Frohlich mass in GaAs-based structures

0.24 MB·English

by  C. Faugeras

Fr¨ohlich mass in GaAs-based structures
C. Faugeras,1 G. Martinez,1 A. Riedel,2 R. Hey,2 K. J. Friedland,2 and Yu. Bychkov1,3
1Grenoble high Magnetic Field laboratory, MPI-FKF and CNRS, B.P. 166, 38042 Grenoble Cedex 9, France
2Paul Drude Institute, Hausvogteiplatz 5-7, D-10117 Berlin, Germany
3L. D. Landau Institute for Theoretical Physics,
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Academy of Sciences of Russia, 117940 Moscow V-334, Russia
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(Dated: February 6, 2008)
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TheFr¨ohlichinteractionisoneofthemainelectron-phononintrinsicinteractionsinpolarmaterials

n originating from thecouplingof oneitinerant electron with themacroscopic electric fieldgenerated
a by any longitudinal optical (LO) phonon. Infra-red magneto-absorption measurements of doped
J GaAs quantum wells structures have been carried out in order to test the concept of Fr¨ohlich

9 interaction andpolaron mass insuch systems. These newexperimentalresultslead toquestion the
1 validity of this concept in a real system.

]
l
l Based on the Fr¨ohlich interaction, polaronic effects structure from the substrate (lift-off process). It is then
a
wereearlystudiedbyLeeandPines[1]andalsobyFeyn- possible to deposit it on an infra-red transparent sub-
h
- man[2]ontheoreticalgrounds: simpleconsequenceswere strate,likeSiforinstance,whichiswedgedwithanangle
s derived leading to the concept of the polaron mass, and of few degrees to avoid interference effects. This struc-
e
m reproduced in many text books [3]. This conceptual in- ture is composed of a single doped GaAs quantum well
teraction also attracted much attention, on the experi- (QW) of width L and areal density n sandwiched be-
. S
t mentalside,ineitherthreeorquasitwo-dimensional(3D tween two GaAs-AlAs superlattices with the Si-n type
a
m orQ2D)semiconductors[4]whichweremodelsystemsto dopingperformedsymmetricallyonbothsidesoftheQW
studytheeffect. ForthesimplestcaseofaQ2Dsemicon- [5]. This enables to grow structures of quasi 2D electron
-
d ductor, the single-particle energy spectrum of electrons gas (2DEG) combining high density and high mobility.
n of mass m∗, in a perpendicular magnetic field B⊥, con- We already reported results [6] on such a sample with
co sists of equally spaced (h¯ωc⊥ = eB⊥/m∗) energy states L = 10 nm,nS = 12×1011cm−2 [7]. The same process
[ knownas Landaulevels(LL)labelledwith anintegerin- has been applied to new samples with similar structures
dex n going from 0 to infinity: one direct consequence but with L = 13 nm and various lower doping levels.
1
of the polaronic concept is the prediction of the reso- Magneto infrared transmission measurements are done
v
nant magneto-polaron coupling (RMPC) which antici- inanabsolutewayforanyfixedmagnetic fieldB: this is
8
3 pates a non linear field variation of the cyclotron res- achieved with a rotating sample holder allowing to mea-
4 onance (CR) energy, h¯ω , when h¯ω ≃ ¯hω , with a sure reference spectra in the same conditions on a par-
c c LO −→
1 pinning of the CR above ¯hω . Such experimental ev- ent wedged piece of Si substrate. With the k vector of
0 LO −→
idence is quite difficult to obtain because the range of the incoming light parallel to B, the growth axis of the
6
energies concerned coincides with that of the strong ab- sample was maintained either parallel to this direction
0
/ sorptionofthephononband(Restrahlen)ofthematerial, (perpendicular Faraday (PF) configuration) or tilted by
t
a thereforeobscuringthedata. Thetheoryitselfraisesalso an angle θ with respect to it (tilted Faraday (TF) con-
m fundamentalquestions because,inrealmaterials,one al- figuration). The light was guided through an over-sized
- ways has many electrons which are known to oscillate guide pipe ending by a cone to minimize the divergence
d
in phase as plasmons developing in turn a macroscopic of the beam. The interpretation of the experimental re-
n
electric field which couples to that of the LO phonon, sultsisdonebycomparingthemwithasimulationofthe
o
c generating hybrid plasmon-phonon modes. What really multi-layer transmission by an appropriate model. This
: happens? Does the polaronic concept remains valid? To isessentialbecause,inthefrequencyrangeofinterest,the
v
i ourknowledgethereisnoquantummechanicaltreatment spectra can be distorted by dielectric interference effects
X
ofthehybridmodesandthereforeitisdifficulttoanswer independent of any specific electron-phonon interaction.
