Orbital mechanism of the circular photogalvanic effect in quantum wells

0.16 MB·English

by  S. A. Tarasenko

Orbital mechanism of the circular photogalvanic effect in quantum wells
S.A.Tarasenko
A.F. Ioffe Physico-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia
It is shown that the free-carrier (Drude) absorption of circularly polarized radiation in quantum
7 well structures leads to an electric current flow. The photocurrent reverses its direction upon
0 switching the light helicity. A pure orbital mechanism of such a circular photogalvanic effect is
0 proposed that is based on interference of different pathways contributing to the light absorption.
2 Calculation shows that themagnitude of the helicity dependentphotocurrent in n-doped quantum
n well structures corresponds to recent experimental observations.
a
J PACSnumbers: 73.50.Pz,73.50.Bk,78.67.De

2
1
I. INTRODUCTION structures can be induced only at oblique incidence of

] the radiation[2, 3]andisdescribedby twolinearlyinde-
ll Theabsorptionofcircularlypolarizedlightinsemicon- pendent constants γ1 and γ2 as follows:
a
h ductor structures may lead to generation of an electric
j = [γ l −γ l ]IP , (1)
- current which reverses its direction upon switching the x 2 x 1 y cicr
s
e light polarization from right-handed to left-handed and jy = [γ1lx−γ2ly]IPcicr,
m vice versa. Such a circular photogalvanic effect (CPGE)
causedbyasymmetryofelementaryprocessesofphotoex- where l = q/q is the unit vector pointing in the light
.
t citationrequiresstructuresoftheappropriatesymmetry. propagation direction, q and I are the wave vector and
a
m It is possible only in media which belong to one of the intensity of the light inside the structure, respectively;
gyrotropic classes [1, 2]. Pcirc is the radiation helicity ranging from −1 (for the
-
d Recently, CPGE has attracted a great deal of at- left-handed circular polarization) to +1 (for the right-
n handed circular polarization); x k [100], y k [010], and
tention. It was observed in low-dimensional struc-
o z k [001] are the cubic crystallographic axes. Phe-
tures: in zinc-blende- and diamond-type quantum wells
c nomenologically, the constant γ is related to the het-
[ (QWs), which are also belong to the gyrotropic me- 1
dia [3]. Circular photocurrents were studied for a va- erostructureasymmetry,i.e.,nonequivalenceofthezand
1 −z directions, while the constantγ originatesfromlack
rietyofopticalranges,including interbandtransitionsin 2
v of an inversion center in the host crystal.
QWs [4, 5, 6], direct intersubband [7] and indirect in-
4
At normal incidence of the light the helicity depen-
trasubband (Drude-like) [8] transitions in n-doped QW
7
2 structures. dentphotocurrentcanbe induced onlyin quantumwells
grown along one of the low-symmetric crystallographic
1 Sofar,themicroscopictheoryofCPGEinzinc-blende-
0 type structures has been developed for the direct inter- directions, such as (110)- or (113)-oriented structures.
7 In this geometry CPGE is described by
band or intersubband optical transitions. It has been
0
shown that the effect originates from the complicated
t/ bandstructureofthesemiconductorcompounds,mostly, jx′ =γ3lz′IPcicr, (2)
a
m from spin-orbit splitting of the electron or hole states,
where x′ is an in-plane axis, whichis pointed along[1¯10]
which is odd in the wave vector k [7, 9, 10], and linear-
- for (110)- and (113)-oriented QWs, and z′ is the growth
d in-k terms in the matrix elements of interband optical
direction. The phenomenological constant γ is entirely
n transitions[9,11]. HereweaddressCPGEcausedbythe 3
o free-carrier absorption of circularly polarized radiation relatedtoabsenceofaninversioncenterinthehostcrys-
c in n-doped QW structures. We show that a pure orbital tal, similarly to γ2 in Eq. (1).
