,thenn(N) 1011cm 2. Forchargedimpuritiesandtakingavaluefortheseparationdbetweentheimpurity
∼ imp ∼ −
and the 2DES as d 20nm, yields n(C) 1.5 1011cm 2. In this case the weak disorder condition takes the form
≈ imp ≈ × −
V(q)/(l2~ω ) πexp( 2πd/l ) 1, that is satisfied. A final remark is related to the radiative electron decay Γ
thatdetBermcine∼sthedir−ectelectBron≪responsetothemicrowaveexcitation(seeEqs. (45)and(23)),followingreferencrea3d4
Γ isrelatedtotheradiativedecaywidththatisinterpretedascoherentdipolere-radiationofelectromagneticwaves
rad
by the oscillating 2Delectrons excited by microwaves. Hence, it is givenby Γ =ne2/(6ǫ m c), using the values of
rad 0 ∗
n and m given above it yields Γ 0.38meV.
∗ rad
Adding the dark and microwave in≈duced conductivities, the total longitudinal σ = σD + σw , and Hall σ =
xx xx h xxi xy
σD + σw conductivitiesareobtained. Itshouldbepointedoutthattheinterferencebetweenthedarkandmicrowave
xy h xyi
contributionsexactlycancels. Thecorrespondingresistivitiesareobtainedfromtheexpressionρ =σ / σ2 +σ2
xx xx xx xy
and ρ = σ / σ2 +σ2 . The relation σ σ holds in general, hence it follows that ρ σ , and the
xy xy xx xy xy ≫ xx xx ∝ (cid:0)xx (cid:1)
longitudinalresistivityfollowsthesameoscillationpatternasthatofσ . Theplotsofthe totallongitudinalandHall
xx
(cid:0) (cid:1)
resistivities as a function of the magnetic field intensity are displayed in Fig. 1. Whereas the Hall resistance presents
the expected monotonous behavior, the longitudinal resistance shows a strong oscillatory behavior with distinctive
NRS. The behavior of the complete ρ is contrasted with the dark contributions that presents only the expected
xx
Shubnikov-de-Hass oscillations.
Fig. 2showsacomparisonofthelongitudinalresistivityasfunctionofω/ω obtainedforthecaseofneutralimpurity
c
scattering using both the approximated expression in the appendix (VIF) as well as the result of the numerical
integration Eq. (35); the electron mobility is selected as µ = 0.25 107cm2/Vs. The approximated analytical
×
9
numerical integration
80 (charged impurity)
(neutral impurity)
60 approximated
)
Ω 40
(
x
x
ρ 20
0
-20
1 2 3 4 5
ω/ω
c
FIG. 2: Longitudinal resistance as a function of ǫ=ω/ωc for neutral and charged impurities. Results for neutral impurities
are obtained from the numerical integration (dotted line) and also using the analytical approximation discussed in appendix
VIF) (dashed line) with the parameters: µ≈0.25×107cm2/Vs, α2n(N) =5×106cm−2. The continuos line corresponds to
imp
thechargedimpuritycasewithparametersµ≈2.5×107cm2/Vs, n(C) =1.5×1011cm−2. Theotherparametersarethesame
imp
as in figureFig. 1.
result shows a good agreement with the one obtained from the numerical integration. It should be remarked that
the approximated expression includes only one-photon exchange processes, while the numerical result includes the
possibilityofmulti-photonexchange,henceitisconcludedthatfortheselectedelectricfieldintensity(E~ 2.5V/cm),
| |∼
the one-photon processes dominate. Results are also presented for the case of charged impurity scattering but for
µ = 2.5 107cm2/Vs. In spite of the very different nature of the two physical processes and that the mobility is
×
increased by an order of magnitude in the charged case, it is observed that the results for the neutral and charged
cases are very similar. The similarity of both results is based on: (i) the increase in the mobility is compensated by
the factorβ in (37) givinga similar broadeningvalue, (ii)althoughfor the neutralcaseV(q) is constantoverallthe
µ
q-range of integration, whereas for charged case V(q) varies according to the expression in (40), the integral in (35)
is dominated by the region in which q 2π/l .
B
≈
One of the puzzling properties of the observed huge magnetoresistance oscillations is related to the fact that they
appear only in samples with an electron mobility exceeding a threshold value µ . The phenomenon is absent in
th
samples in which µ is slightly reduced. This behavior is well reproduced by the present formalism. Fig. 3 displays
the ρ vs. ǫ = ω/ω plot for neutral impurity scattering and three selected values of µ. For µ 0.5 106cm2/Vs
xx c
≈ ×
the previously known, almost linear behavior ρ B is clearly depicted. As the electron mobility increases to
xx
∝
µ 1.5 106cm2/Vs, the magnetoresistance oscillations are clearly observed; however, NRS only appear when the
mo≈bility×is increased to µ 2.5 106cm2/Vs. It is observed that ρ vanishes at ǫ = j for j integer. The period
xx
≈ ×
and phase of the oscillations follow a pattern very similar to the one observed in experiments2,4, with minima at
ǫ=j+δ, and maxima at ǫ=j δ, adjusted with δ 1/5. It should be pointed out that this value of δ depends on
− ≈
the correctrepresentationof the density ofstates, using a Lorentzianforminsteadofthe Gaussianin (37) wouldgive
δ 1/1026. Similar behavior is observed for the charged impurity scattering case, but with the mobility threshold
∼
increasedapproximatelybyandorderofmagnitudeµ 2.5 107cm2/Vs. Theprecisedeterminationofµ depends
th th
≈ ×
of course on the selected values of the other parameters, mainly on the frequency and microwave intensity.
