Observation of An Anisotropic Wigner Crystal

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by  Yang Liu

Observation of an Anisotropic Wigner Crystal
Yang Liu, S. Hasdemir, L.N. Pfeiffer, K.W. West, K.W. Baldwin, and M. Shayegan
Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544
(Dated: January 27, 2016)
We report a new correlated phase of two-dimensional charged carriers in high magnetic fields,
manifestedbyananisotropicinsulatingbehavioratlowtemperatures. ItappearsnearLandaulevel
filling factor ν = 1/2 in hole systems confined to wide GaAs quantum wells when the sample is
tilted in magnetic field to an intermediate angle. The parallel field component (B ) leads to a
||
crossingofthelowesttwoLandaulevels,andanelongatedholewavefunctioninthedirectionofB .
||
Under these conditions, the in-plane resistance exhibits an insulating behavior, with the resistance
6
along B more than 10 times smaller than the resistance perpendicular to B . We interpret this
1 || ||
anisotropic insulating phase as a two-component, striped Wigner crystal.
0
2

n Low-disorder, two-dimensional (2D) systems of base temperature of T (cid:39) 30 mK, and an in-situ rotat-
a
charged carriers, cooled to low temperatures and sub- able sample platform to induce B . As illustrated in
J ||
jected to a strong perpendicular magnetic field (B ) are Fig. 1(c), weuseθ toexpresstheanglebetweenthefield
6 ⊥
host to a plethora of exotic, quantum many-body states and the sample plane normal, and denote the longitudi-
2
[1–3]. Atodd-denominatorfractionalfillingsofthelowest nal resistances measured along and perpendicular to the

] Landau level (LL), they exhibit fractional quantum Hall direction of B by R and R , respectively. We used
l || xx yy
l states (FQHSs), uniform-density, incompressible liquid low-frequency (∼ 30 Hz) lock-in technique to measure
a
h phases for which the resistance vanishes as temperature the transport coefficients.
- T approaches absolute zero [1–3]. On the other hand, Figure 1 (e) highlights our main finding. It shows
es whenthefillingfactorbecomesverysmall(ν <∼1/5),the thelongitudinal(Rxx andRyy)andHall(Rxy)magneto-
m system condenses into an ordered array of electrons, the resistance traces, measured for a 2DHS confined to a 40-
. so-called Wigner crystal, which is insulating because it nm-wideGaAsQWatp=1.28×1011 cm−2 andθ (cid:39)35◦.
t
a is pinned by the ubiquitous disorder potential [2–8]. Yet StartingatB (cid:39)8T,bothRxx andRyy rapidlyincrease;
m another set of states are the anisotropic phases observed note the 50 times change of scale for R and R above
xx yy
- at large even-denominator fillings (e.g., ν = 9/2) which 8T[13–15]. Mostremarkably,nearν =1/2,Rxx is∼25
d are believed to be nematic liquid states [9–12]. The new kΩwhileR (cid:39)10R and,aswewillshowshortly,both
n yy xx
correlated phase we report here is distinct from these R and R exhibit an insulating behavior. To probe
o xx yy
c states as it shows an anisotropic insulating behavior. It the origin of this anisotropic insulating phase (IP), we
[ is manifest at low fillings (near ν = 1/2) in 2D hole present several experimental observations.

1 systems (2DHSs) with a bilayer charge distribution and Data of Fig. 2, which were taken at θ = 0 and 50◦,
v tilted in magnetic field to introduce a field component demonstrate that the anisotropic IP seen in Fig. 1(e)
5 (B||) parallel to the 2D plane. Curiously, the anisotropic occurs near a crossing of the lowest two LLs (Fig. 1(d)).
3 phaseformsinarelativelynarrowrangeoftiltanglesnear The crossing is signaled by a profound weakening of the
1 θ (cid:39) 35◦ when the two lowest energy LLs are very close ν = 1 integer QHS at intermediate θ [16, 17]. As is
7
0 inenergy. Outsidethisrange,the2DHSisnotinsulating evident in Fig. 2, traces taken at both θ = 0 and 50◦
. and exhibits FQHSs at numerous fillings. The condi- show a strong integer QHS at ν = 1, with a very wide
1
0 tions under which the new insulating phase appears sug- resistance plateau and large excitation gap ∆>∼10 K. In
6 gest that it is an anisotropic (striped), two-component, contrast, at intermediate angle θ (cid:39) 35◦ (Fig. 1(e)), the
1 pinned Wigner crystal (Fig. 1(a)). ν =1 QHS becomes much weaker (∆(cid:39)0.22 K) and has
v: Our 2DHSs are confined to 40- and 50-nm-wide GaAs a very narrow plateau.
