On the origin of transversal current in double-layer heterostructures
It is shown that usually used theoretical model for double-layer heterostructures as a pseudospin ferromagnet does not explain the observed two-dimensional spectrum. Its existence is possible when neglecting Coulomb interaction destroying two-dimensional structures and can be realized only in a stro...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2017
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| Цитувати: | On the origin of transversal current in double-layer heterostructures / S.V. Iordanski // Физика низких температур. — 2016. — Т. 43, № 1. — С. 11-14. — Бібліогр.: 11 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860151236309811200 |
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| author | Iordanski, S.V. |
| author_facet | Iordanski, S.V. |
| citation_txt | On the origin of transversal current in double-layer heterostructures / S.V. Iordanski // Физика низких температур. — 2016. — Т. 43, № 1. — С. 11-14. — Бібліогр.: 11 назв. — англ. |
| collection | DSpace DC |
| container_title | Физика низких температур |
| description | It is shown that usually used theoretical model for double-layer heterostructures as a pseudospin ferromagnet does not explain the observed two-dimensional spectrum. Its existence is possible when neglecting Coulomb interaction destroying two-dimensional structures and can be realized only in a strong magnetic field. That is connected also with the plain vortex lattices forming in strong magnetic fields due to thermodynamic instability. This model gives reasonable explanations of various observed effects depending on the filling of the corresponding bands. In particular in this work we show that in double-layer heterostructures a large interlayer conductance really observed can exist.
|
| first_indexed | 2025-12-07T17:51:51Z |
| format | Article |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1, pp. 11–14
On the origin of transversal current in double-layer
heterostructures
S.V. Iordanski
Landau Institute for Theoretical Physics RAS, Chernogolovka 142432, Russia
E-mail: iordansk@itp.ac.ru
Received May 11, 2016, revised June 14, 2016, published online November 25, 2016
It is shown that usually used theoretical model for double-layer heterostructures as a pseudospin ferromagnet
does not explain the observed two-dimensional spectrum. Its existence is possible when neglecting Coulomb in-
teraction destroying two-dimensional structures and can be realized only in a strong magnetic field. That is con-
nected also with the plain vortex lattices forming in strong magnetic fields due to thermodynamic instability.
This model gives reasonable explanations of various observed effects depending on the filling of the correspond-
ing bands. In particular in this work we show that in double-layer heterostructures a large interlayer conductance
really observed can exist.
PACS: 73.21.–b Electron states and collective excitations in multilayers, quantum wells, mesoscopic, and
nanoscale systems;
67.10.–j Quantum fluids: general properties.
Keywords: double-layer heterostructures, two-dimensional spectrum, magnetic fields.
The experiments with double-layer heterostructures have
a long history beginning from the late eighties of the pre-
vious century. The main number of theoretical works [1,2]
used the layer index as an additional quasi-spin index.
However the experimental work [3] showed that the theo-
retical results based on this assumption contradict to the
performed experiment and require some other description
close to the used for one-layer systems [4].
We take the coordinate z perpendicular to the plain of
the layers and the coordinates = ( , )x yr describe a two-
dimensional space with a constant electron potential ener-
gy ( ) = ( )U z U z− (see Fig. 1). We assume the existence of
a constant magnetic field B directed along z large enough
to make the energy of electron Coulomb interaction pro-
portional to B small compared to the magnetic energy
proportional to B . That is very important because Cou-
lomb interaction tends to destroy two-dimensional effects.
Further we shall neglect Coulomb electron interaction as-
suming large enough magnetic field.
It was shown in the theoretical work [5] for this case the
situation is unstable and the creation of plain vortex lattices
gives the thermodynamic gain. That is the case of the sepa-
rate action of the variables z and r, therefore electron
wave functions are the products of two functions ( ) ( ).Z z Φ r
Various types of the vortex lattices are possible. The elec-
tron filling of the proper bands defines the physical proper-
ties. The separation of the variables gives two Shrödinger
equations
2 2
2= ( ) ( )
2 e
Z Zi U z Z z
t m z
∂ ∂
− +
∂ ∂
(1)
where ( )U z is the two-well potential (see Fig. 1).
