Transcript Sophia R. Figarova, Electron transport phenomena in

ELECTRON TRANSPORT PHENOMENA
IN LOW-DIMENSIONAL SYSTEMS
Baku State University
Baku, Azerbaijan
Sophia R. Figarova
2012
CONTENTS
• Introduction: a little of history.
• Different low-dimensional systems. Density of states .
• Basic transport phenomena in low-dimensional systems.
• Electron transport phenomena in superlattices in a
nonquantized magnetic field.
• Electron transport phenomena in superlattices in a quantized
magnetic field taking into account spin splitting.
• Conclusions
In solids, sizes of which are comparable with
 mean free path,
 de Broglie wavelength,
 coherence length,
 localization length,
there appear new physical properties, called size effects, caused
mainly by quantum effects.
Among these effects are
 oscillations of conductivity,
 quantum Hall effect,
 resonant tunneling,
 negative differential conductivity,
 giant resistance,
 spin Hall effects,
which can show itself in specially fabricated nanosystems (from 1 to 100nm), for example in

quantum films,

quantum wires,

quantum dots,

heterojunctions,

superlattices.

Not so many years passed since Leo Esaki was awarded to
the Nobel Prize for discovery of new effects in superlattices.
In fact, the work in Nanophysics has begun since the
Eighties of the XX century. The outstanding achievements
in this field were awarded to the Nobel prizes in Physics
A LITTLE OF HISTORY
Nobel prizes in nano - physics

1973 - The Nobel Prize in Physics 1973 was divided, one half
jointly to Leo Esaki and Ivar Giaever "for their experimental
discoveries regarding tunneling phenomena in effects in
quantum wells and superlattices semiconductors".

1985 - The Nobel Prize in Physics 1985 was awarded to Klaus
von Klitzing "for the discovery of the quantized Hall effect".

1986 - The Nobel Prize in Physics 1986 was divided, one half
awarded to Ernst Ruska "for his fundamental work in electron
optics, and for the design of the first electron microscope", the
other half jointly to Gerd Binnig and Heinrich Rohrer "for their
design of the scanning tunneling microscope".

1998 - The Nobel Prize in Physics 1998 was awarded jointly to
Robert B. Laughlin, Horst L. Störmer and Daniel C. Tsui "for
their discovery of a new form of quantum fluid with fractionally
charged excitations".

2000 - The Nobel Prize in Physics 2000 was awarded "for basic
work on information and communication technology" with one
half jointly to Zhores I. Alferov and Herbert Kroemer "for
developing semiconductor heterostructures used in high-speedand opto-electronics" and the other half to Jack S. Kilby "for his
part in the invention of the integrated circuit".

2007 - The Nobel Prize in Physics 2007 was awarded jointly to
Albert Fert and Peter Grünberg "for the discovery of Giant
Magnetoresistance“.

2010 – The Nobel Prize in Physics 2010 was awarded jointly to
Andre Geim and Konstantin Novoselov "for groundbreaking
experiments regarding the two-dimensional material graphene“.



Experimental observation of size effects is possible thanks to development of
new technological methods such as molecular beam epitaxy.
However, in far 1959 at annual meeting of the American Physical Society
theorist Richard Feynman has given well-known, legendary lecture entitled “
There is Plenty of Room at the Bottom: Invitation to enter a new field of
physics”, where he predicted Nano-Physics.
In March of 1959, Richard Feynman challenged his listeners to build
“Computers with wires no wider than 100 atoms, a microscope that could view
individual atoms, machines that could manipulate atoms 1 by 1.”This
assumption has been realized in the creation of scanning electron
microscopes, which allows one to study microscopic objects and purposefully
manipulate these objects.
Over the past two decades in Physics of low-dimensional
systems a number of great discoveries was made. Let us name
major of them: weak and strong localization of guantum
states; quantization of conductivity in ballistic transport;
Coulomb blockade of tunneling in nanostructures; spin Hall
effect.
The behavior of charge carriers in low-dimensional
systems is determined by the following basic phenomena:

Quantum confinement

Ballistic transport of charge carriers

Tunneling of charge carriers

Spin effects
THREE DIMENSIONAL (3D) SYSTEMS
The energy:
The density of states therewith is proportional to the square
root of energy:
Bulk samples, their energy diagrams and density of states
QUANTUM CONFINEMENT
Quantum confinement arises if the free motion of electrons in
one of the directions becomes confined by potential barriers.Transport of electrons can realize parallel and perpendicular to
the potential barriers. In the case of the motion of carriers
along the barriers ballistic transport is dominant effects. The
passage of carriers through the barriers takes place via
tunneling.
Size quantum effect in infinitely
deep potential well. The wave function is standing ones and equal to:
or
The energy spectrum is
quantized and has the form:
Potential well and electron
wave functions in it
The confinement of the motion of charge carriers, leading
to nonzero minimum energy and discreteness of energy
spect-rum, is called the quantum confinement.
TWO-DIMENSIONAL (2D) SYSTEMS
Quantum films – d ~ are two-dimensional (2D) structures, in
which quantum confinement acts only in one direction-the
direction perpendicular to the film (the z-direction). The film
thickness is of the order of de Broglie wave length. Charge
carriers can freely move in the xy - plane. Their energy equals:
In the k-space, an energy diagram of a quantum film represents
a family of
parabolic bands, which, overlapping, form
subbands. The dependence of the electron density of states on
energy in a quantum film has a step-like character:
Quantum films, their energy diagrams and density of states
Density of state as a function of film thickness
ONE DIMENSIONAL (1D) SYSTEMS
Quantum wires are one-dimensional (1D) structures. Charge
carriers can freely move only in one direction-along the wire axis.
The energy spectrum in this case has the form:
The density of states in a quantum wire is inversely proportional
to the square root of energy -1/2:
Quantum wires, their energy diagrams and density of states
ZERO DIMENSIONAL (0D) SYSTEMS
Quantum dots are (0D) structures, in which the motion of charge carriers is
confined in all three directions. In each of these directions electron energy
turns out to be quantized in accordance with the formula:
The density of states represents a set of sharp peaks:
Quantum dots, their energy diagrams and density of states
Quantum films in a quantized magnetic field – If a quantized magnetic field is
directed along the z-axis of a size-quantized film, the energy spectrum
becomes completely discrete:
SUPERLATTICES
Superlattices are solid periodic structures in which apart with the usual
potential of the lattice, there is additional potential. The semiconductor
superlattice is made of two layers of semiconductors with different band gap ,
which thickness is a few nm. In this case the superlattice can be considered as
a periodic system of quantum wells, separated by narrow barrier. Additional
periodicity leads to the fact that the energy spectrum component, connected
with the motion of an electron along this axis, represents a system of narrow
strips-minibands. Such systems have the very strong anisotropy of the energy
spectrum, at which the motion of electrons in the layer plane is free and is
described in the effective mass approximation. However in the direction
perpendicular to the layer plane, the motion of electrons is strongly hindered
and is described by strong coupling approximation. An electron gas in this
direction can be described by a cosinusoidal dispersion law. In a whole, the
energy spectrum of charge carriers can be written as:
where is i the miniband number, i is its width.
In this case the effective mass in direction perpendicular to the layer plane can
change sign for electrons with the wave vector kz=  / 2d :
Superlattice energy diagrams with minibands  = a,b
The density of states in a superlattice has the following form:
(- z) is a Heaviside function. It is seen that the density of state has a steplike character and if the energy of electron is large than mini-band width the
density of state does not depend on energy, the caracteristic for twodimensional systems.
Density of states in a superlattice vs. energy
Dichalcogenides of transition metals of the NbSe2, TaS2 type, the III-V
layered semiconductor compounds, intercalated compounds (synthetic
metals) can be considered as natural superlattices and the electron gas in