i quantum wells (QWs) flanked by undoped Al Ga As The evolution of the FQHSs in Figs. 1(e) and 2 are
X 0.3 0.7
spacer and C δ-doped layers, and have as grown densi- also consistent with a LL crossing occurring near ν = 1
r ties (cid:39) 1.2×1011 cm−2. The structures were grown by whenθ (cid:39)35◦. InFig. 2tracestherearenumerousstrong
a
molecular beam epitaxy on GaAs (001) wafers and have FQHSs at the well-known, “standard” ν =i/(2i±1) fill-
very high low-temperature mobilities, µ >∼ 100 m2/Vs. ings (i>0 is an integer) [3]. In Fig. 1(e) data, however,
Each sample has a 4 × 4 mm2 van der Pauw geome- near ν = 1 there are uncharacteristically strong FQHSs
try with alloyed In:Zn contacts at its four corners. We at the even-numerator fillings ν = 4/3, 6/5, 6/7, and
thenfititwithanevaporatedTi/Aufront-gateandanIn 4/5. This is similar to what is seen in bilayer 2D elec-
back-gate to control the 2DHS density (p) and keep the tron systems(2DESs)withextremelysmallenergysepa-
QWsymmetric. TheholesintheQWhaveabilayer-like ration between the lowest two LLs [18], and implies that
charge distribution (Fig. 1(b)). The transport measure- these are two-component FQHSs, each component hav-
ments were carried out in a dilution refrigerator with a ing half of the total filling. In Fig. 1(e) we also observe
2
(a) (c) B B 4
┴ (e) θ ≈ 35º 4
θ ÷50
T ≈ 30 mK 6 5
3 7 2
B|| Ryy Ω) 4 65 29 3 3 37
R 1
xx (k 5 3 35 5 2
(b) B|| E(nde) rθg y≈ 35° R &R xx yy 21 2 385 1195 ν = 1 4759 12 49 5 2R (h/e)xy
÷50
ν = 1 0 0
W = 40 nm
4 6 8 10 12 14 16
B
B (T)
FIG. 1. (color online) (a) Conceptual plot of an anisotropic (striped) Wigner crystal. (b) The charge distribution (red) and
potential (black), from calculating the Schroedinger and Poisson’s equations self-consistently at B = 0. (c) Experimental
geometry: R andR denotethelongitudinalmagneto-resistancemeasuredalongandperpendiculartotheparallelmagnetic
xx yy
field (B ), respectively. (d) Schematic diagram, showing the crossing of the lowest two Landau levels at θ (cid:39) 35◦ near ν = 1.
||
(e) Magneto-resistance traces measured at tilt angle θ(cid:39)35◦ for a 2DHS with density p=1.28×1011 cm−2 and confined to a
40-nm-wide GaAs QW. Note the factor of 50 change in the scale for R and R for B >8 T.
xx yy
FQHSs at very unusual fillings such as ν = 19/15 and 2
(a) θ = 0º
29/35. SuchstateswereseeninRef. [18]whenthelowest
T ≈ 30 mK
two LLs are nearly degenerate, and were interpreted as
4 2
“imbalanced” two-component FQHSs: for example, the 2
5 5
ν =19/15FQHShasfillings2/3and3/5foritstwocom- Ω)
k 3 3
2pevDoenEne-Sndstesan.noIdmni2FnDiagHt.oSr2s,ficwloliennfiagnlsνeod=otbo1s/ew2rvi.deTeshtGriosanAFgsQFQHQWSHsiSs[s1sea9et–n2t2hin]e. R &R (xx yy1 2 43 23 3547 12 497 1 2R (h/e)xy
It is likely the Ψ state, a two-component FQHS stabi-
331
ν = 1
lized by strong and comparable interlayer and intralayer
interactions which are prevalent at ν =1/2 [23].
Next we focus on the anisotropic IP seen at θ (cid:39) 35◦.
0 0
Figure 3 captures the insulating behavior of this phase
4 6 8 10 12 14
nearν =1/2. BothRxx andRyy increaseastemperature B (T)
is decreased but, as can be best seen in Fig. 3(b), R 2
yy
(b) θ ≈ 50º
is about 10 times larger than R . Before discussing 4
this anisotropic behavior, it is instxrxuctive to first briefly T ≈ 30 mK 5 2
review the IPs seen in 2D systems at low fillings.
