Role of finite layer thickness in spin-polarization of GaAs 2D electrons in strong parallel magnetic fields

5 Pages·0.18 MB·English

by  E. Tutuc

Role of finite layer thickness in spin-polarization of GaAs 2D electrons in strong
parallel magnetic fields
E. Tutuc, S. Melinte, E.P. De Poortere, M. Shayegan
Department of Electrical Engineering, Princeton University, Princeton, NJ 08544
3
0 R. Winkler
0 Institut fu¨r Technische Physik III, Universit¨at Erlangen-Nu¨rnberg, Staudtstr. 7, D-91058 Erlangen, Germany
2 (Dated: February 1, 2008)

n Wereportmeasurementsandcalculationsofthespin-polarization,inducedbyaparallelmagnetic
a field,ofinteracting,dilute,two-dimensionalelectronsystemsconfinedtoGaAs/AlGaAsheterostruc-
J
tures. The results reveal the crucial role the non-zero electron layer thickness plays: it causes a

7 deformation of the energy surface in the presence of a parallel field, leading to enhanced values for
2 theeffective mass and g-factor and a non-linearspin-polarization with field.

] PACSnumbers: 73.50.-h,71.70.Ej,73.43.Qt
l
l
a
h
The spin-polarization of an interacting, dilute two- B, and C). The samples were all modulation-doped
-
s dimensional (2D) carrier system has been of interest GaAs/AlGaAs heterojunctions with n in the range 0.8
e for decades. It has long been expected that because to 6.5×1010 cm−2. Their low-temperaturemobility var-
m of Coulomb interaction the product g∗m∗, which de- ied depending on the sample and n; atn=2×10 cm−2,
. termines the spin susceptibility of the 2D system, in- it ranged from about 2×105 to 2×106 cm2/Vs. Sam-
t
a creases as the 2D density (n) is lowered and eventually ples were patterned in either van der Pauw or Hall bar
m
divergesas the system makes a transitionto a ferromag- shapes,andwerefittedwithback-orfront-gates. Totune
- netic state at sufficiently low n [1, 2] (g∗ and m∗ are n in samples A, B1, and B2, following illumination with
d
the carrier Land´e g-factor and effective mass, respec- a red LED, we used front-gate bias; for B3 (in the range
n
o tively). Recently, there has been much renewed inter- n<4×1010 cm−2)andCweusedback-gatebiasandno
c est in this problem, thanks to the availability of high- illumination. ForB3,the highestdensity (n=6.5×1010
[ quality dilute 2D systems, and the belief that it may cm−2)wasobtainedafterillumination,followedbyback-

shed light on the controversialissue of a metal-insulator gating to reduce n to 4.5×1010 cm−2. Measurements
2
v transition in 2D [3]. A technique commonly used to were done down to a temperature of 30 mK, and a ro-
7 study the spin-polarization is to measure the response tating platform was used to tune the angle between the
2 of the 2D system to a tilted or parallel magnetic field applied magnetic field and the sample plane.
0 [4, 5, 6, 7, 8, 9, 10, 11]. The results of some of these
1 Figure 1 summarizes our data taken on the different
measurements [8, 9, 10], however, appear to be at odds
0 with whatis theoreticallyexpected [1, 2] for a dilute, in- samples. Plotted are the values of g∗m∗/gbmb, deter-
3 minedfromthe parallelmagneticfield, B ,atwhichthe
teracting 2Dsystemthat is otherwise ideal, i.e., has zero P
/0 layer thickness and is disorder-free. In particular, when 2D system becomes fully spin polarized (mb = 0.067m0
at g∗m∗isdeducedfromparallelmagneticfieldatwhichthe and gb = −0.44 are the band effective mass and Land´e
g-factorforGaAselectrons;m0isthefreeelectronmass).
