of 17
Competing correlated states around the
zero-field Wigner crystallization transition
of electrons in two dimensions
In the format provided by the
authors and unedited
Supplementary information
https://doi.org/10.1038/s41563-021-01166-1
Competing correlated states around the
zero-field Wigner crystallization transition
of electrons in two dimensions
In the format provided by the
authors and unedited
Supplementary information
https://doi.org/10.1038/s41563-021-01166-1
Competing correlated states around the zero field Wigner crystallization transition of
electrons in two-dimensions
J. Falson,
1, 2, 3,
I. Sodemann,
4, 5
B. Skinner,
6
D. Tabrea,
1
Y. Kozuka,
7, 8
A. Tsukazaki,
9
M. Kawasaki,
10, 11
K. von Klitzing,
1
and J. H. Smet
1
1
Max-Planck-Institute for Solid State Research, D-70569 Stuttgart, Germany
2
Department of Applied Physics and Materials Science,
California Institute of Technology, Pasadena, California 91125, USA.
3
Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, California 91125, USA
4
Max-Planck-Institute for the Physics of Complex Systems, 01187 Dresden, Germany
5
Institut f ̈ur Theoretische Physik, Universit ̈at Leipzig, D-04103, Leipzig, Germany
6
Department of Physics, Ohio State University, Columbus, Ohio 43210, USA
7
Research Center for Magnetic and Spintronic Materials,
National Institute for Materials Science, Tsukuba 305-0047, Japan
8
JST, PRESTO, Kawaguchi, Saitama, 332-0012, Japan
9
Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
10
Department of Applied Physics and Quantum-Phase Electronics Center (QPEC), University of Tokyo, Tokyo 113-8656, Japan
11
RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan
S1. SAMPLE PARAMETERS
TABLE
I:
Parameters
of
the
MgZnO/ZnO
2DES
when
n
= 2.3
×
10
10
cm
2
,
m
b
= 0.30
m
0
[Ref.1] and