4
It is well established that in very clean 2D systems of
Ω) 3 1
caaphnrineadnrbgνeeedl<∼dibevyc1ae/tdr3hrietfeoorsrsmb,heaaoltlWlebsvi)uegtrtnyheuerbsmiFcqrQauyliHslttoSνaulss((gνWdivi<∼seCow)r1d/ape5yhratf[o4soer–Is7Pe,tlshe2ca4wtt–rh2oai7cnr]hes. R &R (kxx yy1 5385 75 ν = 1 23 35 47 2 1 2R (h/e)xy
For both 2D electron and hole systems in wide, symmet-
ric GaAs QWs, the charge distribution becomes more
bilayer-like with increasing density and the IP sets in at
0 0
progressivelylargerν [22,28,29]. TheseIPsarebelieved 8 12 16 18
to be bilayer WC states which, thanks to the additional B (T)
layer/subband degree of freedom, are stabilized at rela-
tively large ν compared to the single-layer systems. For FIG.2. (coloronline)Magneto-resistancetracesforthesam-
pleofFig. 1,measuredatθ=0◦ and50◦. BothR andR
example, in 2DHSs confined to a 40-nm-wide QW with xx yy
are much smaller near ν =1/2 than in Fig. 1(e) data.
p>∼1.7×1011 cm−2 [22],anIPisobservednearν =1/2.
All the IPs described above are isotropic, and were ob-
served in the absence of B [30, 31].
||
3
103 8
(a) θ ≈ 353/º54/7 ν = 1/2 4/93/7 2/5 (b) ν = 1/2 (a) Tp ≈= 300.9 m5 Kx 1011 cm-2 3
Ω)102 2/3 4 5 65°
R & R (kxxyy101 ν = 1/2 6 53 3 ν = 1 23 ν = 1/2 W37 = 5250 nm55°
100 Ω)
31100 m mKK νΔ =≈ 20/.54 K R (kyy 4 29
10-1 200 mK 35
8 10 12 14 16 10 20 1
÷ 50
B (T) 1/T (K-1) 19 37°
2 15
FIG. 3. (color online) (a) Rxx and Ryy measured at three θ ≈ 0°
temperaturesinthe40-nm-wideQWnearν =1/2atθ(cid:39)35◦.
(b) T-dependence of R and R at ν = 1/2. To avoid the
xx yy
influence of the relatively sharp ν = 1/2 resistance minima 0
seeninsomeofthetraces,hereweareplottingthebackground 2 4 6 8 10
resistances by interpolating between the shoulders flanking B (T)
┴
theν =1/2minima. AlsoplottedistheT-dependenceofthe
103 (b) p = 1.12 x 1011 cm-2
Rxx minimum for the ν =2/5 FQHS. θ ≈ 35° ν = 1/2 3/7 2/5
Ω) 1/3
k102
To discuss the likely origin of the anisotropic IP we R (yy 11/15
observe near ν =1/2, we focus on its key attributes: R & xx101 2/3
(i) It is a collective state. There are numerous FQHSs
30 mK
nearν =1/2inFig. 1(e),e.g.,atν =2/5,3/7,4/9,3/5, 100 110 mK
4/7, 5/9. These correlated states are much weaker than 200 mK
the FQHSs seen at the same fillings in Fig. 2 traces, but 8 10 12 14 16
B (T)
their mere presence in Fig. 1(e) strongly suggests that
correlations are prevalent near ν = 1/2 where the IP
FIG. 4. (color online) (a) R vs B traces measured for a
reigns. Also worth emphasizing is that in Fig. 2 traces yy ⊥
2DHS confined to a 50-nm-wide GaAs QW at density p =
there are very strong FQHSs near and even at ν = 1/2. 0.95×1011 cm−2 and at different θ. The 2DHS exhibits an
It is very unlikely that at the intermediate tilt angle of insulating behavior near ν =1/2 at θ (cid:39)37◦ and at θ =65◦,
Fig. 1(e) interactions would disappear and the ground butnotatθ=25◦ orθ=55◦. Theinsetshowsthecalculated
state become of single-particle origin. chargedistributionandpotentialatB =0. (b)Rxx andRyy
(ii) It is a two-component state. It is clear that the traces are shown at θ (cid:39) 35◦ and at a slighlty larger density
(p=1.12×1011 cm−2)fordifferenttemperatures. Similarto
anisotropicIPisobservednearaLLcrossing, andFig. 2
the data of Fig. 3, both R R exhibit insulating behavior
traces,whichweretakenfarfromtheLLcrossing,donot xx yy
and the system is anisotropic near ν =1/2.