: Below we present a microscopic theory for the orbital
v mechanism of the effect can predominate in this spec-
i tralrange,whichisrelatedtoneitherspin-orbitcoupling mechanismofCPGEandshowthatthismechanismcon-
X
tributes to γ as well as γ and γ constants.
nor spin-sensitive selection rules for the optical transi- 1 2 3
r
a tions. This mechanism is based on interference of dif-
ferent pathways contributing to the light absorption. A
similar CPGE model for the optical transitions between II. MICROSCOPIC THEORY
quantum subbands was proposed in Ref. [12]. However,
the orbital mechanism is especially efficient for the free- The free-carrier absorption of radiation is always ac-
carrierabsorption,whichis contributedby indirect opti- companied by electron scattering from static defects,
cal transitions. acoustic or optical phonons, etc., because of the need
WeconsiderCPGEin(001)-orientedQWsgrownfrom for energy and momentum conservation. Such indirect
zinc-blende-type semiconductors. Symmetry analysis optical transitions are treated as second-orderprocesses,
showsthatthe helicity dependentphotocurrentjinsuch which involve electron-photon interaction and electron
2
e2 e2 e2 e2

e1 k’ e1 k’ e1 k’ e1 k’
k kx k kx k kx k kx
FIG. 1: Intrasubband optical transitions (e1,k) → (e1,k′) FIG. 2: Intrasubband optical transitions (e1,k) → (e1,k′)
with intermediate states in the subband e1. Dashed circles with intermediate states in the subband e2. Dashed and
and dotted lines represent electron-photon interaction and dotted lines represent electron-photon interaction and elec-
electron scattering, respectively. Panels correspond to two tron scattering, respectively. Panels correspond to two pos-
possible processes: electron-photon interaction followed by sible processes: electron-photon interaction followed by elec-
electron scattering (left panel) and electron scattering fol- tron scattering (left panel) and electron scattering followed
lowed by electron-photon interaction (right panel). byelectron-photon interaction (right panel).
element of these indirect processes has the form
scattering, via virtual intermediate states. The interme-
diate states can be those within the same quantum sub- eA ε ε
M(e2) =i 21 − 21 z e V . (4)
band, e1 in our case, or in other conduction or valence k′k c¯h ε −¯hω ε +h¯ω 21 z 21
subbands. The dominant pathway determining the QW (cid:18) 21 21 (cid:19)
absorbance involve intermediate states within the sub- Here ε21 is the energy separation between the subbands
bande1(see Fig.1). The matrixelementofsuchkindof e2 and e1, z21 is the coordinate matrix element between
processes has the form [2] theenvelopefunctionsinthesubbands,ϕ1(z)andϕ2(z),
+∞
eA
Mk(e′k1) = cωm∗ e·(k′−k)V11. (3) z21 = ϕ2(z)zϕ1(z)dz, (5)
Z
−∞
Here k and k′ are the initial andfinal electronwavevec- and V is the matrix element of inter-subband scatter-
21
tors, e is the electron charge, A is the amplitude of the ing. In contrast to Eq. (3), the matrix element (4) is
electromagnetic wave related to the light intensity by nonzeroonlyforradiationwithnonvanishingout-of-plane
I = A2ω2nω/(2πc), ω is the light frequency, m∗ is the component of the polarization vector e and contains the
effective electron mass, nω is the refractive index of the imaginary unit as a prefactor.
medium, e is the (complex) unit vector of the light po- Lightabsorptionbyfreecarriersiscontributedbypro-
larization,andV11 isthematrixelementofintrasubband cesses with all possible intermediate states. Making al-
scattering. As follows from Eq. (3), optical transitions lowance for the transitions via the subbands e1 and e2
with intermediate states in the subband e1 can be in- one can write for the photocurrent
duced only by radiation with nonzero in-plane compo-
4π
nent of the polarization vector e. They do not lead to j=e [τp(εk′)vk′ −τp(εk)vk] (6)
¯h
any circular photocurrent because the square of the ma- k,k′
X
trix element (3), which determines the transition rate, is
2
independent of the light helicity. × Mk(e′k1)+Mk(e′k2) (fk−fk′)δ(εk′ −εk−¯hω),
Thehelicitydependentphotocurrentarisesifonetakes (cid:12) (cid:12)
into account interference of the processes depicted in where (cid:12)(cid:12)vk = h¯k/m∗ a(cid:12)(cid:12)nd εk = h¯2k2/(2m∗) are the elec-
Fig.1andthosewithvirtualintermediatestatesinother tron velocity and kinetic energy, respectively, τp(εk) is
subbands. the momentum relaxation time, which may depend on
theelectronenergy,fk isthe functionofequilibriumcar-
rier distribution in the subband e1, and the factor 4 in
Eq. (6) accounts for the spin degeneracy.