= 8.5

0
. The electron mobility is assumed to be around
μ
= 600
,
000 cm
2
/Vs.
Parameter
Magnitude
Units
E
F
1.8
×
10
4
eV
T
F
2.1
K
τ
tr
100
×
10
12
s
τ
q
10
×
10
12
s
L
1.7
×
10
6
m
r
s
25
q
TF
1.3
×
10
9
m
1
t
WF
10
×
10
9
m
k
F
t
WF
0.3
Table
I
collects
relevant
parameters
of
the
MgZnO/ZnO 2DES when
n
= 2
.
3
×
10
10
cm
2
.
Here,
E
F
the kinetic energy,
T
F
the Fermi temperature,
τ
tr
the transport scattering time,
τ
q
the quantum
scattering time (estimated from the onset field of
quantum oscillations),
L
the mean free path of carriers,
r
s
the Wigner-Seitz parameter,
q
TF
the Thomas-Fermi
screening wave length, and
t
WF
an estimate of the
wavefunction thickness (based on the data presented
in Ref.2). Note that the band effective mass is used
in these calculations and the value of
m
b
= 0.30
m
0
is
the mean value of cyclotron resonance measurements
in Ref.1. In that work, and uncertainty of
±
0
.
01
m
0
was identified, which provides a 3% uncertainty in our
calculated value of
r
s
. The role of renormalized mass
values will be discussed in the subsequent sections. The
quasi-Hall bar sample is not well suited for accurate
calculations of the electron mobility, and hence we work
with an assumed peak electron mobility of approximately
600,000 cm
2
/Vs. This is calculated based on the mea-
sured resistance of the device, which is assumed to be
the same order of magnitude as the resistivity. Previous
studies
3
reported an electron mobility at
T
= 500 mK
of
μ
= 80,000 cm
2
/Vs when
n
=2
×
10
10
cm
2
. Our
results point towards an approximately four times
enhancement of the electron mobility when reducing the
temperature from 500 mK to approximately 10 mK. The
μ
=80,000 cm
2
/Vs device in Ref.3 would thus translate
to a mobility of approximately
μ
=300,000 cm
2
/Vs when
n
=2
×
10
10
cm
2
at lower temperatures, which is a
value comparable to the device studied in this work.
Furthermore, as a simple consistency check, the MIT
occurs at the 3
h/e
2
in this work, which is comparable
to many reports in the literature of a value of
h/e
2
.
Low-field quantum oscillations begin at approximately
0.07 T, yielding a conservative estimate of the quantum
lifetime of carriers
τ
q
>
10 ps, using the relationship
τ
1
q
= 2
ω
c
where
ω
c
is the cyclotron frequency at the
onset of oscillations using the band mass. Furthermore,
Ohmic contacts were formed by evaporating Ti (10 nm)
followed by Au (50 nm) on the sample surface. Indium
was additionally soldered upon these pads to improve
the contact quality. Qualitatively similar results were
obtained both with and without this additional indium
layer.
S2. MEASUREMENT DETAILS
All measurements have taken place in a
3
He immersion
cell anchored at the base of an annealed silver cold finger
attached to the mixing chamber of a 650
μ
W at 120 mK
dilution refrigerator (base
T
7 mK), as shown in Fig.1.
The cell design is based on the one utilized at the Na-
2
a
b
FIG. 1:
The
3
He immersion cell. a
Closed cell with sample
puck inside (scale bar
1 cm)
b
Open cell displaying the main
heat exchanger and
3
He inlet hole (top) and sample puck
(bottom). The sample is glued upon a metalized chip carrier,
with each contact being connected to a sintered silver heat
exchanger by a gold bonding wire.
tional High Magnetic Field Laboratory High B/T facility
in Gainesville, Florida. This technique has proven pow-
erful in achieving the low electron temperatures required
for studies of delicate fractional quantum Hall features,
4,5
and also insulating phases in low density 2DES devices.
6
The mixing chamber temperature is measured using a
calibrated cerous magnesium nitrate paramagnetic ther-
mometer for
T
120 mK, and a ruthenium oxide ther-
mometer for
T
50 mK. These thermometers have good
overlap in the range 50
< T <
120 mK. We do not mea-
sure the temperature inside the
3
He cell. As electrons at
low temperature are actively cooled primarily via electri-
cal contacts at ultra-low temperatures, the exact electron
temperature is dependent on the contact resistance. The
Kapitza phonon interfacial resistance (
1
/
(
AT
3
)) may
be effectively suppressed by immersing the sample in the
cryogen and attaching large surface area (
A
) sintered sil-
ver heat exchangers to each measurement wire. Electri-
cal signals are carried from room temperature down to
the mixing chamber using individual thermocoax lines,
each with a length of approximately 2 m. These lines
are thermally anchored at the 1 K, still, 50 mK and mix-
ing chamber plates. These cables act as distributed
R-C
filters and are effective in attenuating stray radiation in
cryogenics applications.
7
They consist of a resistive stain-
less steel inner conductor (
R
cable
150 Ω) isolated from
the outer conductor by a MgO nanoparticle dielectric.
The skin effect of the nanoparticles strongly attenuates
high frequency signals (
>
1 MHz). An additional
R-C
filter is employed at the mixing chamber. Superconduct-
ing Nb-Ti loom is used between the mixing chamber and
the measurement puck.
Magnetic flux is generated by a 3-axis (9-3-1 T) su-
perconducting vector magnet. The cryostat is suspended
above a pit carved into a 35 ton concrete block that is
20
30
40
50
0.808
0.810
0.812
0.814
0.816
0.818
0.820
SPWC
AFM WC
SPFL
Para.FL
r
(a.u.)
s
(
E