showinsulatingbehaviornearν =1/2. Thisimpliesthat
thepresenceoftwonearlydegenerateLLsplaysacrucial
role for its stability. Also, theoretical calculations rule
nematic phases are liquid states: while in-plane trans-
out any single-component WC near ν = 1/2 [32]. The
anisotropic IP we observe at θ (cid:39) 35◦ is thus likely to portbecomesanisotropic,Rxx andRyy donotdivergeat
low temperatures; instead they remain finite and in fact
have a two-component origin. In Fig. 3(a), the existence
the resistance along the “easy axis” direction decreases
ofminimaatν =2/5,3/7,3/5,andtheirdeepening(rel-
as temperature is lowered and attains extremely small
ative to the insulating background resistance) at lower
values [9, 10]. This is very different from the insulat-
temperatures, signalaclosecompetitionbetweenareen-
ing behavior and the large values we measure for both
tranttwo-componentWCphaseandtheFQHSs. Weadd
R andR . Second,inthecaseofB -inducednematic
that, whetherisotropicoranisotropic, one-componentor xx yy ||
phases,the“hardaxis”istypicallyalongB [34,35]. We
two-component, the observation of an IP near the cross- ||
observe the opposite behavior: the resistance along B
ing of two LLs is by itself unprecedented. ||
(R )issmaller thanintheperpendiculardirection[36].
(iii)Itisnotanematicliquidstate. Onemightnaively xx
concludethattheanisotropywereportresemblestheone Based on the above observations, we associate the IP
observed at higher half-filled LLs, and believed to signal observedinFig. 1(e)withapinned,anisotropicWC.We
nematic electron phases [9–12, 33]. But this is incorrect, suggest that the anisotropy originates from the strongly
as there are two major, qualitative differences. First, distorted shape of the hole charge distribution induced
4
by B , as schematically depicted in Fig. 1(a). Because discussions, andR.Winklerforprovidingthechargedis-
||
ofthe finitethickness ofthe holelayerinour sample, B tribution and potential calculations shown in Figs. 1(b)
||
couples to the out-of-plane (orbital) motion of the carri- and 4(a). A portion of this work was performed at the
ers, and squeezes the charge distribution in the direction NHMFL, which is supported by the NSF Cooperative
perpendicular to B (see Fig. 1(a)). Such distortions Agreement No. DMR-1157490, the State of Florida, and
||
havebeenrecentlydocumentedforcarriersnearB =0, theDOE.WethankS.Hannahs,G.E.Jones,T.P.Mur-
⊥
and also for composite fermions at high B [37–39]. An phy, E. Palm, A. Suslov, and J. H. Park for technical
⊥
elongated charge distribution can lead to anisotropic in- assistance.
teraction, and provides a natural explanation for the
anisotropic IP we observe in terms of a pinned, striped
WCasshowninFig. 1(a)[40–42]. Moreover,itisconsis-
tentwiththeexperimentalobservationthatthetransport
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yy
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Theresultswereporthereattesttotheextremelyrich xy
T is an artifact due to the mixing of the longitudinal
physics of 2DHSs confined to wide GaAs QWs. In these
resistances which attain very high values [14, 15].
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WeacknowledgesupportbytheDOEBES(DE-FG02- (2015).
00-ER45841) grant for measurements, and the NSF [18] H. C. Manoharan, Y. W. Suen, T. S. Lay, M. B. Santos,
(Grants DMR-1305691 and MRSEC DMR-1420541), and M. Shayegan, Phys. Rev. Lett. 79, 2722 (1997).
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Fradkin, J.K. Jain, and S.A. Kivelson for illuminating
5
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whichcanbeinterpretedastheparticle-holecounterpart small (less than a factor of two) anisotropy observed in
of the ν = 1/2 FQHS. Also, the weak QHS at ν = 1 in Fig. 2(a) (θ = 0◦) near ν = 1/2 possibly comes from
Fig. 1(e) is likely the two-component Ψ state [17]. the sample’s van der Pauw geometry and the contacts’
111
[24] Y. Lozovik and V. Yudson, JETP Lett. 22, 11 (1975). misalignment. The anisotropy becomes larger, about a
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||
M. Shayegan, Phys. Rev. B 46, 13639 (1992). we discuss later in the manuscript.
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been reported in fixed-density 2DESs confined to wide kler, Phys. Rev. B 89, 085304 (2014).
GaAsQWswhenalargeB isapplied[31].Inthiscase, [40] An“insulatingstripe-crystal”phasehasindeedbeendis-
||
B couples to the orbital (out-of-plane) motion of elec- cussed theoretically for an interacting 2D system in the
||
tronsandrendersthesystemprogressivelymorebilayer- excited (N =2) LL [33].
like with increasing B [31]. [41] The role of effective mass anisotropy for 2D WC states
||
[31] S.Hasdemir,Y.Liu,H.Deng,M.Shayegan,L.N.Pfeif- has also been discussed in Ref. [42].
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