Contribution to γ1 The square of the matrix element sum in Eq. (6) con-
tains the interference term 2Re[M(e1)M(e2)∗], which is
k′k k′k
The quantum subband, which is the closest to e1, is odd in the wave vector. Therefore, it leads to an asym-
the electron subband e2. Optical transitions with in- metrical distribution of the photoexcited carriers in k-
termediate states in the subband e2 contributing to the space, i.e., to an electric current. Moreover, the inter-
free-carrierabsorptionaresketchedinFig.2. Thematrix ference term is proportional to components of i[e×e∗].
3
It vanishes for the linearly polarized light and is pro- is the Fermi energy, k is the Boltzmann constant, and
B
portional to the light helicity P for the elliptical or T is the temperature).
circ
circular polarization, because i[e×e∗] = lP . Thus, Atlowtemperaturesthefree-carrierabsorptionisdom-
circ
the sign of the interference term is determined by the inatedbyprocesses,whichinvolveelasticscatteringfrom
light helicity, and the photocurrent reverses its direction staticdefectssuchasimpurities,imperfectionsoftheQW
by changing the light polarization from right-handed to interfaces, etc. For electron scattering from short-range
left-handed and vice versa. defects the parameter ξ has the form
Physically, the orbital mechanism of the circular pho-
+∞ +∞
togalvaniceffectcanalsobeinterpretedasfollows. Under
ξ = ϕ3(z)ϕ (z)w(z)dz ϕ4(z)w(z)dz, (9)
excitation with circularly polarized light the processes 1 2 1
,
Z Z
depicted in Fig. 1 and Fig. 2 are added constructively −∞ −∞
for transitions to positive kx′ (or negative kx′, depending where w(z) is the distribution function of the scatter-
on the light helicity) and destructively for transitions to ers along the growth direction. At higher temperatures
−kx′. It means that the transition rates to the states kx′ electron-phonon interaction predominates over the elec-
and −kx′ are different, resulting in an electric current of tron scattering by static defects. In the case of quasi-
the photoexcited carriers. Direction of the photocurrent elasticscatteringfromacousticphononsthe parameterξ
withrespecttothecrystallographicaxesisdeterminedby isalsogivenbyEq.(9)wherew(z)issettobeaconstant.
the lightpolarizationandthe explicitformofthe matrix In accordance with general symmetry arguments, the
elements of intrasubband and intersubband scattering. helicity dependent photocurrent corresponding to the
The matrixelement(4)is proportionaltoout-of-plane phenomenological constant γ is related to inversion
1
component of the polarization vector. Therefore, the asymmetryoftheheterostructure. Thisfollowsalsofrom
term caused by interference of the optical processes via Eqs. (7) and (9), which demonstrate that the sign and
thesubbande1ande2cancontributetoCPGEatoblique magnitude ofγ is determinedby asymmetryofthe con-
1
incidence only. We consider the effect for (001)-grown finement potential and the doping profile. In particular,
quantum wells. In such structures electron scattering γ ≡ 0 for the absolutely symmetrical structures, where
1
by static defects, acoustic or optical phonons is usually w(z) and ϕ (z) are even functions while ϕ (z) is an odd
1 2
treated as central. Under this assumption the matrix el- function with respect to the QW center.
ementsofintrasubbandandintersubbandscattering,V
11
andV , respectively,takenoaccountofthe lattice sym-
21
metry of the host semiconductor and depend on |k′−k| Contribution to γ2 and γ3
only. Then, the helicity dependent photocurrentflowsin
the direction perpendicular to the light incidence plane. To obtain the circular photocurrents described by the
This contribution correspondsto the terms described by phenomenological constants γ in Eq. (1) and γ in
2 3
thephenomenologicalconstantγ1inEq.(1). Calculation Eq.(2)onehastotakeintoaccountthelatticesymmetry
after Eq. (6) shows that, in the case of elastic scattering oftheQWhostsemiconductor. Thiscanbedoneconsid-
and provided h¯ω ≪ε21, the constant is given by eringinterferenceofopticaltransitionswithintermediate
ωz states in the subband e1 and those via the valence-band
21
γ =−2eτ ξη , (7)
1 p ε k states.