The first reported experiments1,2,3,4 were carried out for a microwave radiation with transverse polarization with
respect to the longitudinal currentflow direction. However,it is clear that the presence of the magnetic field induces
a chirality in the system, the model can be used to test these effects. Fig. 4 shows the results for different E field
−
polarization’swithrespecttothecurrent. InFig. 4aitisobservedthattheamplitudesoftheresistivityoscillationare
slightlybiggerfortransversepolarizationascomparedtolongitudinalpolarization. Thisresultisinagreementwiththe
recentexperiment4,inwhichitis reportedthatthe selectionoflongitudinalortransversepolarizationproducessmall
10
µ =106cm2/Vs
0
40
µ=1.5µ
0
µ=0.5µ
0
)
Ω
( 20
x
x
ρ
0
µ=2.5µ
0
-20
1 2 3 4
ω/ω
c
FIG. 3: Longitudinal resistivity as function of ǫ = ω/ωc for neutral impurity scattering and three values of the electron
mobility: µ≈0.5×106cm2/Vs (dotted line), µ≈1.5×106cm2/Vs (dash-dotted line), and µ≈2.5×106cm2/Vs (continuos
line). Inthetwoformercasestheoscillations follow apatternwithminimaatǫ=j+δ,andmaximaatǫ=j−delta,adjusted
with δ ≈ 1/5. NRS only appear when µ > µ ∼ 1.5×106cm2/Vs. The values of the other parameters are the same as in
th
figure Fig. 1
differences. However,weproposethatthe moresignificantsignatureswillbe onlyobservableforcircularpolarization.
Selecting negative circular polarization (see Fig. 4b), the oscillation amplitudes get the maximum possible value.
Instead, for positive circular polarization an important reduction of the amplitude is observed leading to the total
disappearance of the NRS. These results are understood recalling that for negative circular polarization and ω ω
c
≈
the electric field rotates in phase with respect to the electron cyclotronrotation. Based on the present results, it will
be highly recommended to carry out experiment for circular polarization configurations.
The present formalism can also be used in order to explore the non-linear regime in which multi-photon exchange
plays an essential role. As the microwave radiation intensity is increased, the analytical approximation breaks down
and the numerical expression in (35) with higher multipole (l) terms needs to be evaluated. In the explored regime
convergent results are obtained including terms up to the l = 3 multipole. Fig. 5a displays ρ vs. ǫ plots for
xx
electric field intensities E~ = 2.5V/cm and E~ = 5V/cm respectively, the increase on the field intensity produces a
| | | |
corresponding increase in the minima and maxima of ρ , but apart of this, the qualitative behavior in both cases
xx
is similar. A further increase of the electric field intensity to E~ = 10V/cm and E~ = 30V/cm (Fig. 5b), takes us
| | | |
to the nonlinear regime in which a qualitatively new behavior is observed. For ǫ > 2 the same NRS are observed,
however the width of these regions increase to include practically all the range from ǫ = j to ǫ = j + 1. Notice
2
that the negative resistanceminima has not a monotonousdependence on E , in fact for the strongestfield intensity
| |
the minima approaches zero. Remarkably, for E~ = 30V/cm and ǫ < 2, new negative resistance states associated
with transitions by two microwave photons are| o|bserved near ǫ = 1 and ǫ = 3. The minima of these states are
2 2
centered at ǫ = 0.52 and ǫ =1.52 respectively. Evidence of ZRS associated with multi-photon processes have
min min
been already observed by Zudov etal. (1); they reported structures with maxima near ǫ =1/2 and ǫ =3/2, and the
corresponding minima centered around ǫ = 0.67 and ǫ = 1.68 respectively. Dorozhkin5 and Willett etal. (6)
min min
have also reported ρ minimum associated with ǫ = 1/2. Although the exact position of the minima and maxima
xx
of ρ observed in Fig. 5b are not localized at the same position reported by Zudov etal. (1), the general pattern is
xx
verysimilar,supportingthe interpretationasmulti-photonprocesses. Clearly,amoresystematicanalysisandfurther
experimental studies are necessary.
Comparison with some other theoretical work is obliged. Previous work in references10,11,12,17 analyzed the effects
of the microwave radiation on the electron scattering by impurities in the presence of a magnetic field. Durst etal12
consider an out of equilibrium calculation, instead here a quasi-adiabatic approximation is implemented, assuming
thatthe systemisthermalizedinthosestatescharacterizedbythe Landau-Floquetspectrum. Thesimilaritybetween