r these questions on theoretical grounds. The previous study [6] reported on PF configuration
a
Experimentsperformedinsuitableconditionsarehow- measurements where no sign of RMPC was observed.
ever able to bring new light in that matter. Overcom- This was not necessary a proof that the effect was not
ing the problem of the Restrahlen band is indeed tech- existing because it could have been screened by the high
nically very difficult in 3D semiconductors. In that re- value of n : indeed theoretical simulations [8], includ-
S
spectthe Q2Dstructures,grownbymolecularbeamepi- ing the RMPC, of data obtained with such high density
taxy (MBE), are more appealing because, besides the QW was still showing a small interaction. New exper-
factthat,withthe remotedopingtechnique, carrierscan iments with a lower n , for which this simulation pro-
S
remain quasi-free at the lowest temperatures, one can vided clear effects, were necessary to check the concept
remove, by selective chemical etching, the whole MBE oftheRMPC.Thedopinglevelofthenewlift-offsamples
2
1.0 40
Abs. Trans.100000......002468 a) hw TO Bqh= w=L 60O0 To hw TO hw LO gy (meV) 33332468 hhww LO b)
Abs. Trans.00000.....02468 b) C qB=R^ =22 02o0 .7 T CR ener 223680 TO Sample 1211 qqq === 012°56°.5°
1.0 40
ns.0.9 Sample 1200 V) 38 hw LO a)
Tra0.8 c) q= 20o me 36
el. 0.7 B^ = 20.7 T y ( 34
R0.625 30CR1 35 40 45 50 energ 32 hw TO q = 0°
Energy (meV) R 30 q = 14°
C 28 q = 20°
Sample 1200 q = 25°
FIG. 1: Experimental transmission spectra (empty dots) of 26
sample 1200. a) absolute transmission for B = 0T and θ = 18 20 22 24 26
60o;b)andc)absolutetransmissionandrelativetransmission B (T)
respectively for B⊥ = 20.7T and θ = 20o. The continuous ^
redcurvesarethefitofdatawith themodeldescribed inthe
FIG.2: Observedcyclotronenergiesforsample1200(a)and
text.
sample 1211 (b) for different tilt angles . The crosses and
black down triangles are the measured LO and TO energies
respectively and the empty squares the cyclotron energies in
ranges between n =7.0×1011cm−2 (sample 1200) and
S the PF configuration. Other points are the observed CR en-
nS =9.0×1011cm−2(sample1211)withrespectivemobil- ergies in theTFconfiguration: (a) fulldots: θ=14o,empty
ities of µ = 260m2/V/sec and µ = 280m2/V/sec dots : θ = 20o, stars : θ = 25o and (b) full dots : θ = 15o,
DC DC
[7]. All experimental results reported here have been empty dots : θ = 26.5o. Results in (b) for B⊥ lower than
obtained at a fixed temperature of 1.8 K. We will con- 18T correspond to filling factors > 2 when non parabolicity
effects are apparent.