21
To calculate this contribution to CPGE we neglect
where ξ is a dimensionless parameter, which depends on spin-orbitsplittingofthevalencebandforsimplicityand
the structure design and mechanisms of scattering, and assume that the effective hole masses in the QW plane
ηk is the QW absorbance for radiation polarized in the are larger than the effective electron mass. Then, the
QW plane. We note that both the absorbance ηk and matrix element of the intrasubband optical transitions
the momentum relaxation time τp are governed by the via the valence-band states can be presented as
matrixelementofintrasubbandscatteringV . Thus,the
11 eA¯hω
product τ η is independent of the scattering rate. For M(v) =i P e V . (10)
p k k′k c¯h E2 j SRj
quasi-elastic scattering from acoustic phonons or short- g j
X
range static defects, the product is given by
Here E is the band gap energy, P = i(h¯/m )hS|p |R i
g 0 z z
2πα κ¯h isthe Kanematrixelement,m0 isthefreeelectronmass,
τ η = N , (8)
p k n m∗ω2 e S andRj (j =x,y,z)aretheBlochfunctionsofthecon-
ω
duction and valence bands at Γ-point of the Brillouin
where α = e2/¯hc ≈ 1/137 is the fine-structure constant, zone, respectively, and V are the matrix elements
SRj
N isthecarrierdensity,andκisaparameterdepending of inter-band scattering. For scattering from acoustic
e
on the carrier distribution. The latter is equal to 1 and phononsthematrixelementsV havetheform[13,14]
SRj
2 for the cases ¯hω ≫ ε¯ and h¯ω ≪ ε¯, respectively, with
V =Ξ hk′|u |ki, (11)
ε¯being the mean kinetic energy of equilibrium carriers SRx cv yz
V =Ξ hk′|u |ki,
(ε¯=EF/2foradegeneratetwo-dimensionalgasandε¯= SRy cv xz
k T foranon-degeneratetwo-dimensionalgas,whereE V =Ξ hk′|u |ki,
B F SRz cv xy
4
where Ξ is the interband deformation-potential con- upon the replacement k → −k and k′ → −k′. Such an
cv
stant,hk′|u |ki arethe matrix elements ofcouplingbe- anisotropy in scattering can be modeled, e.g., by impu-
αβ
tween the electron states k and k′ in the subband e1 rity potential of the form [15]
caused by the deformation, and u (α 6= β) are the
αβ
phonon-induced off-diagonal components of the strain Ui(r)=U0(|r−ri|)+U3(|r−ri|)(x−xi)(y−yi)(z−zi),
tensor. The interband constant Ξ originates from lack (14)
cv
ofaninversionsymmetryinzinc-blende-typecrystals(Td where ri is the impurity position. The antisymmetric
point group) and vanishes in centrosymmetric semicon- part of the impurity potential does not violate inherent
ductors [13]. isotropy of electrical properties of bulk cubic materials
Calculation shows that in (001)-oriented QWs the and, therefore, can not be easily detected [16]. How-
matrix element M(v) contains a term proportional to ever,in(001)-orientedQWsitaddsatermtothe matrix
k′k
element of electron scattering, which is proportional to
(kx′ −kx)(ky′ −ky)ez. Interferenceofthis termandMk(e′k1) (kx′ −kx)(ky′ −ky) and, therefore, contributes to γ2. We
gives rise to the circular photocurrentdescribed byγ in
2 assumethattheimpurityrangeismuchshorterthanthe
Eq.(1). Forthefree-carrierabsorptionassistedbyquasi-
QWwidthandtheinequalityh¯ω ≪ε isfulfilled. Then,
21
elastic scattering from acoustic phonons the constant γ
2 thecontributiontoγ causedbythescatteringanisotropy
2
has the form
has the form
γ2 =−2eτpΞΞccv ωEPg2 ζηk, (12) γ2′ =eτpuu30ωεz2211m¯h∗2E Qη, (15)
where Ξc is the intraband deformation-potential con- where u0 and u3 are related to the symmetric and anti-
stant,whichdeterminesthe lightabsorbanceηk,andζ is symmetric parts of the impurity potential by
aparameterdependingontheQWwidth,thecarrierdis-
tributionandthephotonenergy. Forarectangularquan-
u = U (|r|)dr, u = U (|r|)x2y2z2dr; (16)
tum well with the infinitely high barriers, ζ = 8πk¯a/3 if 0 0 3 3
Z Z
the photon energy is much less than the mean electron
E is an energy given by E = 6ε¯if h¯ω ≪ ε¯and E = h¯ω
energy and ζ = k a/12 in the opposite limiting case.