The functions ( )Φ r are defined by two-dimensional
equation
2
eff
( ) 1= ( ) ( ).
2 e
ei i
t m c
∂Φ ∂ − − Φ ∂ ∂
r A r r
r
(2)
Here the effective vector potential effA includes not on-
ly the vector potential of the external magnetic field
(1/ 2)[ ]Br but also the contribution of vortices (with the
Fig. 1. Interwell potential.
© S.V. Iordanski, 2017
S.V. Iordanski
lowest energy) with the minimal circulation 2π in each unit
cell. The solutions of the Eq. (2) have simple band energy
spectrum only at the rational values of the total magnetic
flux through the unit cell area of the vortex lattice (see,
e.g., [6]). Having in mind the experiments [7] where the
electron density was close to that at the half filling of Lan-
dau level, we assume two vortices in each unit cell. Therefore
the total magnetic flux through the area of the unit cell σ will
be 0 02 = ( / )B l nσ− Φ Φ , where 0 = 2 / | |c eΦ π is the flux
quantum and l, n are coprime integers. We see that the half
filling of Landau level is achieved at = 0l , and the steady-
state solutions correspond to the representation of Abelian
group of the periodic translations with the unit cell area
02 4= = .
| |
c
B e B
Φ π
σ
(3)
The situation is similar to that of graphene. It was
shown in the paper [5] that the lattice have the hexagonal
symmetry which is determined by the vortex lattices with
the periodic vector potential in the plain r. Having in mind
the experiments [7] we assume that the vortex lattice has
the hexagonal symmetry with two vortices of the minimal
circulation 2π in each elementary zone. It was shown in [5]
that in this case there are two nonequivalent points 0k and
0′k on the boundary of the two-dimensional Brillouin cell
where the representation of the space group for the vortex
lattice is two-fold and the full filling of the lowest band
corresponds to the density of the half filled Landau level.
We assume that the maximal energy in the lowest band
corresponds to the energy 0( ) = ( )′ε ε0k k at the critical
points. We have to add to this energy the lowest energy of
the size quantization 1E , corresponding to the symmetric
wave function 1 1( ) = ( )Z z Z z− and the chemical potential µ
is assumed to satisfy inequality
1( ) < .Eε + µp (4)
We have defined the filling only for the lowest level of size
quantization. For the next level 2E of size quantization
with the antisymmetric wave function 2 2( ) = ( )Z z Z z− −
we will also assume that
2( ) < .E′ε + µp (5)
Therefore we have two hole Fermi circles with
1 2( ) =Fp E Eε − around the critical points 0k , 0′k . In what
further we shall consider only the case of zero temperature.
Let us consider the stationary problem without the exter-
nal electric field (details can be found in [8]). Both energies
1E and 2E are close to the oscillation energy 0E in one well
without tunneling. The symmetry consideration of the tun-
neling process gives (in semiclassical approximation) two
different states with the energies 1E and 2E , where
0
2 1
1= exp | | ,
a
a
E
E E p dz
−
− −
π
∫
(6)
where a− and a are the positions of the turning points,
the integral is in the classically forbidden region,
0| | = = 2 ( )e ep m m U E−v . Therefore 2E and 1E are ex-
ponentially close.
In the presence of the external electric field there is an
additional term in the Hamiltonian
2ˆ ˆ= | |H e z dzd rδ − γ ρ∫ (7)
where ρ̂ is the operator of the electron density. We choose
the electric field > 0γ . Therefore we will have the current
from the left well to the right one. There is no possibility to
avoid the classically forbidden region. Thus we have to
consider the subbarrier current. The oscillations in the left
well with the energy 0E give the ingoing wave at the be-
ginning of the classically forbidden region
0
in = exp .
2 4
a
b
E i ipdz
− π ψ −
π
∫
v
(8)
At the point =z a we get the outgoing wave in the right
well out in= ( )t Eψ ψ where ( )t E is the transmission ampli-
tude, 2| ( ) | =t E D ,
2= exp | |
a
a
D p dz
−
−
∫
.