centrate on the results obtained for B⊥ around 20 T for
which the CR span the Restrahlen band of GaAs. For
both samples, in that range of fields, the filling factor
of the structure ν =n h/(eB) remains essentially lower nation of four experimental ones and therefore the noise
S
than 2 and, in that case, non-parabolicity (NP) effects is more important. Howeverthe only remaining features
can be ignored. are those related to the contribution of the 2DEG. The
Typical spectra, for sample 1200, are displayed in energies of the CR1 and CR2 structures for each sam-
Fig.1. The absolute transmission (AT) at B = 0T and ple, traced as a function of B⊥, are displayed in Fig. 2
θ = 60o is shown in Fig. 1a. The strong absorption for different angles. Whereas in the PF configuration
features are those related to the TO phonons of GaAs (emptyblacksquares),nosignofinteractionisobserved,
and AlAs whereas the weak ones are those related to inthe TFconfigurationapronouncedanticrossingofthe
the LO modes which are known to become active in ab- CR1 and CR2 components occurs, which increases with
sorption for thin slabs [9] in the TF configuration. They θ, one of the components remaining pinned at an energy
clearly originate from the layers of the superlattices sur- of 35.6±0.05 meV for the sample 1200 and 35.5±0.05
rounding the doped QW. They are slab modes and give meVforthesample1211,bothenergiesbeinglower than
a lower bound of the energy of LO modes in the whole ¯hωLO(GaAs). Other samples with different densities of
structure. In Fig. 1b, the absolute transmission of the electrons(nS =5.8×1011cm−2 andnS =8.8×1011cm−2
sample in a TF configuration is displayed for θ = 20o and the same L of 13 nm or with a smaller L of 10 nm
and B⊥ = 20.7T. In practice with the available mag- andnS =12×1011cm−2,havealsobeeninvestigatedand
netic fields, we are led to work with angles where the demonstrate exactly the same qualitative features.
LO phonons are less visible but they are still clearly To interpret these results, one has to calculate the
present as indicated by the vertical dashed lines. Two transmission of the whole structure. This requires a full
CR transitions, labelled CR1 and CR2 (vertical dotted derivationofthemultilayerstructuretransmissioninthe
lines), are now present, the CR2 feature being, for that TF configuration, a rather complicated problem which
field,atalower energythanh¯ω (GaAs). InFig. 1cwe we have succeeded to solve in some specific cases and
LO
show the relative transmission (RT) obtained by divid- which will be detailed in a further publication [10]. We
ing the AT spectrum of Fig. 1b by the AT spectrum at just here outline the principles. We used a standard ap-
B =0T forthesameangle. TheRTspectraareacombi- proachwhichconsistsinevaluatingforeachlayerN,with
3
↔
a dielectric tensor εN(ω), the transfer matrix MN after B = 25.3 T
−→i ^
findingtheappropriatemodesofpropagationoflight k
N
inside the layer, by solving the Maxwell equations. Tak- B = 24.4 T
^
ingthez-axisperpendiculartothelayer,anddefiningthe
B = 23.5 T
y-z plane as the plane of incidence which also contains n ^
−→B, the angle θ is defined as the angle (−→z,−→B). There sio
are four modes ki for each layer. The 4×4 transfer mis B^ = 21.2 T
z,N
s
matrixisbuiltbywritingtheconservationofthetangen- n
a B = 20.8 T
tialcomponentsofthe electricandmagneticfields ofthe Tr ^
light at eachinterface of the layer. If ↔ε (ω) is diagonal, e
like in layers which do not contain freeNelectrons (or at ativ B^ = 19.9 T
B =0T)the transfermatrix decomposesinto twoblocks el
R
B = 18.1 T
of 2×2 matrices related to the TE and TM modes re- ^
spectively. This result is well known and has been used
[11] to measure the LO phonons frequencies of semicon- 50 % hw Sample 1200
ductorsinawaysimilartotheresultsreproducedinFig. TO q = 25.0°
1a. For the doped QW, in the TF configuration, all el- 26 28 30 32 34 36 38 40 42
↔
ements of ε (ω) and therefore of the transfer matrix
QW
Energy (meV)
arenonzero andthe TE andTM modes aremixed. The
resulting total transfer matrix has the same properties
FIG. 3: Fit of the relative transmission curves for sample
and a special treatment [10] needs to be developed to 1200 at θ = 25o. The curves are shifted for clarity. Empty
extract the transmission of the structure. This can be
dots are experimental points and full continuous curves the
done as soon as the dielectric tensor of the QW in the corresponding calculated relative transmission spectra. The
TF configuration has been defined. hatchedregioncorrespondstothatofh¯ωTO(GaAs)(seetext).
