Electron accumulation layer in ultrastrong magnetic field
When a three-dimensional electron gas is subjected to a very strong magnetic field, it can reach a quasi-onedimensional state in which all electrons occupy the lowest Landau level. This state is referred to as the extreme quantum limit (EQL) and has been studied in the physics of pulsars and bulk...
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irk-123456789-1293722018-01-20T03:03:29Z Electron accumulation layer in ultrastrong magnetic field Sammon, M. Han Fu Shklovskii, B.I. К 100-летию со дня рождения И.М. Лифшица When a three-dimensional electron gas is subjected to a very strong magnetic field, it can reach a quasi-onedimensional state in which all electrons occupy the lowest Landau level. This state is referred to as the extreme quantum limit (EQL) and has been studied in the physics of pulsars and bulk semiconductors. Here we present a theory of the EQL phase in electron accumulation layers created by an external electric field E at the surface of a semiconductor with a large Bohr radius such as InSb, PbTe, SrTiO₃ (STO), and particularly in the LaAlO₃/SrTiO₃ (LAO/STO) heterostructure. The phase diagram of the electron gas in the plane of the magnetic field strength and the electron surface concentration is found for different orientations of the magnetic field. We find that in addition to the quasi-classical metallic phase (M), there is a metallic EQL phase, as well as an insulating Wigner crystal state (WC). Within the EQL phase, the Thomas–Fermi approximation is used to find the electron density and the electrostatic potential profiles of the accumulation layer. Additionally, the quantum capacitance for each phase is calculated as a tool for experimental study of these phase diagrams. 2017 Article Electron accumulation layer in ultrastrong magnetic field / M. Sammon, Han Fu, B.I. Shklovskii // Физика низких температур. — 2017. — Т. 43, № 2. — С. 283-290. — Бібліогр.: 39 назв. — англ. 0132-6414 PACS: 71.10.−w, 73.20.Qt, 75.70.−i http://dspace.nbuv.gov.ua/handle/123456789/129372 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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К 100-летию со дня рождения И.М. Лифшица К 100-летию со дня рождения И.М. Лифшица |
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К 100-летию со дня рождения И.М. Лифшица К 100-летию со дня рождения И.М. Лифшица Sammon, M. Han Fu Shklovskii, B.I. Electron accumulation layer in ultrastrong magnetic field Физика низких температур |
| description |
When a three-dimensional electron gas is subjected to a very strong magnetic field, it can reach a quasi-onedimensional
state in which all electrons occupy the lowest Landau level. This state is referred to as the extreme
quantum limit (EQL) and has been studied in the physics of pulsars and bulk semiconductors. Here we present
a theory of the EQL phase in electron accumulation layers created by an external electric field E at the surface
of a semiconductor with a large Bohr radius such as InSb, PbTe, SrTiO₃ (STO), and particularly
in the LaAlO₃/SrTiO₃ (LAO/STO) heterostructure. The phase diagram of the electron gas in the plane of the
magnetic field strength and the electron surface concentration is found for different orientations of the magnetic
field. We find that in addition to the quasi-classical metallic phase (M), there is a metallic EQL phase, as well
as an insulating Wigner crystal state (WC). Within the EQL phase, the Thomas–Fermi approximation is used
to find the electron density and the electrostatic potential profiles of the accumulation layer. Additionally,
the quantum capacitance for each phase is calculated as a tool for experimental study of these phase diagrams. |
| format |
Article |
| author |
Sammon, M. Han Fu Shklovskii, B.I. |
| author_facet |
Sammon, M. Han Fu Shklovskii, B.I. |
| author_sort |
Sammon, M. |
| title |
Electron accumulation layer in ultrastrong magnetic field |
| title_short |
Electron accumulation layer in ultrastrong magnetic field |
| title_full |
Electron accumulation layer in ultrastrong magnetic field |
| title_fullStr |
Electron accumulation layer in ultrastrong magnetic field |
| title_full_unstemmed |
Electron accumulation layer in ultrastrong magnetic field |
| title_sort |
electron accumulation layer in ultrastrong magnetic field |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2017 |
| topic_facet |
К 100-летию со дня рождения И.М. Лифшица |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/129372 |
| citation_txt |
Electron accumulation layer in ultrastrong magnetic field / M. Sammon, Han Fu, B.I. Shklovskii // Физика низких температур. — 2017. — Т. 43, № 2. — С. 283-290. — Бібліогр.: 39 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT sammonm electronaccumulationlayerinultrastrongmagneticfield AT hanfu electronaccumulationlayerinultrastrongmagneticfield AT shklovskiibi electronaccumulationlayerinultrastrongmagneticfield |