ω
Here k¯ is the mean value of |k| (k¯ = 2k /3 for a degen- if h¯ω ≫ε¯; Q is a parameter,which is determined by the
F
QW confinement potential and the doping profile,
erate two-dimensionalgas and k¯ = πm∗k T/(2h¯2) for
B
a non-degenerate two-dimensional qgas, where kF is the +∞dϕ3(z)ϕ (z) +∞
Fermi wave vector), k = 2m∗ω/¯h, and a is the QW Q= 1 2 w(z)dz ϕ4(z)w(z)dz. (17)
ω dz 1
width. Z , Z
p −∞ −∞
In(110)-orientedQWsinterferenceoftheopticaltran-
sitions with intermediate states in the subband e1 and The parameterQ equals to 2π/a for a rectangularquan-
in the valence band leads to the helicity dependent pho- tum well with infinitely high barriers where the layer of
tocurrent even at normal incidence of the light. Calcu- impurities is placed in the QW center.
lation shows that for this particular mechanism the con-
stant γ has the form
3
III. DISCUSSION
γ
2
γ =− . (13)
3
4ζ
We have shown that the free-carrier absorption of cir-
It is independent of the QW width and is determined cularly polarized radiation in quantum well structures
solely by the carrier density, the scattering mechanism leads to an electric current, which reverses its direction
and the band parameters. upon switching the light helicity. The proposed pure or-
As is mentioned above, the phenomenological con- bital mechanism of the circular photogalvanic effect is
stantsγ andγ arerelatedtolackofaninversionasym- based on interference of different pathways contributing
2 3
metryinthebulksemiconductorratherthantothestruc- to the light absorption and does not involve spin of free
ture asymmetry. This follows also from Eqs. (12) and carriers.
(13), whichshowthatbothγ andγ donotvanisheven In QW structures grownalong[110]or [113]crystallo-
2 3
in symmetrical QWs grown from zinc-blende-type com- graphicdirectionthecircularphotocurrentcanbeexcited
pounds. at normal incidence of the radiation. Following Eqs. (2)
Another orbital contribution to γ can originate from and (13) we can estimate the magnitude of the circular
2
interference of the processes via the subbands e1 and photocurrent density. It gives j ∼ 2×10−10A/cm for
e2 (depicted in Figs. 1 and 2, respectively) in the pres- the photon energy h¯ω = 10meV, the light intensity I =
enceofscatteringanisotropy. Indeed,theTd pointgroup 1W/cm2, the carrier density Ne = 1012cm−2, and the
of zinc-blende-type semiconductors does not contain the band parameters P/¯h ≈ E /(2m∗) ≈ 1.3×108cm/s,
g
space inversion and allows for an antisymmetric term in |Ξ /Ξ | ≈ 0.36 [13], which are appropriate to GaAs-
cv c
p
the matrix element of scattering, which changes the sign based structures.
5
In (001)-oriented QWs the helicity dependent pho- considerably smaller than the orbital term.
tocurrent can be induced at oblique incidence of the Additional experiments, such as measurements of the
radiation only, and the current magnitude varies in a frequency and temperature dependencies of the pho-
wide range depending on the QW width and asymme- tocurrent, can be useful in establishing the roles of the
try. Equations(1)and(7)givej ∼10−8A/cmforaQW orbital and spin-related mechanisms of CPGE more re-
of the width a = 150˚A, the asymmetry degree ξ = 0.2, liable. In particular, the ratio γ /γ is expected to be
1 2
l = 0.2, and the photon energy, the light intensity and independentofthephotonenergyh¯ω forthespin-related
x
the carrierdensity presentedabove. The estimatedmag- mechanism, while it does depend on h¯ω for the orbital
nitudeofthephotocurrentcorrespondstothatmeasured mechanism provided h¯ω > ε¯. Besides, increase of the
in experiments on (001)-oriented structures [3]. temperaturefromliquidheliumtotheroomtemperature
In addition to the orbital mechanism, a contribution leads to changing the dominant scattering mechanism.
to CPGE in quantum wells structures may come from It can strongly affect the orbital contribution to CPGE,
k-linearspin-orbitsplittingofthequantumsubbandsto- which is inherently related to details of scattering.
getherwiththespin-sensitiveselectionrulesforthe opti- Acknowledgments. The author acknowledges use-
cal transitions. However,estimations show that the spin ful discussions with E.L. Ivchenko, L.E. Golub, and
contributionto the circularphotocurrentinduced by the N.S. Averkiev. This work was supported by the RFBR,
free-carrier absorption in n-doped structures vanishes to programs of the RAS, President Grant for young scien-
the first order in spin-orbit interaction and, therefore, is tists, and the Russian Science Support Foundation.