In the PF configuration, for any finite value of B, the
z-part of the 2DEG wave function is decoupled from the
and ω2 are the solutions of the equation:
x-y part. When θ 6=0, this decoupling is no longer valid 2
and as we shall see below is indeed responsible for the
ω4−(Ω2+ω2)ω2+ω2 Ω2 =0 (3)
observed anticrossing. One has then to reproduce the c c⊥
mixing of the z and x-y part of the 2DEG wavefunction with ωc = eB/(h¯m∗). When Ω >> ωc⊥, ω1 → ωc⊥
with the best accuracy. For the square QW which is un-
and ω → Ω. In the present case, however, the exact
2
derstudy,thismixingcanonlybeevaluatedinperturba-
values for the poles are introduced in the calculation.
tion theory [12]. On the other hand, if we assume the z-
This explains why in Fig. 2, far from the anticrossing
confining potential to be parabolic,with a characteristic
region the energies of the CR lines do not coincide with
energyh¯Ωseparatingtheelectricsubbands,thecoupling
the values obtained in the PF configuration.
canbe evaluatedanalyticallyatallorders. Ofcoursethe
Different fitting steps have to be performed. We first
strength of the coupling is modified, the dipole oscilla-
have to fit ε with standard values as already done [6]
L
tor strengths along the z-axis being different between a
and shown in Fig. 1a (red continuous lines). In fact, in
square QW and a parabolic one. It is this model, based
the region of h¯ω (GaAs), interference effects, between
TO
on the Drude formalism, that we have used to evaluate,
↔ phonon and CR absorptions become important and this
neglectingretardationeffects,allcomponentsof ε (ω)
QW isthereasonwhyweprefertointerprettherelativetrans-
and then simulate the transmission. Before discussing
missionspectra(Fig. 1candFig. 3)forwhichthesespu-
these simulations it is instructive to look at the struc-
↔ rious effects are minimized. They are however not elimi-
ture of two characteristic components of ε (ω). The
QW natedand,forthisreason,weignoreinthefittingprocess
realpart of ε and ε are expressed, ignoring damping
xx zz a region of ±0.5 meV around h¯ω (GaAs), sketched as
TO
effects, as [10] :
thehatchedregioninFig. 3. Inthis figure,experimental
data (empty dots) and their simulations (red continuous
ω2(ω2−Ω2) curves)for sample 1200are displayedfor θ =25o. There
p
εxx =εL− (ω2−ω2)(ω2−ω2) (1) are different fitting parameters to adjust: nS, m∗, τCR
1 2
, Ω and θ, τ being the scattering time entering the
ω2(ω2−ω2 ) CR
εzz =εL− (ω2−p ω2)(ω2c−⊥ω2) (2) Drude model . In the PF configuration, nS is fitted at
1 2 lower fields with the model extended for NP effects [10]
∗
together with m and τ . Outside the energy region
CR
where ε is the contribution of the lattice, ω2 = around h¯ω (GaAs) where there are signs of some in-
L p TO
4πe2n /(Lm∗) the square of the plasma frequency, ω2 teraction (at least for the sample 1200) which are not
S 1
4
∗
discussed here, the values of m fitted correspond quite frequency,thereisaclearsignofinteractionespeciallyin
well to the expected variationinduced by NP effects [13] sample1200. This is aninteractionwhichmaybe driven
∗
whereas the mobility µ = eτ /m ranges around by the mechanism of deformation potential which is at
CR CR
150m2/V/sec revealing the good quality of the samples. the origin, for instance, of the Raman response of such
The angle θ has also to be fitted because, for technical structures.
reasons, the Si substrate being wedged, θ can differ by
a few degrees from the mechanical angle. It is fitted by In conclusion, we have performed careful magneto-
setting the m∗ value constant for a given value of B⊥. optical measurements of the infra-red absorptionrelated
This is notcompletely independent ofthe value ofΩ but toquasitwo-dimensionalelectrongas,withdifferentden-
the loop easily converges. The value of B⊥ indicated in sities, in a range of magnetic fields which allows the cy-
the figures are calculated values of B ×cos(θ). Finally clotronresonancetospantheopticalrangeofphononen-
Ω is fitted in such a way the splitting of the CR1-CR2 ergies. All singularities observedcan be explained quan-
lines can be best reproduced. We impose also,in the fit- titatively without any electron-phonon coupling of the
ting process, the value of Ω to be constant for all angles Fr¨ohlich type. We argue that this support the idea that
and QW having the same width L. As an example, the the concept of the Fr¨ohlich polaron mass has to be re-
results, shown in Fig. 3 for sample 1200, are obtained examined in a real material.