The electron flux to the right well (see Fig. 1) at the turn-
ing point =z a is (see [8])
0
out out
2= exp | | .
2
a
a
E
p dz+
−
ψ ψ −
π
∫
v (9)
When obtaining this formula we use only ( )Z z compo-
nent of the electron wave function. For a more complete
expression it is necessary to perform the second quantiza-
tion, using the two-dimensional representation of the space
group for the vortex crystal ( ) exp( / ),iΦ p pr
( ) exp( / ),i+Φ −p pr where ( )Φ p is the column of two
functions, ( )+Φ p is the line, mutually orthogonal and nor-
malized (see also [5]). Therefore the full electron operators
of the second quantization are
( , , ) = ( ) exp ( ) ( , )i i i
it z Z z a t ψ Φ
∑
p
prr p p
(10)
and
( , , ) = ( ) exp ( ) ( , )i i i
it z Z z a t+ + + +− ψ Φ
∑
p
prr p p
(11)
here = 1,2i and ( , ), ( , )i ia t a t+p p are the usual Fermi opera-
tors in Heisenberg representation with the energy
1 2( ) <p E Eε − +µ , here µ is the chemical potential equal
to the total energy in Dirac points.
The quantities ( ),iZ z ( )iZ z+ do not depend on the func-
tions ( )Φ p , ( )+Φ p , but their number is essential for the
12 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1
On the origin of transversal current in double-layer heterostructures
counting of the different filled states. The number of the
various states in the crystal band coincides with the num-
ber of the elementary cells. For the two-dimensional case it
gives = /N S σ , where S is the sample area, and σ is the
area of the elementary cell. Generally speaking, we should
count only the filled states and exclude the states in the
hole Fermi circles near Dirac points. Their quantity is ex-
ponentially small according to Eq. (6) and we neglect them.
Using Eq. (9) we get for the electron transition rates
from the left well to the right well
0 2= exp | | .
2
a
l
a
dN ES p dz
dt
−
−
σ π
∫
(12)
However besides this channel of the electron transitions
there is also a channel of the electron transitions from the
right well to the left well with the opposite direction of the
current
0 2= exp | | .
2
a
r
a
EdN S p dz
dt
−
−
σ π
∫
(13)
The full change of the electron number in the right well is
given by a sum / = / /t l rdN dt dN dt dN dt+ . In the ab-
sence of the external electric field and ( ) = ( )U z U z− there
is no current between the wells / = 0tdN dt . In the pres-
ence of the electric field the direction of the current coin-
cides with the direction of the electric field and therefore
=t ldN dN
dt dt
and we have to calculate the subbarrier current for a small
value of the electric field γ . The effective potential energy
is eff ( ) = ( ) | |U z U z e z− γ (see Fig. 2). The subbarrier im-
agine momentum at > 0z is given by the expression
0| |= 2 [ ( ) | | ]ep m U z e z E− γ − and we see that the elec-
tric field results in the decrease of the barrier height. Sup-
posing the symmetry at ( )z z→ − one can replace | |z z→
resulting in the decrease of the barrier height in the whole
domain of integration. At small electric field γ we obtain
in the linear approximation
0
0
| || | 2
| | 2 [ ( ) ] .
2 ( )
e
e
e z m
p m U z E
U z E
γ
− −
−
The appearance of | |z instead of z indicates only one
direction of the subbarrier current.
Substituting this expression in the Eq. (12) in linear on
γ approximation one gets
0
2 2exp | | exp 2 [ ( ) ]
a a
e
a a
p dz dz m U z E
− −
− − − ×
∫ ∫
1
1
1 0
| || | 211 .
( )
a
e
a
e z m
dz
U z E−
γ
× +
−
∫
The term without γ does not contribute to the current and
can be omitted. The turning points in the linear approxima-
tion coincide with a and a− in the absence of electric field.
Thus we get for the net current to the right well
0
0
2= exp 2 [ ( ) ]
2
a
t
e
a
dN ES dz m U z E
dt
−
− − ×
σ π
∫
1
1
1 0
| | 2 | |1 .