+ 1.106103/
r
)
r
s
s
3/
2
(a.u.)
FIG. 2:
Quantum Monte Carlo results adapted from
Ref.8.
The energy
E
per electron of competing unpolarized
paramagnetic Fermi liquid (Para. FL), spin polarized Fermi
liquid (SPFL), unpolarized AFM Wigner crystal (AFM WC)
and polarized Wigner crystal (SPWC) as a function of
r
s
.
subsequently isolated from the main building structure
by vibration dampening pads. Circulation pumps are an-
chored to the outer wall of the superstructure to reduce
vibrations. The measurement room is electrically iso-
lated from the outside environment by a Faraday cage de-
livering -60 dBm attenuation. Measurement electronics
are powered by a dedicated phase of a three-phase power
supply, operate on individual isolating transformers, and
are isolated from the data gathering measurement com-
puter by an optical isolator. Significant efforts, such as
electrically isolating the cryostat from pumps and diag-
nostics electronics, have been taken to minimize ground-
loops within the measurement circuitry. We strive to
utilize low-noise electronics for measurements, such as
all analogue lock-in amplifiers (PAR 124A) (only used
for the data presented in Fig.11), Yokogawa 7651 DC
voltage sources and DL Instruments voltage (1201) and
current (1211) preamplifiers.
DC measurement techniques often incorporate offsets
in both the current and voltage signal. These offsets can
also fluctuate in time. We have tried to reduce this un-
certainty by ensuring excellent temperature stability of
the laboratory, while maintaining electrical power to the
electronics at all times. The current offset is addressed
by adjusting the zero offset of the current preamplifier at
zero applied electric field. The voltage offsets are more
challenging to eliminate with features order of nano- to
microvolts between individual IV sweeps being observed.
The only point where we have adjusted the data in this
manuscript is in Fig.4
a
-
c
and 15, where we take the mea-
sured voltage at zero current at a set temperature and
charge density and subtract it as an offset for each trace.
3
28
30
32
38
35
25
26
27
a
1
1.2
1.4
1.6
1.8
2
2.2
n
(10
10
cm
-2
)
1
0.1
0.01
B
x
(T)
-0.2 -0.1
0
0.1 0.2
B
x
(T)
0.2
-0.1
0
0.1 0.2
B
x
(T)
-0.1
0
0.1 0.2
B
x
(T)
-0.1
0
0.1 0.2
c
d
e
r
s
B
x
(T)
-0.1
0
0.1
b
AFM
WC
AFM
WC
SP
WC
SP
WC
SP
WC
SP
FL
SP
FL
SP
FL
SP
FL
SP
FL
SP
FL
Para.
FL
Para.
FL
Para.
FL
Scenario I
Scenario II
Scenario III
B
*
B
c
B
*
B
c
Fig. 2
c
Fig. 3
h
B B
/
=sat
r
r
h e
/
2
/
2.5nA
r r
10
-1
10
4
FIG. 3:
Comparison of experimental data with QMC ground states.
8
a
ρ
B
B=sat
in the (
B
x
,
n
)-plane.
b
Nonlinearity
in IV, defined as
ρ/ρ
2
.
5nA
in the (
B
x
,
n
)-plane. Phases associated with
c
scenario I,
d
scenario II and
e
scenario III, as discussed
in Sec. S3 A.
The process of differentiating data eliminates this offset
as it is only sensitive to the slope of the data.
S3. MONTE-CARLO PHASE DIAGRAMS OF 2D
JELLIUM
The analysis we present here relies on the variational
Monte-Carlo study of Ref.8. This study compared en-
ergies of four states: a spin unpolarized paramagnetic
Fermi liquid (Para. FL), spin polarized Fermi liquid (SP
FL), spin unpolarized Wigner crystal (AFM WC) with a
stripe-like spin antiferromagnetic order and a triangular
lattice and a spin polarized Wigner crystal (SP WC) with
triangular lattice. The main results of Ref.8 are adapted
in Fig. 2, and imply that as
r
s
increases, the ground state
progresses from Para. FL, to AFM WC, to SP WC, in
the vicinity of
r
s
= 30. Importantly, Ref.8 ruled out the
possibility of the traditional Stoner transition from Para.
FL to a SP FL.
We use this study to estimate the phase diagram as a
function
B
and
r
s
. Because electrons in ZnO have weak
spin-orbit coupling, the total spin
S
z
of the system is a
good quantum number. This allows us to know the ex-
act energy of each of the four states previously described,
which are all either fully spin polarized or fully spin un-
polarized. However, to have a complete phase diagram
we need to estimate the energy of potential states with
different partial polarization. To do this, it is unavoid-
able to make certain assumptions about the dependence
of the energy on spin polarization. We therefore analyze
the competition of these phases within three distinct sce-
narios. As we will see the main conclusions are relatively
robust to the assumptions within different scenarios.
A. Scenario I: phase diagrams without a partially
polarized crystal
The appearance of a possible competing intermediate
crystal is a relatively new development within the se-
ries of Monte-Carlo studies of the ideal Jellium model,
8
although indications of such an intermediate state ap-
peared in a previous study,
9
and also in self-consistent
Hartree-Fock calculations.
10,11
Since most previous stud-
ies did not find or ignored the possibility of these partially
polarized crystalline states, it is natural to try to deter-
mine what would be the phase diagram if such states are
not considered as part of the competing states. To do so
we begin by approximating the ground state energy per
electron of the Fermi liquids as a function of their spin
polarization polarization
p
, given by
p
n
n
n
+
n
, p
[
1
,
1]
,
(S1)
with a simple parabolic dependence, as follows:

FL

Para.FL
(
r
s
)
+
p
2
(

SPFL
(
r
s
)

Para.FL
(
r
s
))
B
B
x
2
p.
(S2)
Here

Para.FL
and

SPFL
are the energies of the un-
polarized FL and spin polarized FL phases from Ref.8
and the
g
-factor is
g
2.
12
Notice from Fig. 2 that
since

Para.FL
(
r
s
)
< 
SPFL
(
r
s
) for our range of interest
r
s
.
40, there is always an energy penalty for polarizing
the Fermi liquid, and no Stoner transition. In this first
scenario, these states compete only with a PWC, whose
4
energy is

WC

SPWC
(
r
s
)
B
|
B
x
|
2
,
(S3)
where

SPWC
is the energy of spin polarized WC from
Ref.8. For each
r
s
we find the FL state with the optimal
polarization and determine its energy competition with
the WC. The resulting phase diagram is shown in Fig. 3
c
.
B. Scenario II: phase diagrams with a partially
polarized crystal with a first order transition
We now include the possibility of a partially polar-
ized (AFM) WC and model its energy using the same
parabolic dependence we used for the FL, as follows:

WC

AFMWC
(
r
s
)
+
p
2
(

SPWC
(
r
s
)

AFMWC
(
r
s
))
B
B
x
2
p
(S4)
The phase diagrams obtained by comparing these en-
ergies from those of the FL states from Eq. (S2) is shown
in Fig. 3
d
. Notice that since there is a crossing of the
energies of SPWC and AFM WC around
r
s
38 (see
Fig. 2), the above form predicts a first-order spin-flop-
type transition of the spin polarization among the WC
states at
r
s
38. There is
a priori
no reason favoring
this transition to be first order rather than continuous,
which leads us to consider this scenario separately.
C. Scenario III: phase diagrams with a partially
polarized crystal with a continuous transition
In order to account for a possible continuous transition
of the spin polarization of the partially polarized Wigner
crystal into the fully polarized crystal, we parameterize
their energy dependence on spin polarization with a sim-
ple quartic Ginzburg-Landau (GL) form:

WC

AFMWC
(
r
s
) +
p
2
b
(
r
s
) +
p
4
c
(
r
s
)
B
B
x
2
p
(S5)
As customary for GL quartic functionals, we take
the coefficient
c
(
r
s
) to be positive for all
r
s
.
The
crystal starts developing a non-zero spontaneous spin-
polarization at a certain density
r
s
1
, for which the co-
efficient
b
(
r
s
) changes from positive to negative. For
r
s
> r
s
1
the GL functional predicts a monotonically in-
creasing spontaneous polarization, until the system sat-
urates at
p
= 1 at some second density
r
s
2
. With these
inputs the coefficients of the GL functional can be fixed
to be:
b
(
r
s
) = 2(

AFMWC
(
r
s
)

AFMWC
(
r
s
1
))
×

AFMWC
(
r
s
2
)

SPWC
(
r
s
2
)

AFMWC
(
r
s
2
)

SPWC
(
r
s
1
)
,
(S6)
c
(
r
s
) =

SPWC
(
r
s
)

AFMWC
(
r
s
)
b
(
r
s
)
.
(S7)
Since the Monte Carlo study
8
did not explore partially
polarized states, the precise values of
r
s
1
and
r
s
2
are un-
known, although they should satisfy
r
s
1
<
38
< r
s
2
. The
phase diagram shown in Fig. 3
e
has been made by choos-
ing
r
s
1
= 27 and
r
s
2
= 45. The precise of form of the
phase diagram does not have strong sensitivity on the
precise values of
r
s
1
and
r
s
2
, provided they are chosen
in the vicinity of
r
s
38 up to some distance of about
r
s
10. In fact, in the limiting case in which the po-
larization changes rapidly from unpolarized to polarized,
r
s
1
approaches
r
s
2
and one recovers the Scenario II of a
first order transition. The main difference is a reduction
in size of the region where the partially polarized crys-
tal is energetically favored, but the overall shape of the
two phase diagrams still looks reasonably similar in both
scenarios II and III, shown in Fig. 3
d
and
e
.
S4. LANDAU FERMI LIQUID PARAMETERS
The energy density of the metallic system at fixed den-
sity in the presence of an in-plane Zeeman field,
B
x
, is
given by
E
E
0
+
s
2
z
2
χ
g
0
μ
B
~
s
z
B
x
+
O
(
s
4
z
)
,
(S8)
where
s
z
= (
~
/
2)(
n
n
) is the spin density,
χ
=
m
/
4
π
the spin susceptibility,
g
0
is the
g
-factor, and
μ
B
is the
Bohr magneton. Therefore, the critical field,
B
c
, at which
the system fully spin polarizes at a given total density,
s
pol
z
= (
~
/
2)
n
, provides an approximate measure of the
spin susceptibility of the system:
B
c
~
2
n
2
μ
B
1
g
0
χ
.
(S9)
Notice that because the dependence of energy of the
electron system on spin polarization is not strictly a
parabola, the measured
χ
inferred from the above for-
mula is not strictly speaking the differential susceptibil-
ity at
B
=0, but rather a finite difference susceptibility
characterizing the energy cost to fully spin polarize the
paramagnetic electron fluid. However, these two suscep-
tibilities can be shown to be identical in the case of the
free electron gas, and are expected to be close to each
other in general whenever the electron system has energy
density that is a smooth function of the spin polarization.
In Landau Fermi liquid theory, this susceptibility is ex-
pressed in terms of the quasiparticle mass (
m
), the bare
band mass (
m
b
), the bare spin susceptibility (
χ
0
) and the
F
s
(0) Landau parameter as follows
13
:
χ
=
χ
0
1
1 +
F
s
(0)
m
m
b
.
(S10)
5
0
B
x
(T)
1
1.2
1.4
1.6
1.8
2
2.2
0.2
-0.2
0
B
x
(T)
0.2
-0.2
0
B
x
(T)
0.2
-0.2
0
B
x
(T)
0.2
-0.2
0
B
x
(T)
0.2
-0.2
0
B
x
(T)
0.2
-0.2
0
B
x
(T)
0.2
-0.2
T
= 7.5 mK
8.8 mK
12 mK
16 mK
19 mK
27 mK
36 mK
n
(10
10
cm
-2
)
a
b
c
d
e
f
g
1
0.1
0.01
B B
/