these systems are described by a cosinusoidal dispersion law. For narrow
minibands wave functions of electrons along axis of superlattices are overlapped and
electron spectrum consists of discrete levels. The electron gas behaves itself as a twodimebsional gas.
Semiconductor heterostructure AlGaAs
Conduction band around a heterostructure between n-AlGaAs and undoped GaAs.
Electrons are separated from their donors to form a two-dimensional electron gas
The peculiarity of transport phenomena in superlattices is the fact that there
takes place longitudinal and transverse transport of charge carriers. In
additional depending on the degree of filling of the mini-cand the electron gas
can be either two-dimensional or quasi-two-dimensional.
Basic type of ideal nanostructures
DOS in Low-Dimensional Electron Systems
We in detail consider the density of states in low dimensional
structures since its behavior essentially influences many
physical characteristics. Moreover, there are physical quantities
which are directly proportional to the density of states and their
behavior is completely dictated by it, e.g. Entropy, Heat
capacity, Thermopower and Magnetization. As an example,
consider a superlattice in a quantized magnetic field.
Entropy and electron heat capacity. In the case of a degenerate
electron gas, these quantities depend on the density of states at
the Fermi surface and have the form:
2.5
30
2.0
S2D ∕20
S3D
S2D 1.5
∕
S3D
1.0
10
0.5
0.0
0
2
4
6
8
10
12
B (T)
Fig.1. Ratio of entropy of a two-dimensional
degenerate electron gas to entropy of a threedimensional gas vs. a magnetic field.
0
1
2
3
4
Z0
Fig.2. Ratio of entropy of a two-dimensional
degenerate electron gas to entropy of a threedimensional gas vs. the degree of band filling.
The parameters a = 10nm, m = 0.067m0, 0= 0.1meV for the GaxAl1-xAs superlattices are used.
From the Figures it is seen that depending on the magnetic field, entropy
oscillates and in the quantum limit, the entropy of a two-dimensional electron
gas S becomes larger than the entropy of a three-dimensional electron gas:
This fact, apparently, is connected by the fact that the radius of a cyclotron
orbit in a two-dimensional case is larger (since m>m ). Therefore cyclotron
orbits are crossed and the confusion in a two dimensional gas becomes larger
[Askerov B.M., S.R.Figarova S.R., Mahmudov M.M., Figarov V.R., Jap. J.
Appl. Phys., 50, 05FE10, 2011].
Thermo-electrical power is determined by entropy from the formula  = - S/en, the
behaviour of the thermo electrical power is completely determined by the density of
states .
Thermo-electrical power of quantum films and superlattices oscillates in a strong
magnetic field [B.M.Askerov, S.R.Figarova, V.R.Figarov, Nanotechnology, 18, 424024,
2007]. Such a behaviour of thermo-electrical power was experimentally confirmed in
superlattices of the GaAlAs type at low temperatures [W. Zawadzki, Physica B+C, 127,
388, 1984].
Magnetization in the case of a degenerate electron gas is directly proportional to the
density of states, too.
600
50
400
M
(Z0)
,(А/
m)
M(B)( A/m)
25
0
200
0
25
200
50
1
1.5
2
2.5
3
3.5
Z0
Fig. 1.6. Diamagnetic magnetization of a degenerate quasi-twodimensional electron gas versus the degree of band filling at the
following parameters: 0 = 1meV, a = 10nm, n = 1023m-3.
400
0
1
2
3
4
5
6
B(T)
Fig.1.7. Diamagnetic magnetization of a degenerate quasi-twodimensional electron gas versus the magnetic field at the
following parameters: 0 = 1meV, a = 10nm, n = 1023m-3.
From the Figures it is seen that diamagnetic magnetization
of a quasi-two-dimensional electron gas depending on the degree
of band filling changes the sign and in the two-dimensional case
becomes positive. In a magnetic field magnetization oscillates.
Such a behaviour of magnetization in superlattices is
explained by existence of a negative-effective-mass region in the
mini-band. Therewith a conduction electron moves in the
direction opposite to the free electron motion. In a magnetic field
the conduction electron rotates in the opposite direction; this fact
leads to the positive magnetization [B.M.Askerov, S.R.Figarova,
M.M.Mahmudov, V.R.Figarov, Proc. Royal Soc. A, 464, 3213, 2008].
The sign change of magnetization in superlattices was observed
in the experimental work [S.D.Prado, M.A.de Aguiar, Phys. Rev.,
E, 54, 1369, 1996]. The sign change essentially influences optical
phenomena.
BALLISTIC TRANSPORT OF CHARGE CARRIERS