| first_indexed |
2025-07-09T11:14:53Z |
| last_indexed |
2025-07-09T11:14:53Z |
| _version_ |
1837167748975689728 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2, pp. 283–290
Electron accumulation layer in ultrastrong magnetic field
M. Sammon, Han Fu, and B.I. Shklovskii
Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA
E-mail: sammo017@umn.edu
Received August 30, 2016, published online December 26, 2016
When a three-dimensional electron gas is subjected to a very strong magnetic field, it can reach a quasi-one-
dimensional state in which all electrons occupy the lowest Landau level. This state is referred to as the extreme
quantum limit (EQL) and has been studied in the physics of pulsars and bulk semiconductors. Here we present
a theory of the EQL phase in electron accumulation layers created by an external electric field E at the surface
of a semiconductor with a large Bohr radius such as InSb, PbTe, SrTiO3 (STO), and particularly
in the LaAlO3/SrTiO3 (LAO/STO) heterostructure. The phase diagram of the electron gas in the plane of the
magnetic field strength and the electron surface concentration is found for different orientations of the magnetic
field. We find that in addition to the quasi-classical metallic phase (M), there is a metallic EQL phase, as well
as an insulating Wigner crystal state (WC). Within the EQL phase, the Thomas–Fermi approximation is used
to find the electron density and the electrostatic potential profiles of the accumulation layer. Additionally,
the quantum capacitance for each phase is calculated as a tool for experimental study of these phase diagrams.
PACS: 71.10.−w Theories and models of many-electron systems;
73.20.Qt Electron solids;
75.70.−i Magnetic properties of thin films, surfaces, and interfaces.
Keywords: Wigner crystal state, extreme quantum limit, ultrastrong magnetic field, Thomas–Fermi approxi-
mation.
1. Introduction
When a degenerate electron gas at low temperature T is
subjected to an ultrastrong magnetic field, its properties
undergo dramatic changes. Particularly, when the external
field B is so strong such that
2 1/3
, ,c F B
e nE k Tω
κ
(1)
the cyclotron energy becomes the dominant energy scale in
the system. Here *= /c eB m cω is the cyclotron frequency,
*m is the effective mass, 2 2/3 / 2FE n m≈ is the Fermi
energy at = 0B , Bk T is the thermal energy, κ is the die-
lectric constant, and n is the three-dimensional concentra-
tion of electrons. When Eq. (1) is satisfied, we say that the
system is in the “extreme quantum limit” (EQL).
Under the influence of a magnetic field B , the kinetic
energy of electrons in the direction perpendicular to B is
quantized into Landau levels. In the EQL, the gap between
adjacent levels becomes very large and electrons occupy
the lowest Landau level only. As a result, the energy of the
electron gas depends only on the momentum in the direc-
tion parallel to B , creating a quasi-one-dimensional state.
It has been proposed that under such conditions, various
instabilities such as charge density waves, spin density
waves, or Wigner crystallization occur [1–4].
What conditions are necessary to reach the EQL exper-
imentally? From Eq. (1), it follows that in order for the gas
to remain metallic, one must have 3 1na . Here
2 2*= /a m eκ (2)
is the effective Bohr radius of the material. Additionally,
the strong magnetic field condition, c FEω , requires
3 1nλ , where we have introduced = /c eBλ as the
magnetic length. Combining 3 1na and 3 1nλ , we
find that in order to reach the EQL, we require aλ . At
10 T, 10λ ≈ nm, and so we require materials in which
10a nm. There are special materials such as InSb and
Hg1 x− CdxTe in which a ranges between 60 120− nm, so
that the EQL is achievable at reasonable magnetic fields
[5–7]. In particular, bulk transport studies of InSb have
found an experimental phase diagram that consists of
a metal, EQL, and insulator phase [7].
Another material in which the EQL may be reached is
bulk SrTiO3 (STO). STO is a semiconductor with a rela-
tively heavy effective mass * = 1.5 em m [8], but a dielectric
constant that becomes very large, 4= 2 10κ ⋅ , at liquid heli-
um temperatures [9,10]. As a result, the Bohr radius of
© M. Sammon, Han Fu, and B.I. Shklovskii, 2017
M. Sammon, Han Fu, and B.I. Shklovskii
STO becomes = 700a nm. This should create an ideal
situation to study the EQL and several studies of the bulk
magnetic properties of the material have been conducted
[11–14]. Despite such effort, attempts to observe the EQL
in bulk STO have not met much success, presumably due
to disorder effects [14].