=sat
r
r
FIG. 4:
Temperature dependence of in-plane magnetic field mapping. a-g
Mapping of the normalized differential
resistance
ρ/ρ
B=sat
as a function of
B
x
and
n
at different temperatures.
10
10
10
11
10
12
n
(cm
-2
)
Coincidence
method
In-plane
polarization
= 0.6
15
10
5
20
30
r
s
g m m
b b
/
0
5
10
15
20
25
This
work
Falson,
.,
Nat. Phys.
,
347 (2015).
et al
11
Falson,
.,
Sci. Advances ,
eaat8742 (2018).
et al
4
Kozuka,
.,
PRB
, 075302 (2012).
et al
85
0
Tsukazaki,
.,
PRB
, 233308
(2008).
et al
78
g m m
* */
0
FIG. 5:
Summary of renormalization effects in ZnO
devices.
g
m
/m
0
as a function of
n
. The band value
g
b
m
b
/m
0
=0.6 is displayed as a horizontal dashed line. Data
have been gathered from devices analyzed in this text, previ-
ous publications,
14–17
and other unpublished measurements.
1.2
1.6
2.0
0
1
2
3
4
5
0
0.1
0.2
0.3
0.4
2.4
T
(K)
F
m
* (
m
)
0
m
= 0.3
m
b
0
10
-2
n
(10
cm
)
FIG. 6: Estimated renormalized effective mass and Fermi
temperature as a function of
n
. See text and also Ref.18
for an extended discussion.
The spin susceptibility of carriers can be evaluated by
various means. A strategy we have employed in previous
studies is the coincidence method which relies on rotating
the sample within a magnetic field to study transport fea-
tures associated with the crossing of opposing spin Lan-
dau levels.
14,16,17
In addition to the coincidence method,
it is possible to polarize carriers into a single spin band by
applying an in-plane magnetic field. According to Eq.S9,
identification of a critical field
B
c
leads to the quantifi-
cation of
g
m
/m
0
. This measurement is discussed in
the main text in the context of Fig. 3 with saturation of
the magnetoresistance at a critical field
B
c
being taken
as evidence for full spin polarization.
For the sake of completeness, here we present the full
raw data set used to produce the analysis presented in
Fig. 3 of the main text. This data set comprises mul-
tiple maps of the differential resistance in the (
n
,
B
x
)-
parameter space at a number of set temperatures. These
are all presented in Fig. 4. As in Fig. 3
b
, we plot the
ratio
ρ/ρ
B=sat
, where
ρ
B=sat
is the saturated differential
resistance at a set charge density at high magnetic field
(corresponding to full spin polarization). The charac-
teristic positive magnetoresistance is evident for all
T
at
high
n
, but is difficult to discern in the higher
T >
20 mK
data upon depletion of the 2DES.
The renormalization of band parameters in the ZnO-
based 2DES as a function of charge density is summarized
in Fig. 5. This figure is compiled from values reported
in previous publications,
15–17
the devices presented in
this manuscript, along with other unpublished results
gathered on samples characterized in the course of this
project. Values obtained through the coincidence method
are displayed as circles, with squares representing those
gained through analyzing the positive magnetoresistance
of a polarizing in-plane magnetic field. The error associ-
ated with each point is no larger than the symbol size. A
good agreement between these two methods is obtained
for the range of densities shown. We additionally display
the corresponding
r
s
value at each
n
on the top axis.
Previous studies on Si-based devices identified that the
enhancement of the spin susceptibility as the critical den-
sity is approached is associated with an enhancement of