In low-dimensional structures the size of which is less than the mean free
path, transport of charge carriers occurs without the carrier scattering.
Such a transport is called the ballistic transport. Main effects, related to
ballistic transport, are determined by the ratio between structure sizes and
mean free path at the elastic and the inelastic scattering of carriers, the
phase coherence length and Fermi wavelength.
.
If the structure size is comparable with these lengths, quantization of energy
becomes essential. The ideal ballistic transport of charge carriers in
nanostructures is characterized by the universal ballistic conductance, which
is independent of the material type and is determined only by fundamental
constants. The conductivity is quantized in terms of 2e2/h. This fact is
observed at quantum point contacts. In the conductivity curve there appear
steps (see Fig). As the electron motion becomes coherent, its wave function
conserves its phase. Therewith various interference effects arise.
Conductance at ballistic transport: a - scheme of quantum point contacts,
conductance.
b -
TUNNELING OF CHARGE CARRIERS IN
LOW-DIMENSIONAL SYSTEMS
Tunneling means transport of particles through the region,
confined by a potential barrier, the height of which is larger
than the total energy of the given particles. Such an effect is
impossible from the point of view of classical mechanics,
however it takes place for quantum particles. The interaction of
quantum particles with various potential barriers was illustrated
in the Figure.
Existence of the wave having
passed through the barrier,
corresponding to a quantum
particle with energy that less
than the barrier height is called
the tunneling effect.
In low-dimensional structures tunneling has specific features, the fact which
distinguishes it from effects in bulk systems. One of these features is connected
with the discrete nature of the charge carriers and is called “single-electron
tunneling” [Tinkham Am.J.Phys., 1996, N 64, p.343].
Current–voltage characteristic
(CVC) at Coulomb blockade
Another feature, determined by discreteness of energy states of
charge carriers in semiconductor nano-structures is called resonant
tunneling. Resonant tunneling occurs if the following conditions are
satisfied: 1. de Broglie wavelength should be comparable with the
width of the quantum well, 2. the free mean path should be large
and the electron is scattered specularly at the edges of the quantum
well, 3. electron energy should coincide with the energy of quantum
levels in the well. In the current-voltage characteristics (CVC) there
appears the region with a negative differential resistance, which
promotes to the generation of energy and leads to light
amplification. An increase in the tunneling current occurs if the
Fermi level coincides with the discrete level of the quantum well.
The CVC for the resonant tunneling is shown in the figure below.
[Chang L.L., Esaki L., Tsu R., Appl. Phys. Lett., 1974, v.24, p.593]
SPIN EFFECTS
Taking into account spin of charge carrier in low-dimensional
structures leads to new features of transport. The spin effects
in low-dimensional systems manifest itself through Hall effect
and magnetoresistance. In nonmagnetic materials spin effects
represent spin splitting of energy levels in a magnetic field and
Rashba spin-orbital splitting. In these cases splitting of energy
levels schematically has the shape:
In the Hamiltonian additional terms appear:
here g is the spin splitting factor,  = e/2m is the Bohr magneton,
 = 1/2 is the electron spin quantum number, В is the magnetic field
induction. . Each Landau level is split into two sublevels. The magnitude of g
-factor depends on the band gap width g, spin-orbital interaction , number
of the Landau level and the magnetic field magnitude.
here =3x10-11eVm is the Rashba constant, p is the impulse operator in the
confinement direction. In the energy spectrum, there appears a term
proportional to the wave vector.
For a two-dimensional electron gas, placed in a perpendicular magnetic
field, the energy spectrum with regard to these two mechanisms has the
form :
here
Spin effects in magnetic materials arise if there is the spin misbalance of
population of the Fermi level. Such a misbalance is presented in
ferromagnetic materials, where the densities of vacant states for electrons
with different spins are identical, however these states are distinguished by
energy, as it is schematically shown in Fig.

