Table Of ContentInterplay between Quantum Well Width and Interface Roughness for
Electron Transport Mobility in GaAs Quantum Wells
D. Kamburov, K. W. Baldwin, K. W. West, M. Shayegan, and L. N. Pfeiffer
Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA
(Dated: January 20, 2017)
We report transport mobility measurements for clean, two-dimensional (2D) electron systems
confined to GaAs quantum wells (QWs), grown via molecular beam epitaxy, in two families of
structures, a standard, symmetrically-doped GaAs set of QWs with Al Ga As barriers, and
0.32 0.68
one with additional AlAs cladding surrounding the QWs. Our results indicate that the mobility
7 innarrowQWswithnocladdingisconsistentwithexistingtheoreticalcalculationswhereinterface
1 roughnesseffectsaresoftenedbythepenetrationoftheelectronwavefunctionintotheadjacentlow
0 barriers. In contrast, data from AlAs-clad wells show a number of samples where the 2D electron
2 mobility is severely limited by interface roughness. These measurements across three orders of
magnitudeinmobilityprovidearoadmapofreachablemobilitiesinthegrowthofGaAsstructures
n
of different electron densities, well widths, and barrier heights.
a
J
8
Two-dimensional electron systems (2DESs) in engi- nificant, deviations from this trend should be expected
1
neered quantum structures have proven to be a fruit- [17].
] ful tool for discovering new fundamental physical phe- In this study we present results for 2DESs confined
l
l nomena, includingtheinteger(IQHE)andthefractional to symmetrically delta-doped, MBE-grown GaAs QWs
a
h quantum Hall effect (FQHE) [1, 2]. Enabled by the on (001) GaAs substrates. Our work focuses on two
- progress in molecular beam epitaxy (MBE) technology sets of samples: (i) simple GaAs QWs symmetrically
s
e [3], the GaAs system has become the benchmark for the modulation-doped to a nominal electron density of n (cid:39)
m highest material quality and has paved the way to very 1.2 × 1011 cm−2, and (ii) 5.66 nm AlAs-clad QWs, in
. longcarriermean-freepathsandhighmobilities[4]. The which the symmetrically modulation-doped GaAs QWs
at present state of the MBE art allows precise control of a have a nominal electron density of n (cid:39) 3.0 × 1011
m numberofgrowthparameters, suchassubstratetemper- cm−2. The 2D electrons in (i) are located 398 nm un-
- ature,rate,materialcomposition,andgrowthinterrupts, der the surface and are flanked on each side by undoped
d which critically affect scattering mechanisms and carrier Al Ga As setback layers. The AlAs-clad samples
n 0.32 0.68
mobilities [5–9]. Generally, quantum wells (QWs) doped areshallower,withthe2Delectronsat270nmunderthe
o
c from both sides have to be narrower than triangular surface, flanked on each side by 5.66 nm AlAs cladding,
[ modulation-doped single-sided GaAlAs/GaAs QWs to varying-thicknessundopedAl Ga Assetbacklayers,
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avoidsecondsubbandoccupationathighcarrierconcen- andδ-dopedSilayers. Thevariationinthesetbackthick-
1
v trations [10]. In such QWs, small variations of the QW nessinbothsetsofsamplesisnecessaryinordertomain-
6 width have a profound effect on the energy eigenvalues. tain the 2DES density near either of the nominal values
5 Thusinterfaceroughnesstakesonadditionalimportance n(cid:39)1.2×1011cm−2orn(cid:39)3.0×1011cm−2tocompensate
2 for MBE growth of high density, high mobility 2DESs. for changes in the energy eigenvalues with QW width.
5
Despite the immense progress in MBE techniques, how- The AlAs cladding steepens the QW barriers, thus lim-
0
. ever, GaAs structures where interface roughness domi- iting the electron wave function penetration outside the
1 natesthecarrierscatteringhavenotbeenfullydelineated QWs. The QW widths range from W = 2−48 nm for
0
systematically [11–15]. More specifically, the emergence the simple wells at n(cid:39)1.2×1011 cm−2 and W =0−29
7
1 of significant interface scattering has been demonstrated nmfortheAlAs-cladQWsatn(cid:39)3.0×1011 cm−2. Each
: only in isolated cases in samples with sufficiently narrow sample was measured in a van der Pauw configuration
v
i QWs [11] and for 2D systems with very low density [13]. with annealed InSn contacts in a 3He refrigerator with
X Thelackofsystematicreportsofcarriermobilitiesissur- base temperature of T (cid:39) 0.3 K. Measurements were car-
r prising in view of the important role of interface rough- ried out using standard low-frequency lock-in amplifier
a
ness at high wave function confinement and its effect on techniques. Mobilityvalueswereextractedusingthevan
carrier mobility. In QWs with AlAs cladding, where the der Pauw’s method [18] in a square geometry using the
carrier wave function is expected to have very little pen- resistancemeasurementsfromthefourcornersorflatsof
etration in the barrier, the mobility was experimentally the samples. Density values were measured using stan-
shown in a landmark paper by Sakaki et al. [11] to go as dard transport measurements at perpendicular magnetic
µ∝W6. Thisagreeswellwiththeconceptthatinterface field up to 14 T.
scattering is the major mobility-limiting factor [11–16].