Much attention has been devoted in recent years to
LaAlO3/SrTiO3 (LAO/STO) heterostructures in which the
“polar catastrophe”[15,16] creates an electric field that
causes a high mobility electron gas to form at the interface.
Recent magnetotransport studies of these structures have
reported integer quantum Hall effect steps in xyρ that may
be evidence of the gas approaching the EQL [17–19].
In this paper, we study the conditions under which one
can observe the EQL in electron accumulation layers in
semiconductors with a given dielectric constant κ and
Bohr radius a. Such an accumulation layer can be created
in many ways. One example already mentioned is the polar
catastrophe in LAO/STO heterostructures which creates
an accumulation layer at the interface. Other common
techniques include ionic liquid gating [20] of the semicon-
ductor surface and δ-doping by donors in the bulk of the
sample [21]. In all such cases the end result is an external
electric field E along the direction perpendicular to the
surface that causes electrons to accumulate near the surface
(see Fig. 1). We can always relate E to the surface concen-
tration N of electrons in the accumulation layer by
4= .eNE π
κ
(3)
In our discussion below, all results are expressed through
the surface concentration N rather than the external field E.
Within the EQL phase we calculate the Thomas–Fermi
profiles of the electron density and electrostatic potential
as a function of the distance from the surface. By compar-
ing the parameters found in the EQL metal with those of
the quasi-classical metal (M), we determine the strength of
the magnetic field at which the electrons enter the EQL.
On the other end, these parameters are compared to those
of the Wigner crystal (WC) phase at a large magnetic field
to find the upper limit of the magnetic field at which the
EQL metal is still valid. Figure 2 summarizes our results
for the case of ⊥B E as a phase diagram in dimensionless
units of 0/B B and 2Na . Here
2 3 2 3*0 = / ( )B m e c κ (4)
is the magnetic field such that = aλ . To preserve the uni-
versality of Fig. 2 for different semiconductor parameters,
we introduce the material specific constant
*
= .em
m
κ
κ (5)
In this notation, 5 2
0 = (2.5 10 )B −⋅ κ T.
Let us discuss what is achievable experimentally. The
strongest static magnetic fields available in laboratories are
approximately max 45B T, from which it follows that
4 2
max 0/ 1.8 10B B −≈ ⋅ κ . (6)
Given the values of κ and a in STO at liquid helium tem-
peratures, we find 4= 1.3 10κ ⋅ , and so max 0/B B ≈ κ , while
for InSb, 3= 1.1 10κ ⋅ and 3/4
max 0/B B ≈ κ . These values
are indicated in Figs. 2 and 4 by the dashed and dotted
lines respectively.
Fig. 1. (Color Online) Schematic energy diagram of an accumula-
tion layer in a lightly doped n-type semiconductor, where ε is the
energy and x measures the distance from the surface. Electrons
(blue/dark grey layer) are attracted to the surface by an external
electric field E , where they form the accumulation layer with
a characteristic width d . In the bulk of the semiconductor, the
Fermi level Fε lies near the bottom of the conduction band.
Fig. 2. (Color online) Phase diagram of the electron gas for
⊥B E in the dimensionless plane of 0/B B and the surface con-
centration 2Na plotted in a log–log scale. The regions are the
quasi-classical metal (M), the metallic EQL phase (EQL), and the
insulating Wigner crystal state (WC). The dashed line indicates
the ratio max 0/B B in STO, while the dotted line is the same quan-
tity in InSb. See Eqs. (2), (4), and (5) in the text for the definitions
of a, 0B , and κ .
284 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2
Electron accumulation layer in ultrastrong magnetic field
Below we focus on STO. Figure 3 presents that phase
diagram for ⊥B E in STO at liquid helium temperatures.
The lower EQL border defined by Eq. (20) intersects maxB
in STO at a concentration 128 10N ⋅ cm 2− . Surface con-
centrations as low as 121 10⋅ cm 2− with high mobility have
been achieved in modified LAO/STO interfaces and δ-dop-
ed STO [17,19], so that the lower critical magnetic field is
reachable. Additionally, this range of surface concentra-
tions 12 12= 1 10 8 10N ⋅ − ⋅ cm 2− corresponds to bulk concent-
rations /N d ranging between 173 10⋅ cm 3− and 18 31 10 cm ,−⋅
where according to the data [22] a reasonably large mobili-
ty can be maintained making the EQL achievable in this
range of concentrations. Here d is the characteristic width
of the accumulation layer.