m 2Dsystembecomesfully spin-polarized,thenthe experi-
The parallel magnetic field (B ) leads to the formation
mentalresultsforGaAs2Delectrons[8]andholes[9,10] k
- of two energy subbands, one for each spin, and sepa-
d suggest a decreasing value of g∗m∗ with decreasing n,
n opposite to the theoretical predictions. rated by the Zeeman energy, EZ = |g∗|µBBk, where µB
o is the Bohr magneton. The 2D system becomes fully
c Here we report a combination of measurements and spin-polarized above a field B at which E equals the
P Z
: calculations for the parallel magnetic field-induced spin- Fermienergy. TheequalityleadstoanexpressionforB :
v P
i polarization of 2D electrons at the GaAs/AlGaAs het- BP =(h2/2πµB)·(n/|g∗|m∗), from which we determine
X erojunction. The results highlight the importance of the g∗m∗ that are plotted in Fig. 1.
r finitethicknessoftheelectronlayerandthe resultingde-
a The procedures we have used to experimentally de-
formation of the energy surface E(k ), where k is the
k k termine B have been described elsewhere [5, 8]; here
in-plane wave vector, that occurs in the presence of a P
we give a brief summary. We determine B from two
strong parallel field. This deformation induces an en- P
independent sets of experiments: Shubnikov - de Haas
hancement of both m∗ and g∗ and leads to a non-linear
(SdH) measurements in a nearly parallel magnetic field,
spin-polarization in a parallel field. We find that, once
and magnetoresistance measurements in a strictly par-
the effect of the finite layer thickness and interaction is
allel field. In the first type of experiment, we apply a
taken into account, there is reasonable agreement be-
constant magnetic field (B ) whose initial direction is
tween the experimental data and calculations. tot
parallel to the 2D electron plane, and then slowly rotate
We used five samples from three different wafers (A, the sample while recording the sample resistance as a
2
experimental 2D electrons non-ideal, namely their finite
layerthickness,appearstoberesponsibleforthesample-
9
A dependentg∗m∗ andthedifferencebetweentheobserved
8 and expected density dependence of g∗m∗.
InthepresenceofalargeB ,whenthemagneticlength
k
7
mb B1 B3 (= ¯h/eBk)becomescomparabletoorsmallerthanthe
*m/gb 6 thicpkness of the electron layer, the energy surface E(kk)
*g 5 of the electrons gets deformed in the in-plane direction
B2 perpendicular to Bk. The deformation leads to an in-
4 C creaseofthein-planeeffectivemass,m∗,which,insecond
order perturbation theory, is given by [12]
3
20 1 2 n (31010 cm4-2) 5 6 7 m∗(Bk)=mb(cid:30)vuu1− 2em2Bbk2 Xj6=0 E|jhz−i0Ej|0 (1)
t
where the sum runs over all excited subbands j and z
FIG. 1: Values of g∗m∗/gbmb, determined from the paral-
is the quantization axis [13]. Data of Fig. 2 provide an
lel magnetic field BP at which the 2D electrons in GaAs
experimentaldemonstrationofthiseffectinoursamples.
become fully spin-polarized, are shown as a function of 2D
electrondensity. Resultsareshownforfivesampleswithlow- In Fig. 2(a) we show, as a function of Bk, the measured
temperature mobilities (at n = 2×1010 cm−2) of A: 4, B1: magnetoresistance of samples A and B1, both at a den-
16, B2: 16, B3: 7, C: 2×105 cm2/Vs. The closed symbols sity of 2.7×1010 cm−2. The magnetoresistance for each
representBP determinedfromSdHmeasurementsinanearly sample shows a clear change in its dependence on Bk at
parallel magnetic field while open symbols are for BP from a field marked by a vertical arrow as B . As demon-
P
magnetoresistance measurements in a parallel field. strated previously [5, 8], the field B marks the onset of
P
fullspin-polarization. Note inFig.2(a)thatB islarger
P
for sample B1 than A even though they have the same
function of the angle between the 2D plane and the field density. We also measured m∗ in the two samples as a
direction. Ifwelimitourselvestosmallangles,thefield’s functionofB ,asshowninFig.2(b)[14]. Wedetermined
k
parallelcomponent (Bk) remains almost constant (equal m∗ fromthetemperaturedependenceoftheamplitudeof
to Btot typically to better than 1%) during the rotation, the SdH oscillations, measured as the sample was slowly
while its perpendicular component (B⊥) changes suffi- rotatedinanalmostparallelfield. We performedastan-
ciently to probe the SdH oscillations. We then Fourier dard analysis, fitting the amplitude of the SdH oscilla-
analyzethe SdHoscillationsto obtainthepopulationsof tions(∆R)totheDingleformula,∆R∼ξ/sinhξ,where
the two spin subbands. These densities provide a direct ξ ≡2π2k T/¯hω andω =eB/m∗. ItisclearinFig.2(b)
B c c
measure of the spin-polarization of 2D electron system thatm∗ for both samples exhibits a strongenhancement
and allow us to determine the field BP above which the with increasing Bk, consistent with Eq. (1). Moreover,
system becomes fully spin-polarized. In the second type m∗ has a larger enhancement for sample A than for B1.