[1] B.I.SturmanandV.M.Fridkin,ThePhotogalvanicEffect W. Wegscheider, D. Weiss, and W. Prettl, Conversion
andRelativePhenomenainNon-CentrosymmetricMedia of spin into directed electric current in quantum wells,
(inRussian)(Nauka,Moscow,1992)[Englishtranslation: Phys. Rev.Lett. 86, 4358 (2001).
The Photovoltaic and Photorefractive Effects in Non- [9] Yu.B. Lyanda-Geller and G.E. Pikus, Circular photo-
Centrosymmetric Materials (GordonandBreachScience galvanic effect in deformed nongyrotropic crystals, Fiz.
Publishers, New York,1992)]. Tverd. Tela 31(12), 77 (1989) [Sov. Phys. Solid State
[2] E.L. Ivchenko, Optical Spectroscopy of Semiconductor 31, 2068 (1989)].
Nanostructures (AlphaScienceInt.,Harrow, UK,2005). [10] L.E.Golub,Spin-splitting-inducedphotogalvaniceffectin
[3] S.D. Ganichev and W. Prettl, Spin photocurrents in quantum wells, Phys.Rev.B 67, 235320 (2003).
quantum wells, J. Phys.: Condens. Matter 15, R935 [11] J.B. Khurgin, Quantum interference control of electri-
(2003). cal currents and THz radiation in optically excited zinc-
[4] V.V. Bel’kov, S.D. Ganichev, Petra Schneider, C. Back, blende quantum wells, Phys.Rev. B 73, 033317 (2006).
M. Oestreich, J. Rudolph, D. H¨agele, L.E. Golub, [12] L.I. Magarill and M.V. Entin, Photogalvanic effect in
W. Wegscheider, and W. Prettl, Circular photogalvanic quantum-sizedsystem,Poverkhnost’1,74(1982)(inRus-
effect at inter-band excitation insemiconductor quantum sian).
wells,Solid State Commun. 128, 283 (2003). [13] G.E.PikusandA.N.Titkov,inOptical Orientation,eds.
[5] M. Bieler, N. Laman, H.M. van Driel, and A.L. Smirl, F. Meier and B.P. Zakharchenya (Elsevier Science, Am-
Ultrafast spin-polarized electric currents injected in a sterdam, 1984).
strainedzincblendesemiconductor bysinglecolorpulses, [14] E.L. Ivchenko and S.A. Tarasenko, Monopolar optical
Appl.Phys.Lett. 86, 061102 (2005). orientation of electron spins in bulk semiconductors and
[6] C.L. Yang, H.T. He, L. Ding, L.J. Cui, Y.P. Zeng, heterostructures, Zh. E´ksp. Teor. Fiz. 126, 426 (2004)
J.N. Wang, and W.K. Ge, Spin photocurrent and con- [JETP 99, 379 (2004)].
versespinpolarizationinducedinaInGaAs/InAlAstwo- [15] L.S.Levitov,Yu.V.Nazarov,andG.M.E´liashberg,Mag-
dimensional electron gas, Phys. Rev. Lett. 96, 186605 netoelectric effects in conductors with isomer symmetry,
(2006). Zh. E´ksp. Teor. Fiz. 88, 229 (1985) [Sov. Phys. JETP
[7] S.D. Ganichev, V.V. Bel’kov, Petra Schneider, 61, 133 (1985)].
E.L. Ivchenko, S.A. Tarasenko, W. Wegscheider, [16] Impuritypotential of theform (14) can giverise to non-
D. Weiss, D. Schuh, E.V. Beregulin, and W. Prettl, diagonalcomponentsoftheconductivitytensor,σxy and
Resonantinversionofthecircularphotogalvaniceffect in σyx, in asymmetrical (001)-grown QWs.
n-doped quantum wells,Phys.Rev.B68,035319 (2003).
[8] S.D. Ganichev, E.L. Ivchenko, S.N. Danilov, J. Eroms,