The rest of the paper is organized as follows. In Sec. 2
the new density profile ( )n x in the EQL is derived using
the Thomas–Fermi approximation and the critical magnetic
field at which the gas enters the EQL is found. In Sec. 3 we
finish constructing the phase diagram for different direc-
tions of magnetic field and arrive at Figs. 2 and 4. In Sec. 4
we calculate the quantum capacitance in all phases and
present the plot of the effective width of the inverse ca-
pacitance as a function of the magnetic field ( )B T in
Fig. 6. In Sec. 5 we show how our results map to the prob-
lem of heavy atoms in ultrastrong magnetic fields which
has been intensely studied in astrophysics.
2. Thomas–Fermi theory of the accumulation layer
2.1. Quasi-classical metal
In an accumulation layer, an electric field E applied
perpendicular to the the surface causes electrons to accu-
mulate with a three-dimensional concentration ( )n x , where
x is the distance measured from the surface. Here we as-
sume that the semiconductor is such that the Fermi level in
the bulk of the material lies at the bottom of the conduction
band, and the electron concentration tends to zero at large
distances. This can be true if the semiconductor is lightly
doped by donors [23]. This problem was first solved in the
absence of a magnetic field by Frenkel [24], and we repeat
his argument below.
In order to find the density profile, we make use of the
Thomas–Fermi approach in which the local potential ( )xϕ
is related to the local chemical potential ( )xµ such that
( ) ( ) = = 0Fe x xϕ +µ ε . In a normal metal, the chemical
potential is related to the density such that
2
2 2/3
0 0
( ) = [3 ( )] .
2
x n x
m
µ π
(7)
Here we use the subscript 0 to denote quantities when B = 0.
When the dielectric response is linear, the potential and
density are related through Gauss’s law, such that
2
0
02
4= ( ).
d e n x
dx
ϕ π
κ
(8)
Combining Eqs. (7) and (8) with the above equilibrium
condition, we obtain the Thomas–Fermi equation
3/22 7/2 1/2
0 0
2 2
2= .
/ /3
d
e a e adx a
ϕ ϕκ
π
(9)
The solution of this equation that satisfies the condition
lim ( ) = 0
x
x
→∞
ϕ is known to be
3
0 1 4
0
( ) = ,
( )
e ax C
x d
ϕ
κ +
(10)
and the associated density is
3
0 2 6
0
( ) =
( )
an x C
x d+
(11)
where 2
1 = (225 / 8) 278C π and 2 = (1125 / 8) 442C π .
To determine the characteristic width 0d , we use the
definition of the two-dimensional electron density
0
= ( )N n x dx
∞
∫ . (12)
Combining Eqs. (11) and (12) we find that for the quasi-
classical metal
1/5
0 3 2
1= ,d C a
Na
(13)
where ( )1/5
3 = 225 / 8 2.45C π .
2.2. Extreme quantum limit
The main purpose of this paper is to understand how the
above distribution changes when the gas is subjected to
such strong magnetic fields that it is in the EQL.
Fig. 3. (Color online) Phase diagram for ⊥B E in STO at liquid
helium temperatures. The diagram is presented in a log-log scale.
All regions and borders are identical to those in Fig. 2
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 285
M. Sammon, Han Fu, and B.I. Shklovskii
As stated above, when in the EQL, the kinetic energy in
the direction perpendicular to the field is quantized and
electrons occupy the lowest Landau level. This means that
the density of electrons in the direction perpendicular to
the field is fixed by the density of the lowest Landau level
21/ (2 )πλ . In addition, the magnetic field aligns the spins
in the direction of the field, lifting the spin degeneracy.
The remaining direction has a density determined by the
wavevector k . We can relate the maximum value of this
wave vector to the three dimensional density of electrons by
2
1 = ( ).
2
k n x
π πλ
(14)
As a result, the local chemical potential changes from
Eq. (7) to
2
2 2 2( ) = [2 ( )] .
2
x n x
m
µ π λ
(15)
Proceeding in the same way as before, we arrive at the
EQL Thomas–Fermi equation
2 3/2 1/2
1/2
2 4 1/2
2= .