ofexperimentwemeasurethesampleresistanceasafunc- This stronger enhancement correlates with the smaller
tion of a magnetic field applied strictly in the 2D plane. B measured for sample A. We believe that the main
P
Asshownelsewhere[5,8],thein-planemagnetoresistance difference between the two samples in Figs. 2(a) and (b)
showsa markedchangeinits functionalformatthe field is that sample A has a larger layer thickness: because
BP. For a given sample, the value of BP obtained from of a decreasein subband separation,the larger thickness
the two types of experiments (open and closed symbols leads to the larger enhancement of m∗, consistent with
in Fig. 1) are in agreement. the smaller B for sample A compared to B1.
P
Data of Fig. 1 illustrate that the product g∗m∗, de- To further substantiate the connection between layer
duced from B as described in the last two paragraphs, thicknessandm∗enhancement,inFig.2(c)weshowdata
P
deviatessubstantiallyfromwhatisexpectedforanideal, for sample A at a higher carrier density of 4.2 × 1010
interacting 2D electron system, namely a monotonically cm−2. The measured m∗ enhancement is smaller for the
increasing g∗m∗ as n is lowered [1, 2]. The data also re- higher n state. This is consistent with layer thickness
veal that the measured g∗m∗ is sample dependent and being responsible for the m∗ enhancement: as we use a
not a unique function of n. A possible reason for this morepositivefront-gatebiastoincreaseninthissample,
non-uniqueness may be the sample disorder that indeed the electron wavefunction is squeezed more towards the
varies between different samples. An examination of the interface so that the layer thickness is reduced [see the
data, however, argues against this hypothesis: consider- insetsto Fig.2(c)], consistentwiththe smallermeasured
ingthedataatagivendensity,itisclearthatthereisno m∗ enhancement.
simpletrendlinkingthesampledisorder,asdeducedfrom Inordertoquantitativelyunderstandtheexperimental
thelow-temperaturemobility,tothemeasuredg∗m∗. As data,wehavedone self-consistentdensity-functionalcal-
we demonstrate below, another factor that renders the culationsofthe subbandstructureinthe presenceofB .
k
3
10
(a) 4 sample A (a)
T=30 mK n = 4.2×1010 cm-2
B
P
B=0)|| -2cm) 3
R(B) / R(|| sample B1 sample A 10n, n (10 “” 2
1
experiment
1 BP n=2.7×1010 cm-2 calculations
0
2.5 6.0
2.5
(b) (b)
2.0 5.0
2.0 BP mb 1.5 4. 0 gb
mb sample A *m/ 1.0 *g/
*m/ 1.5 sample B1 3.0
0.5
2.0
B
P 0.0
1.0 0 3 6 9 12 15 18
n = 2.7×1010 cm-2 B (T)
||
2.5
sample A (c) 2.7×1010 cm-2 FIG.3: Resultsofcalculationsareshownfor(a)spinsubband
densities n and n , (b) effective mass, and the g-factor for
↑ ↓
2.0 BP a GaAs 2D electron system (density 4.2×1010 cm−2) with
realistic finite layer thickness. Also plotted (circles) are the
mb experimentally measured m∗, n and n at this density.