Our findings from the set of simple wells with
Ontheotherhand,inQWswithnoAlAscladding,where
Al Ga As barriers are summarized in Fig. 1. The
0.32 0.68
wave function penetration into the barriers is more sig-
measured electron mobility of each sample is plotted
2
107 107
n ≈ 1.2 x 1011 cm-2 µ ~ W 2
n ≈ 3.0 x 1011 cm-2
µ ~ W 0.6
QWs with low barriers QWs with high barriers
106 106
) )
Vs Vs W = 0
2/ µ ~ W 3 2/ GaAs
m m µ ~ W 6
(c 105 (c 105
y y
bilit bilit µ ~ W 0.9
o o
M M
104 104
our data
µ ~ W 6
Sakaki et al. [8]
103 103
0.1 1 10 100 0.1 1 10 100
Well width (nm) Well width (nm)
FIG. 1. Electron mobility as a function of QW width in FIG. 2. ((cid:4)) Electron mobility for AlAs-clad GaAs QWs
GaAs QWs without AlAs cladding and with 32% AlGaAs dopedfrombothsideston(cid:39)3.0×1011 cm−2. Threeregimes
barriers doped from both sides to an electron density of n(cid:39) are prominent in the mobility curve: W > 11 nm, where
1.2×1011 cm−2. We identify two regimes, for W < 10 nm interface scattering is insignificant; 8.65 nm < W < 11 nm,
where µ ∝ W3, and W > 10 nm for which µ ∝ W0.6. The whereinterfacescatteringissignificantandµ∝W6;andW <
straight lines are best fits to the experimental data. 8.65 nm, for which the width of the GaAs QW dependence
is fairly flat (see text). (◦) Reproduced data from Ref. [11],
showingtheimportantroleofinterfaceroughnessasalimiting
mechanism in the electron mobility. All straight lines are
againstitsQWwidth. MobilitiesmeasuredinwideQWs best fits to the experimental data. Note that in the region
are quite high, reaching (cid:39) 12×106 cm2/Vs for the 48 5.65 nm < W < 10 nm both data sets can be approximated
nmQW.Tworegionsofdifferentslopescanbeidentified by parallel lines with µ∝W6.
in Fig. 1. For W > 11 nm, the mobility dependence is
µ ∝ W0.6, while with decreasing well width, mobilities
degrade more quickly as approximated by µ∝W3. The points, are included for reference in Fig. 2 with open
gradual decrease of µ in narrow QWs is consistent with circles.
numerical calculations based on the theoretical model of
Electron mobilities from the AlAs-clad QWs in Fig.
Refs. [11, 17], in which the QW barrier is treated as
2 show a complex dependence on the GaAs QW width.
finite, fully taking into account the penetration of the
In wide QWs (W > 11 nm), where monolayer-scale in-
electron wave function into the barriers. Effectively, the
terface roughness effects are expected to be less signifi-
absence of AlAs cladding allows the electron wave func-
cantcomparedtothescaleoftheQWwidth,ourexperi-
tion to leak outside the QW and leads to a reduced sig- mentsshowµ∝W2. Suchbehaviorisnotunexpectedas
nificanceoftheinterfaceroughnessandthustoanoverall
theinterfaceroughnessshouldbecomeimportantonlyin
highermobility. Ourdataqualitativelymatchthebehav-
narrower QWs, where local QW width variations would
ior of the theoretical model in Ref. [17].