( )
d e
dx a
ϕ
ϕ
π λ κ
(16)
The solution gives the potential
4
4 4
( )
( ) = ,
x dex C
a
λ
−
ϕ
κ λ
(17)
and density as
2
5 4
( )
( ) = ,
x d
n x C
a
λ
−
λ
(18)
where 2
4 = 1/ (18 ) 0.006C π and 3
5 = 1/ (6 ) 0.005C π .
Using Eq. (12), the characteristic width is determined to be
4 1/3
6= ( ) ,d C Na
λ
λ (19)
where 3 1/3
6 = (18 ) 8.23C π .
This result is valid when 0<d d
λ
. We find then that the
magnetic field compresses the accumulation layer closer to
the surface. Equating Eqs. (13) and (19), and going back to
the magnetic field, we find that the EQL is achieved when
2 4/5
1 7 0> = ( ) ,cB B C B Na (20)
where 3/2
7 6 3= ( / ) 6.15C C C ≈ . If 0 1< < cB B B , ( )n x
obeys Eq. (11) until ( ) = cxµ ω where the gas enters the
EQL. The distance from the surface at which this occurs is
given by
1/2
l ( ) .x a≈ λ (21)
At this distance, the remaining electrons are in the EQL
and the electron density is sharply cut off.
We emphasize that the direction of the magnetic field
has played no role in our discussion so far. Therefore, we
see that our boundary given by Eq. (20) is independent of
the field direction. This line is shown in both Figs. 2 and 4.
For 1cB B , these diagrams lose their universality and we
discuss them separately in the following section.
3. Phase diagrams for different magnetic field
directions
Below we address the role of the magnetic field direc-
tion and complete the phase diagrams Figs. 2 and 4. Let us
assume that the electric field is strong enough such that
2 > 1Na . The Thomas–Fermi approximation is only valid
as long as the electrons can be treated semiclassically. We
can make this condition quantitative by requiring that
0
1 ( ) > 1xk x dx
∞
π ∫ (22)
which is a generalization of the 1d particle in a box. This
condition depends on the direction of the magnetic field
relative to the electric field, and so below we consider sep-
arately the two cases B E and ⊥B E .
3.1. Magnetic field parallel to electric field
When B E , 2 2( ) = 2 ( )xk x n xπ λ . From this we find that
the approximation breaks down when
2
2 0= = (2 ).cB B B Naπ (23)
Fig. 4. (Color online) Phase diagram of the electron gas for B E
in the dimensionless plane of 0/B B and the surface concentra-
tion 2Na plotted in a log–log scale. The regions are the quasi-
classical metal (M), the metallic EQL phase (EQL), and the insu-
lating Wigner crystal state (WC). The dashed line indicates the
ratio max 0/B B in STO, while the dotted line is the same quantity
in InSb. See Eqs. (2), (4), and (5) in the text for the definitions of a,
0B , and κ . Numerical values of N and B for STO at liquid heli-
um temperatures can easily be recovered from comparison of
Figs. 3 and 4.
286 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2
Electron accumulation layer in ultrastrong magnetic field
This is the boundary between regions EQL and WC in
Fig. 4. As the magnetic field is increased beyond this val-
ue, the Thomas–Fermi approximation becomes invalid
everywhere. Instead, the electron gas forms a WC consist-
ing of single electron cylinders of radius λ and height L
(see Fig. 5(a)). The height of the cylinders can be deter-
mined as follows. At the border 2cB , the electron gas is
confined to the first sub-band of a triangular potential well.
The kinetic energy is then
2
2=
2
K
mL
(24)
while its potential energy is = / 2U eEL , where E is relat-
ed to N by Eq. (3). Equating the kinetic and potential en-
ergies we find that
1/3= ( / 2 ) .L a Nπ (25)
The height of the cylinders L should agree parametri-
cally with the size of the accumulation layer d
λ
along the
EQL–WC phase boundary Eq. (23). Let us confirm this.
Along the boundary, we know that Eq. (23) gives 2= 1/ ,N λ
so that 2 1/3= ( )L aλ . If instead we are coming from the
EQL region, we use Eq. (19) and find that 2 1/3= ( )d a
λ
λ .
Equation (25) is the same as the width of the first sub-
band wave functions obtained for an inversion layer in
an electric field E [25,26]. However, contrary to the inver-
sion layer where electrons are delocalized in the plane per-
pendicular to the field E , electrons here are strongly local-
ized by the magnetic field in a cylinder of size λ. This is
also the simplest case of quantum screening [27–29].