*m/ 1.5 BP ↑ ↓
1.0 4.2×1010 cm-2 Figure 3 provides an example of the results of the cal-
culations for n = 4.2 × 1010 cm−2; shown are (a) the
0 3 6 9 12 15 18 spin-subband densities n↑ and n↓, (b) m∗, and (c) g∗,
B(T) as a function of B . The calculations were done using
|| k
the parameters of sample A (spacer thickness and bar-
FIG.2: (a)Parallel-fieldmagnetoresistanceofsamplesAand rier height), assuming a p-type background doping of
B1at2.7×1010 cm−2. (b)Effectivemassesmeasuredforthe 2.7×1013 cm−3, anda binding energyof90meV for the
two samples of panel (a). (c) Effective masses measured for dopants (Si) in the barrier. These are reasonable values,
sampleAattwodifferentdensities. Thedensityistunedviaa consistent with our estimate of the unintentional (resid-
front-gatebiaswhichleadstoanarrowingofthewavefunction
ual) doping in our molecular beam epitaxy system and
at higher density, as shown schematically in the insets. In
the binding energies quoted in the literature [16]. The
all three panels, the fields BP above which the 2D electrons
calculations predict a non-linear but smooth increase of
become fully spin-polarized are marked by vertical arrows.
the spin-polarization as a function of B .
k
In Figs. 3 (a) and (b) we have also included our mea-
We used the recent parameterization of the exchange- sured spin-subband densities and m∗, determined from
correlation energy by Attaccalite et al. [2]. The en- SdHoscillationsinanearlyparallelmagneticfield. Over-
ergy and length scales for electrons in a semiconduc- all, there is good agreement between the experimental
tor are characterized by the effective Rydberg and the data and calculations for both spin-polarization and m∗
Bohr radius according to the effective mass m∗ and the as a function of Bk. The calculated BP =14.5 T agrees
dielectric constant of the material. In our calculations well with the measured BP ∼=16 T, and is much smaller
it was crucial that m∗ was determined as a function of than BP ∼= 48 T expected for an interacting GaAs 2D
B from the self-consistently calculated subband disper- electronsystemwithzerolayerthicknessatn=4.2×1010
siokn E(k ,B ). These calculations confirm the qualita- cm−2 (see dashed curve in Fig. 4).
k k
tivetrendsexpectedfromEq.(1). Bandstructureeffects To understand the density dependence of B for a
P
beyond the effective-mass approximation are not consis- given sample, we also calculated B as a function of n,
P
dered here. We have checked that these effects are of which is tuned either with a front- or back-gate bias.
minor importance [15]. In the calculations, the field B For these calculations, we kept the sample parameters
P
is defined as the smallest value of B for which the fully fixed, and only changed the boundary conditions for the
k
spin-polarized configuration has the lowest total energy. Hartree potential thus simulating the effect of the gate
4
80 zero-thickness 2D system is approximately (within 5%)
8 linear with B in this density range or, equivalently, the
k
mb Zeemansplitting canbe expressedwellintermsofanef-
6 gb
* m/ fective g-factorindependent of Bk. This implies that the
4*g finite layer thickness is the key factor that leads to the
2 sample B1 observed non-linearity in spin-polarization with Bk and
0 1 2 3 4 5
the resulting reduction of B . The mechanism responsi-
T) 10 n (101 0 cm-2) ble for this non-linearity canP be summarized as follows:
B(P n (10 10 cm-2) B inducesanincreaseofm∗ duetothefinitelayerthick-
k
0.1 0.5 ness. This has a twofold effect. It directly reduces B
P
1 because B ∝ 1/m∗. Furthermore, the increase of m∗
P
T) reduces the effective Bohr radius and thus increases rs,
sample A B(P0.1 the average electron spacing measured in units of effec-
tive Bohr radius. The increase in r in turn yields an
s
1 0.01 increase of g∗ due to the Coulomb interaction. Second,
0.5 1 5 the solid and dashed curves merge as the density is low-
n (1010 cm-2)
ered,consistentwiththeexpectationthat,becauseofthe
smaller B , the finite layer thickness induced g∗m∗ en-
FIG. 4: Comparison between the measured BP for samples P
hancement becomes less important. Third, at ultra-low
A and B1 (same symbols as in Fig. 1) and calculations that
densities the dashed and dotted curves start to diverge
take the finite layer thickness of the GaAs 2D electrons into
(see the lower inset), implying that the interactionalone
account(solidcurve). Thedensitywasvariedviaafront-gate
bias in both the experiments and calculations. For contrast, caninduceasignificantnon-linearityofspin-polarization
wealsoshowBP from twosetsofcalculations thatarebased with Bk in a very dilute 2D system [2].