for a value of Ω = 63 meV (with an uncertainty of ±3
The GHMFL is ”Laboratoire conventionn´e l’UJF et
meV).Theseresultsclearlydemonstratetheanticrossing
l’INPGdeGrenoble”. Theworkpresentedherehasbeen
features observed experimentally. The fit reveals some
supportedin partby the EuropeanCommissionthrough
slightdeficiencywhichissampledependentbuttheover-
the Grant HPRI-CT-1999-00030. Yu. B. acknowledges
all agreement is surprisingly good. It is interesting to
the support of the grant No. RFFI-03-02-16012.
note that if one calculates, at B = 0T, the zeros of ε
zz
(Eq. 2) with the fitted value of Ω, one finds for the low
energy modes 35.62 meV for the sample 1200 and 35.49
meV for the sample 1211, values which agree very well
with the experimental findings. These modes are the so [1] T. D.Lee and D.Pines, Phys.Rev., 92, 883 (1953)
called plasmon-phonon-intersubband (PPI) modes [14]. [2] R. P. Feynman,Phys.Rev., 97, 660 (1955)
Therefore the observed anticrossing occurs between the [3] C. Kittel, Quantum Theory of Solids (John Wiley and
CR mode and this hybrid PPI mode. They are coupled Sons, New York,1963) pp. 130-142
by symmetry but not by any specific electron-phonon [4] For a review, see e.g. J. T. Devreese, Polarons in Ionic
CrystalsandPolarSemiconductors(North-Holland,Am-
interaction. We think, though approximate, the model
sterdam, 1972)
keepsallthe physicalandsymmetry aspectsofthe prob- [5] K. J. Friedland et al., Phys. Rev.Lett., 77, 4616 (1996)
lem. One direct consequence is that, when Ω increases, [6] A. J. L. Poulter et al., Phys.Rev.Lett., 86, 336 (2001)
the oscillator strength of the PPI-like mode rapidly de- [7] The carrier densities reported in this paper, for all sam-
creasestozero[10]. Therefore,forapure2DEG,thereis ples, is obtained from the fit of data with the model de-
nolongeranyinteractionwhateveristheangle. Thedata tailed in thefollowing includingnon-parabolicity effects.
They are systematically lower by 6 to 8 per cent with
in the PF and TF configurations do not show any sign
respecttothevalueobtainedfromtransportdataonthe
of interaction related to the Fr¨ohlich coupling: we are
parent nonlift-off sample. Thevaluesofmobilities given
therefore led to conclude that the concept of polaronic
here are however those obtained from these data.
mass, related to this interaction, is no longer effective. [8] X. G. Wu et al.,Phys. Rev.Lett., 84, 4934 (2000)
All the discussion is based on the assumption that we [9] D. W. Berreman, Phys.Rev., 130, 2193 (1963)
are dealing with carriers for which a plasma frequency [10] Yu.Bychkov et al, tobe published.
can be defined. It is known that for bound electrons, [11] M. D.Sciacca et al., Phys. Rev.,B 51, 7744 (1995)
like the ones found in neutral shallow impurities, the [12] See for instance G. Bastard, Wave mechanics applied to
1s−2p+ absorption reveals [15] the Fr¨ohlich interaction semiconductorheterostructures(LesditionsdePhysique,
Paris, 1992)
around the energy of the LO phonon in GaAs but then [13] C. Hermann and C. Weisbuch, Phys. Rev. B 15, 823
the mass of such an electron is not defined. The same (1977)
situation holds for electrons in quantum dots [16]. It is [14] A. Pinczuk et al., Solid StateComm., 36, 43 (1980)
also true that there is still some electron-phonon inter- [15] S. Huant et al., Europhys.Lett., 7, 159 (1988)
action present in such system : indeed, around the TO [16] S. Hameau et al., Phys. Rev.Lett., 83, 4152 (1999)