3.2. Magnetic field perpendicular to electric field
If ⊥B E , = 1/xk λ . As a result, we find instead of
Eq. (23) that the Thomas–Fermi approximation fails when
2 2
0> ( )B B Na . We show below that the EQL phase forms
a WC at a somewhat smaller field
2 2
0
3 2 2
( )
.
ln ( )
c
B Na
B
Na
≈ (26)
This is the boundary given in Fig. 2.
The structure of the WC phase for ⊥B E is markedly
different than when B E . We can imagine the electrons as
cylinders of radius λ oriented along B which lie on their
sides in the plane of the surface (See Fig. 5(b)).
To describe the WC, one can imagine that E is replaced
by a uniform positive surface charge density eN which is
partitioned into Wigner–Seitz (WS) cells with charge e,
length L , and width = 1/w NL so that each cell contains
exactly one electron. We assume that the energy of each
WS cell is approximately given by the sum of the kinetic
energy Eq. (24), and the electrostatic energy
2 2= ( / ) ln ( )U e L NL− κ . Optimization of this energy with
respect to L gives
2 .
ln ( )
aL
Na
(27)
As the magnetic field is reduced, it is natural to assume
that the WC–EQL transition occurs when the electron is
the same size as the WS cell. Setting =w λ , and using
Eq. (27), we arrive at the border Eq. (26). We see that the
logarithmic term in the denominator of Eq. (26) resembles
those obtained previously for the metal–insulator transition
in the bulk of a doped semiconductor in a strong magnetic
field [7,30].
Up until now our theory is generic and is valid for
any semiconductor material with a linear dielectric constant.
In STO, however, the dielectric response becomes nonline-
ar at sufficiently high surface concentrations [31]. It was
shown the dielectric response becomes nonlinear when
2
2 2 3/2
1
0
1= = .c
aNa N a
a
≈ κ
κ
(28)
Here
0
3.9a Å is the lattice constant in STO. We see in
Figs. 2 and 4 that at this concentration, maxB is such that
the gas is still in region M, where the magnetic field only
acts to cut the tail of the distribution. Thus, the EQL phase
is unachievable experimentally when the dielectric re-
sponse is nonlinear and so we limit Figs. 2 and 4 to
2 3/2<Na κ .
Fig. 5. (Color Online) Schematic of the electron structure in
the WC phase for B E (a) and ⊥B E (b). Each electron
(red/dark grey) forms a cylinder of radius λ oriented along the
direction of the magnetic field on the surface (yellow/light grey)
inside the semiconductor.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 287
M. Sammon, Han Fu, and B.I. Shklovskii
4. Magnetocapacitance
In this section we calculate the capacitance of an accu-
mulation layer as a function of the magnetic field for all
phases. Our results can be used as tools for an experi-
mental study of Figs. 2 and 4. For the setup we imagine
that the accumulation layer is either created by the electric
field of a metallic gate, or by a built in electric field E to
which a metallic gate adds a relatively small field E′. Ex-
amples of such devices include the gating of an intrinsic
semiconductor by an ionic liquid and the application of
a metallic gate to the top LAO surface in the LAO/STO
heterostructure. In both cases one can study the differential
capacitance per unit area = ( ) /C d eN dV , where V is the
gate voltage. The inverse capacitance 1C− may be written
as the sum of the inverse geometrical capacitance and the
inverse quantum capacitance
1 4
= .q
q
d
C− π
κ
(29)
Below we calculate qd for our phases M, EQL, and WC in
both B E and ⊥B E cases.
Let us discuss some of these results. In Sec. 2 the
Thomas–Fermi potential profiles were found for the metal-
lic M and EQL phases. From Eq. (10), we find that in re-
gion M the potential difference from = 0x to =x ∞ at
a given concentration is
1/5
2 4/5225( ) = ( ) .
8
eN Na
a
π ϕ π κ
(30)
Taking the derivative /d dNϕ and using Eq. (29), one
finds
0( ) = .
5q
d
d M (31)
In the EQL we know that the potential is instead given
by Eq. (17) where dλ is given by Eq. (19). Proceeding in
the same way, we find that the EQL changes the capaci-
tance to
( ) = .
3q
d
d EQL λ (32)
Equations (31) and (32) are not surprising. In both cas-
es, qd is the width of the accumulation layer in the direc-
tion of the electric field E , up to some numerical prefactor.
To put another way, the effective width of the quantum
capacitor is the width of the accumulation layer.