( )
a
e
a
e m z
dz
U z E−
γ
×
−∫
(14)
According to definition the coefficient in front of a
small electric field γ gives the conductance of the two-
layer heterostructure. As it is usual in linear response, we
do not calculate the nonlinear entropy production in the right
well. Let us analyze our final expression (14). Proportion-
ality to the sample area is confirmed by the experiment [9].
The important factor / = | | / (4 )S S e B cσ π which is large
in a strong magnetic field and may result in a large value
of the product
0
2exp 2 [ ( ) ] ,
a
e
a
S dz m U z E
−
− −
σ
∫
thus explaining the high currents observed in the experi-
ment [7]. The other factors are strongly dependent on the
semiclassical approximation used in our calculations. The
considered physical problem gives an example of the con-
ductance due to subbarrier currents.
The vortex model is the inevitable consequence of the
thermodynamic instability in our system and it is connect-
ed with the magnetization in a strong magnetic field. The
electric field arises due to the change of the magnetic field
and it induces electric current because the static magnetic
field itself can not produce the work on the electrons. Us-
ing of the vortex lattice model makes unnecessary the addi-
tional constructions like Chern–Simons field or the compo-
site fermions. In the recent works [10,11] authors use more
complicate models assuming exact electron-hole sym-
metry. Another difficulty is connected with the observed Fig. 2. The interwell potential modified by the electric.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1 13
S.V. Iordanski
electron spectrum which is far from the standard Landau
levels. In the vortex model the spectrum is closely con-
nected with the separation of the variables like in the case
of one-layer structures.
Author express his gratitude to E. Kats, I. Kolokolov,
and D. Khmelnitskii for the discussions. The work was
supported by grant RSc#14–12–00898.
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14 Low Temperature Physics/Fizika Nizkikh Temperatur, 2017, v. 43, No. 1
https://doi.org/10.1103/PhysRevLett.94.146804
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|
| id | nasplib_isofts_kiev_ua-123456789-129348 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T17:51:51Z |
| publishDate | 2017 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Iordanski, S.V. 2018-01-19T13:49:18Z 2018-01-19T13:49:18Z 2017 On the origin of transversal current in double-layer heterostructures / S.V. Iordanski // Физика низких температур. — 2016. — Т. 43, № 1. — С. 11-14. — Бібліогр.: 11 назв. — англ. 0132-6414 PACS: 73.21.–b, 67.10.–j https://nasplib.isofts.kiev.ua/handle/123456789/129348 It is shown that usually used theoretical model for double-layer heterostructures as a pseudospin ferromagnet does not explain the observed two-dimensional spectrum. Its existence is possible when neglecting Coulomb interaction destroying two-dimensional structures and can be realized only in a strong magnetic field. That is connected also with the plain vortex lattices forming in strong magnetic fields due to thermodynamic instability. This model gives reasonable explanations of various observed effects depending on the filling of the corresponding bands. In particular in this work we show that in double-layer heterostructures a large interlayer conductance really observed can exist. Author express his gratitude to E. Kats, I. Kolokolov,
 and D. Khmelnitskii for the discussions. The work was
 supported by grant RSc#14–12–00898. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур К 100-летию со дня рождения И.М. Лифшица On the origin of transversal current in double-layer heterostructures Article published earlier |
| spellingShingle | On the origin of transversal current in double-layer heterostructures Iordanski, S.V. К 100-летию со дня рождения И.М. Лифшица |
| title | On the origin of transversal current in double-layer heterostructures |
| title_full | On the origin of transversal current in double-layer heterostructures |
| title_fullStr | On the origin of transversal current in double-layer heterostructures |
| title_full_unstemmed | On the origin of transversal current in double-layer heterostructures |
| title_short | On the origin of transversal current in double-layer heterostructures |
| title_sort | on the origin of transversal current in double-layer heterostructures |
| topic | К 100-летию со дня рождения И.М. Лифшица |
| topic_facet | К 100-летию со дня рождения И.М. Лифшица |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/129348 |
| work_keys_str_mv | AT iordanskisv ontheoriginoftransversalcurrentindoublelayerheterostructures |