on Ref. [2] and are done for interacting, zero layer thickness
2
2D electrons in GaAs (see text). The lower inset shows the Finally, if we use the expression B = (h /2πµ ) ·
P B
calculated curves of the main panel at lower densities. The (n/|g∗|m∗) to determine g∗m∗ from B as frequently
P
upperinset shows g∗m∗ based on different calculations. done in the literature, we obtain the curves shown in
Fig. 4 upper inset. These plots emphasize that ”g∗m∗”
deducedfromB fora2Dsystemwithfinitelayerthick-
P
bias. Theresultsforthecasewherethefront-gateisused
ness (solid curve) is significantly enhanced with respect
are shown in Fig. 4 (solid curve), along with the experi-
to the ideal 2D system and can show a non-monotonic
mental data for samples A and B1. There is reasonable
dependence on n. The results therefore caution against
agreementbetweenthecalculationsandtheexperimental extracting values for g∗m∗ in the limit of zero magnetic
data [17]. field from measurements of B at large parallel fields
P
In Fig. 4 we also plot BP vs. n (dashed curve) calcu- [18]. Moreover, B and g∗m∗ are not unique functions
P
lated based on Ref. [2] where an interacting 2D electron
of n; they depend on the electron layer thickness which
system with zero layer thickness is assumed, as well as
in turn depends on sample parametersand experimental
”B ”(dottedcurve)determinedfromthecalculatedspin
P conditions.
susceptibility in the limit B =0 [2] and assuming a lin-
k
earspin-polarizationasafunctionBk. Threenoteworthy We thank NSF and DOE for support, D.M. Ceperley
trends are observed in Fig. 4. First, for n > 0.5×1010 and S. Moroni for helpful discussions. Part of this work
cm−2 thedashedanddottedcurvesareveryclosetoeach was done at NHMFL; we thank T. Murphy and E.Palm
other, meaning that the spin polarization of an ideal, for support.
[1] B. Tanatar and D.M. Ceperley, Phys. Rev. B 39, 5005 [11] E.P.DePoortere et al.,Phys.Rev.B66,161308(2002).
(1989), and reference therein. [12] F. Stern,Phys.Rev. Lett. 21, 1687 (1968).
[2] C.Attaccaliteet al.,Phys.Rev.Lett.88,256601(2002). [13] The role of finite layer thickness in determining the B -
k
[3] E. Abrahamset al.,Rev.Mod. Phys.73, 251(2001). dependence of the magnetoresistance of 2D carrier sys-
[4] T. Okamoto et al.,Phys. Rev.Lett. 82, 3875 (1999). tems was recently reported [S. Das Sarma and E.H.
[5] E. Tutucet al.,Phys. Rev.Lett. 86, 2858 (2001). Hwang, Phys.Rev.Lett. 84, 5596 (2000)].
[6] S.A.Vitkalov et al.,Phys.Rev.Lett.87, 086401 (2001); [14] Inourm∗ measurements,wechoseBk >BP sothatonly
A.A.Shashkinetal.,Phys.Rev.Lett.87,086801(2001). one spin subband is occupied. In this case the Landau
[7] V.M.Pudalovetal.,Phys.Rev.Lett.88,196404(2002). levels are simply separated by ¯hωc ≡¯heB⊥/m∗.
[8] E. Tutucet al.,Phys. Rev.Lett. 88, 036805 (2002). [15] R.WinklerandU.R¨ossler,Phys.Rev.B48,8918(1993).
[9] Y.Y. Proskuryakov et al., Phys. Rev. Lett. 89, 076406 [16] E.SchubertandK.Ploog,Phys.Rev.B30,7021(1984).
(2002). [17] Inbothexperimentaldata(Fig. 1)andcalculations (not
[10] HwayongNoh et al.,cond-mat/0206519. shown), BP decreases less quickly when a back-gate
5
(sample B3), rather than a front-gate (samples A and the opposite is true when the back-gateis used.
B1), is used to decrease n. This is consistent with the [18] Also see J. Zhu et al.,cond-mat/0301165.
finitelayerthicknesseffect. When we usea front-gate to
reduce the density, the wavefunction gets thicker while