In order to find the point at which the capacitance tran-
sitions from that of the quasi-classical metal to the EQL
metal, we equate Eqs. (31) and (32). We find then that the
EQL becomes observable in capacitance measurements at
3/2
1 1= (5 / 3)c cB B′ , which is slightly larger than Eq. (20).
At this field we should see the effects of the EQL begin to
emerge, and so we use this as the field at which the gas
transitions.
Equations (31) and (32) are valid for both B E and
⊥B E cases. As the magnetic field is increased, the two
cases separate because of their different WC structure. We
first discuss the B E case.
As the magnetic field is increased, we see from Eq. (32)
that qd will decrease. When 2= cB B (Eq. (23)), a Wigner
transition occurs and the gas enters the WC region. In this
state, the width of the accumulation layer in the direction
of E is approximately given by Eq. (25) and no longer
depends on the magnetic field. In order to find the value of
the capacitance, we combine Eqs. (23) and (32) and find
8 2 1/3(WC ) =
( )
q
ad C
Na
(33)
where 1/3
8 = ( / 6) 0.81C π ≈ .
In the ⊥B E case, the transition to the WC phase hap-
pens at a much larger B given by Eq. (26). At this value of
the field, the width of the accumulation layer is such that
1/qd Na . If the field is increased further, then qd con-
tinues to decrease as the negative energy due to correlation
effects of the WC begin to dominate [32]. However at such
large fields, the distance between electrons may become
comparable to the distance between the WC and the gate,
and the coupling of electrons to their image charge be-
comes the dominant factor in the determination of the ca-
pacitance [33]. Despite this, such magnetic fields are too
high to reach experimentally, and so we refrain from any
further discussion of this limit.
We summarize these results in Fig. 6 as a plot of
(nm)qd vs. ( )B T for STO samples with a surface concen-
tration 12 –2= 10 cmN . At this concentration, the transition
into the EQL occurs when 1= 18 TcB B′ ≈ while the EQL–WC
transition occurs at 2= 46 T.cB B ≈ We see then that the
EQL phase is within the realistic range of magnetic fields
and so capacitance measurements at this concentration
provide an opportunity in which the first border will be
observed. However if one wishes to see the splitting be-
tween the two directions, one needs to go to lower concen-
trations than 12 –2= 10 cmN .
Fig. 6. Log–log plot of (nm)qd as a function of ( )B T for the
phases of Figs. 2 and 4 in STO at liquid helium temperatures with
a surface concentration 1 22 = 10 cm .N − The region in which qd
is the same for both B E and ⊥B E is illustrated by a thick line.
The numbers in parentheses correspond to equations in the text.
288 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2
Electron accumulation layer in ultrastrong magnetic field
5. Heavy atoms in pulsars
So far, we have restricted our discussion to electron ac-
cumulation layers in semiconductor materials with such
large Bohr radii that the EQL is achievable experimentally.
For atoms, the EQL is achieved when the magnetic field is
larger than 5
0 = 2.34 10B ⋅ T and is completely unattainable
in a laboratory setting. However, in rotating neutron stars,
or pulsars, the magnetic fields at the surface range from
8 910 10− T [34], so that the EQL can be achievable even
for atoms. The effect of the large magnetic field on the
structure of the surface layer of neutron stars has been
studied extensively [35].
It is believed that within the surface of neutron stars
there exists a layer enriched by iron atoms [36]. Motivated
by this, Kadomtsev studied heavy atoms in ultrastrong
magnetic fields, where he used an EQL Thomas–Fermi
equation which is the spherically symmetric analog of our
Eq. (16) [37,38] He found that the EQL Thomas–Fermi
description of the atom is valid as long as the magnetic
field is in the range
4/3 3
0
BZ Z
B
(34)
where Z is the nuclear charge of the atom. When
4/3
0/B B Z , the magnetic field has only a perturbative
effect on the atomic structure, while for 3
0/B B Z , the
atom is elongated along the direction of the magnetic field.
We will now show that there exists a mapping between the
nuclear charge Z and the surface concentration N showing
that the 4/3Z and 3Z borders are in agreement with those
we found for the EQL phase when B E .
We can imagine that within the accumulation layer,
electrons are bound at a distance d away from the surface
by the positively charged plane with charge density eN .
We can think that this plane consists of positive squares
(nuclei) of length d and charge
2= .Z Nd (35)
At the lower critical field 1cB , we know that the character-
istic width of the gas is given by Eq. (13). Using Eq. (35)
we find that the nuclear charge at this field is 2 3/5= ( )Z Na .