severelychangethelocalQWeigenvalue. AsW decreases
While the data in Fig.1 characterize the general case to about 10 nm, µ is affected more significantly by the
in which the electron wave function penetrates into the interface roughness. In the range W = 5.6 − 11 nm,
barriers, a more interesting scenario can be achieved the mobility trend is well approximated by µ ∝ W6,
whenelectronsaretightlyconfinedintheQWsbyadding in accord with previous experiments [11, 12] and with
AlAs cladding. Due to the larger band gap of AlAs, the slopes parallel to our data. The mobilities in our sam-
cladding acts as a steeper potential barrier that confines ples are almost two orders of magnitude higher, likely
the electrons to a profile more closely resembling an in- becauseweusedgrowthtemperatureof647◦C,whilethe
finite QW. In Fig. 2 we present data from a number of ones from Sakaki et al. [11] were grown at 590◦C. For
AlAs-cladQWs,includingastructureinwhichtheGaAs QWs narrower than W = 5.66 nm, µ increases slightly,
QW is removed altogether, so that the 2DES resides in then decreases gradually again [19]. At W =0, the
GaAs
an AlAs QW. The earlier experimental results by Sakaki mobility is much higher. This behavior can be quali-
et al. [11], which comprised only four low mobility data tatively understood as follows: for narrower GaAs QW
3
well-width (W = 4.53 nm and lower), the tight confine-
n ≈ 3.0 x 1011 cm-2
mentcausesthewavefunctiontosubstantiallypenetrate
T = 300 mK
into the AlAs barriers and make an AlAs bilayer sys- =5/3 4/3
tem [20] with a GaAs barrier. This is the failing point
20 Ω
of the infinite-barrier model because the wave function
effectively begins to reside more in the AlAs cladding
layers and therefore relatively less in the GaAs layer. 16.98 nm
The mobility at W = 4.53 nm increases slightly com-
pared to the W = 5.66 nm QW, possibly because the
11.23 nm
penetration into the AlAs cladding provides net addi-
tional room for the electron wave function. For smaller
)
GaAs QW widths, the mobility starts to drop because Ω
(
thetotalthicknessoftheAlAs/GaAs/AlAssystemisde- e
c 2 1
creasingfrom4.53nm+11.32nmto1.13nm+11.32nmand n
the electrons are again becoming more confined. As the a 5.66 nm
t
s
GaAs QW disappears altogether, µ increases, reaching si
values of mobility of a simple 2DESs confined to AlAs re 4.53 nm
o
QWs [21]. The wave function then fully resides in an
t
e
AlAs QW of WAlAs = 11.32 nm, i.e. twice the width n
g
of the AlAs cladding on each side of the original GaAs a 2 1
well[22]. TheadditionalGaAs/AlAsinterfaceshavealso M 2.83 nm
disappeared and with them the extra scattering caused
by the GaAs QW.
The results from the AlAs-clad QWs corroborate the 0 nm ÷2
previous experimental studies and theoretical models for
narrow systems with W < 10 nm with strong confine-
0.0 0.2 0.4 0.6 0.8 1.0
ment [11, 12] and very low electron density [13], while
1/
adding an unexpectedly rich behavior of carrier mobili-
ties away from this regime. In particular, when the in-
FIG. 3. Magnetoresistance traces vs 1/ν, where ν is the
terface roughness is insignificant on the scale of W, µ is
Landau filling factor, for different AlAs-clad QWs, vertically
only slightly affected by surface scattering because the
offset for clarity. The electron density in all samples is n (cid:39)
ground-state eigenvalue of the wide QW does not locally 3.0×1011 cm−2.
vary much. In cases when the QW is too narrow to con-
tainthewavefunction,the2DESmigratesintotheAlAs
cladding and the system resembles an AlAs bilayer. Fi- GaAs QW is removed, the sample becomes a wide AlAs
nally, whentheGaAsiscompletelyremoved, µimproves QW of W = 11.32 nm, and the magnetoresistance
AlAs
to the point that the sample with no GaAs well has sim- trace again shows FQHE states [21]. This improvement
ilar quality to the GaAs well with W = 7.6 nm. This resultsfromremovingthenarrowGaAsQWasascatter-
complex trend of µ as a function of the GaAs QW width ing mechanism and an intervening barrier. We note that
is important for understanding the interplay between in- for all AlAs-clad samples with W < 5.66 nm, we see bi-
terfaceroughnessandquantumconfinement. However,µ layerAlAs-likecharacteristics,namelyaweakν =1state
is not necessarily the only relevant characteristic in the and absence of minima at odd fillings [20, 23]. The mag-
study of 2D phenomena in engineered GaAs QWs. netoresistancedatafromthelowerdensitysetofsamples
Inordertoprovideamorecomprehensiveunderstand- (notshown), ontheotherhand, lackanyunusualbehav-
ing of the implications of the sample quality for a given ior because of the deep penetration of the wave function
value of µ, in Fig. 3 we plot representative magnetore- into the low barriers and the effective larger quantum
sistance traces taken for the AlAs-clad QWs whose mo- well width.
bilities are summarized in Fig. 2. The samples with To summarize, our data reveal: (1) a µ ∝ W3
W >10 nm have high quality and their magnetoristance dependence of mobility in simple GaAs QWs with
traces show signatures of FQHE states, at Landau level Al Ga As barriers, consistent with muted interface
0.32 0.68
filling factors ν =5/3 and 4/3, marked with vertical ar- roughness, typical of systems with only modestly high
rows. In the W = 5.66 nm QW trace, the FQHE states potential barriers, (2) a µ ∝ W6 relationship for AlAs-
disappear and the trace shows strong and wide IQHE clad narrow GaAs QWs occurs where interface rough-
minima. For just slightly narrower QWs, W =4.53 nm, ness is significant, but only in a narrow range of QW
the ν =1 minimum gets strikingly narrower, a signature widths, in general agreement with previous experiments,
of AlAs bilayer effect [20]. In sharp contrast, when the (3) unusually rich W-dependence of µ in narrow AlAs-
4
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