From here, we find then that the lower critical field 1cB
given by Eq. (20) is related to the nuclear charge by
2 4/5 4/31
0
= ( ) =cB
Na Z
B
(36)
in agreement with the lower border of Eq. (34). On the other
hand, we know that if the magnetic field is applied parallel
to the electric field, the Thomas–Fermi approximation fails
when 2= cB B , where 2cB is given by Eq. (23). At this field
strength, the width of the layer d is given by Eq. (19),
from which it follows that 2 2 5/3 4/3
0= ( ) ( / )Nd Na B B .
Solving this equation for B , and using Eq. (23), we find
that the EQL region ends when
3
0
=B Z
B
(37)
in agreement with Kadomtsev’s second border.
Note that in our comparison, we have used the case of
B E . In a heavy atom, the magnetic field can be both par-
allel and perpendicular to the electric field of the nucleus.
However, we know from Sec. 3 that when the B E , the
Thomas–Fermi approximation fails at a smaller B than for
the case of ⊥B E . It is for this reason that the mapping from
the accumulation layer to the heavy atoms needs B E .
Let us conclude with a discussion about the structure
of the atom when 3
0/B B Z . At such fields all electrons
are in the lowest Landau level and occupy a single sub-
band in the direction of the field B . From the above map-
ping, it would seem natural to expect the structure to be si-
milar to a WC where the same limits apply. Actually, due
to the strength of the Coulomb field of the point charge Z ,
the electrons instead compress into a single uniformly
charged cylinder of radius 1/2=R Zλ and height
3= / ln( / )L a Z B Z . The compression of the cylinder is
stopped by the kinetic energy 2 2/ (2 )mL along the direc-
tion of B . One can think that in our Fig. 4, the atom be-
comes “frozen” at the EQL–WC border when 3
0/B B Z .
6. Conclusion
In this paper, electron accumulation layers induced by
an electric field perpendicular to the surface are studied for
semiconductors with a large Bohr radius under the influ-
ence of a very strong magnetic field. Phase diagrams in the
plane of magnetic field strength and two-dimensional elec-
tron concentration are found for two orientations of mag-
netic field with respect to the electric field (Figs. 2–4).
Each diagram is found to have three phases: the quasiclass-
ical metal, the extreme quantum limit electron gas, and the
Wigner crystal. We showed that in the case of STO, with
the largest known Bohr radius = 700a nm, all of these
phases may be reached in the available magnetic fields.
We calculate the width of the accumulation layer for all of
the phases and predict how the quantum capacitance of the
accumulation layer changes with the magnetic field when
we cross from one phase to another. This provides a tool
with which the phase diagrams can be studied experimen-
tally with the help of magneto-capacitance measurements.
In the future, it will be interesting to explore the transport
properties of the extreme quantum limit of STO accumula-
tion layers.
Above we assumed that the spectrum near the bottom of
the conduction band in STO consists of a single isotropic
band with an effective mass of * 1.5 em m≈ . In reality, the
band structure of STO consists of three degenerate 3d or-
bitals of the Ti atoms which are split by the spin-orbit in-
teraction and the tetragonal distortion of STO at low tem-
peratures [8,39]. At concentrations 18< 10n cm 3− , only the
lowest band is occupied and Shubnikov–de Haas oscilla-
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2 289
M. Sammon, Han Fu, and B.I. Shklovskii
tions measurements show the band has an effective mass of
(1.5 1.8) em− [8]. It was mentioned above that with surface
concentrations 12 –210 cmN one can explore the EQL
with the available magnetic fields. Such N correspond to
bulk concentrations 17 3 18 310 1 m cm0c n − −
, where the
approximation of an effective mass of * 1.5 em m≈ is justi-
fied.
Acknowledgments
We are grateful to Eugene Kolomeisky, Nini Pryds, and
Brian Skinner for helpful discussions. M. Sammon and
Han Fu were supported by the Fine Theoretical Physics In-
stitute.
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290 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 2
1. Introduction
2. Thomas–Fermi theory of the accumulation layer
2.1. Quasi-classical metal
2.2. Extreme quantum limit
3. Phase diagrams for different magnetic field directions
3.1. Magnetic field parallel to electric field
3.2. Magnetic field perpendicular to electric field
4. Magnetocapacitance
5. Heavy atoms in pulsars
6. Conclusion
Acknowledgments
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