Two main transport effects, namely giant magnetoresistance and
tunneling magnetoresistance are connected with electron spin in lowdimensional systems.

Besides in low-dimensional systems owing to spin-orbital interaction of
the Rashba type, Hall spin effect occurs.
The spin-Hall effect

It turns out that the Rashba Hamiltonian gives rise to a
pure transverse spin current in response to a charge current

2DEG

The associated spin-Hall conductivity has a universal value
in the 2D plane, and is of much interest to spintronics.
Transistors
SOME DEVICES BASED ON CONSIDERED
EFFECTS IN LOW-DIMENSIONAL SYSTEMS

MOS (metal-oxidesemiconductor) field
transistors are based on
quantum confinement.


Resonant-tunnel diodes are
based on resonant tunneling
Quantum interference transistors
are based on interference of
electron waves and ballistic
transport of current carriers.
ELECTRON TRANSPORT PHENOMENA IN SUPERLATTICES
IN A NONQUANTIZED MAGNETIC FIELD




In superlattices, anisotropy of the structure, the energy spectrum of
conduction electrons and the scattering mechanisms lead to
fundamentally new phenomena. For example:
resistance oscillations depending of the orientation of the magnetic
field
negative magnetoresistance, if a magnetic field is situated in the layer
plane [A.A.Bykov, G.M.Gusev, J.R.Leite, A.K.Bakarov, A.V.Goran,
V.M.Kudryashev, A.I.Toropov. Phys. Rev. B, 65, 035302, 2001] and
perpendicular to it [N.M.Sotomayor G.M.Gusev, J.R.Leite, A.A.Bykov,
A.K.Kalagin, V.M.Kudryashev, A.I.Toropov, Phys. Rev. B, 70, 235326,
2004].
Such a behaviour of magnetoresistance was usually associated with
with strong scattering of electrons by inhomogeneities. However, as we
demonstrate, main causes of these special effects in superlattices are
form of the energy spectrum and quantum confinement.
To construct a theory of electron transport phenomena first we should
determine the relaxation time in superlattices.
Relaxation time at scattering of current carriers by
phonons and impurity ions
For a cosinusoidal energy spectrum:
we calculated the relaxation times at scattering by different types of phonons and
impurity ions. At scattering by different types of phonons, the relaxation time can be
generalized and written as [B.M.Askerov, B.I.Kuliev, S.R.Figarova, I.R.Gadirova, J.
Phys.: Cond. Matt., 7, 843, 1995]:
where =(, II) are transverse and longitudinal components of the relaxation time, k
are transverse and longitudinal components of the wave vector.
At low temperatures the current carrier scattering by impurity ions is one of dominant
scattering mechanisms. At the weak screening the Coulomb potential of impurity ions
for components of the inverse relaxation time tensor we have [B.M.Askerov,
G.I.Guseynov, V.R.Figarov, S.R.Figarova, Physics of the Solid State, 50, 780, 2008]:
In the case of the strong screening we have:
Galvanomagnetic effects in superlattices
Due to the strong anisotropy of the energy spectrum in quasi-twodimensional systems, the character of the motion of charge carriers parallel
and perpendicular to the layers is essentially distinct. An external magnetic
field binds the current carrier motion in the layer plane and in the direction
perpendicular to it. In connection with this fact, transport phenomena can
be divided into two classes: longitudinal and transverse ones. The
magnetoresistance and the Hall coefficient strongly depend on orientation
of the magnetic field. We consider a two direction of the magnetic field:
a)the magnetic field is perpendicular to the layer plane, b) the magnetic field
is in the layer plane.
Geometry of the problem
For the first geometry of the problem, Hall coefficient R and
specific resistance in the layer plane  are expressed through the
components of the galvanomagnetic tensor as follows:
For the second geometry of the problem, Hall coefficient RII
(Ez=RjxB) and specific resistance  in a magnetic field situated
along the layer plane are determined by the formulas:
where components of the electric conductivity tensor ik are given
by the following expressions:
if B II oz
where
and the angle brackets denote
if B  oz
where
and the angle brackets denote
Separately consider cases of a two-dimensional gas, if the Fermi surface is open-the
corrugated cylinder and a quasi-two-dimensional gas with the closed Fermi surface of
a shape like a rugby ball
Hall coefficient
Hall coefficient of a two-dimensional electron gas (the open Fermi surface) is
determined only by the effective concentration of charge carriers with the formula:
where the plus sign corresponds to the case when the magnetic field is perpendicular
to the layer plane of a superlattice, and the minus sign does when it is parallel to the
layer plane. The concentration, entering in the expression of Hall coefficient is not the
i.е. Hall
full one, but it is the effective concentration of current carriers:
coefficient in the two-dimensional case depends only on parameters of the
superlattice.
Depending on the geometry of the problem Hall coefficient changes its sign. When a
magnetic field is situated in the layer plane, Hall coefficient of a two-dimensional
electron gas is positive. From the dependence of Hall coefficient of a quasi-twodimensional electron gas (the closed Fermi surface) on the degree of band filling (Fig.)
it is seen that Hall coefficient of a quasi-two-dimensional electron gas can be both
positive and negative . Such a behaviour of Hall coefficient is connected with the sign
of the effective mass [Figarova S.R., Figarov V.R. , Phil. Mag. Lett., 2007, v.87, p.373378]. The positive sign of Hall coefficient in quasi-two-dimensional electron systems
with the cosinusoidal dispersion law is due to the existence in the electron miniband a
region with the negative effective mass. To the existence of regions of the negative
effective mass in the superlattice miniband, it was pointed out in the work
[Yu.A.Romanov, Physics of the Solid State 45, pp. 559–565 (2003)]. Singularity in the
behavior of Hall coefficient takes place at z = /2.
Transverse magnetoresistance (TMR) in a
perpendicular magnetic field, B II oz
Magnetoresistance is change in resistance in a magnetic field. Positive magnetoresistance corresponds to an increase in resistance, and negative one does to its
decrease. When the direction of an external magnetic field is perpendicular to the
current the magnetoresistance is called transverse magnetoresistance.
TMR of a quasi-two-dimensional electron gas vs. the degree
of band filling Z0 in a strong magnetic field perpendicular
to the layer plane at phonon scattering
(B)/(0) vs. the ratio between the miniband width and
Fermi level k = 20/ in a strong magnetic field at
impurity ion scattering. The two-dimensional electron gas
From the Figures it follows that for a magnetic field directed perpendicular to the layer
plane in a strong magnetic field TMR of a two-dimensional electron gas is negative
for scattering by phonons and impurity ions. Whereas for the quasi-two-dimensional
case depending on the scattering mechanism and the degree of band filling, TMR
can become positive and negative. In a weak magnetic field TMR is positive.
TMR of a quasi-two-dimensional electron gas vs. the degree of
band filling Z0 in a strong magnetic field perpendicular to the
layer plane at impurity ions (r0 /a=5). Quasi-two-dimensional
electron gas.
TMR of a quasi-two-dimensional electron gas vs. the
degree of band filling Z0 in a weak magnetic field
perpendicular to the layer plane at impurity ions (r0 /a=5).
Quasi-two-dimensional electron gas.
Transverse magnetoresistance in a magnetic field
situated in the layer plane, B II oy
With such a geometry of the problem, in contrast to preceding, in a strong
magnetic field, TMR of a two-dimensional electron gas is positive, and in a
weak one, TMR is negative. [S.R.Figarova, V.R.Figarov, Euro Phys. Lett.,
89, 37004, 2010].
TMR vs. the magnetic field parallel to the layer
plane 0 for a quasi-two-dimensional electron gas.
From the Figures it is seen that transverse
magnetoresistance of a quasi-two-dimensional
electron gas in the magnetic field parallel to the
layer plane is negative in the weak field and is
positive in the strong field. Therefore changing
the direction and magnitude of the magnetic
field, one can change the TMR sign. The fact
that TMR is positive in a strong magnetic field
parallel to the layer plane is explained by the
fact that conduction electrons moving in the z direction become localized, and resistance
grows.
TMR vs. the degree of band filling Z0 in the weak
magnetic field parallel to the layer plane (0 =0.1)
for quasi-two-dimensional electron gas.
TMR vs. the degree of band filling Z0 in the
strong magnetic field parallel to the layer plane
(0 =6) for quasi-two-dimensional electron gas.
Negative TMR in superlattices was experimentally revealed in the work [D.N.Bose, S.Pal,
Phys. Rev. B 63, 235321, (2001)] where negative magnetoresistance in the magnetic field in
the layer plane was observed at 10K for fields of 0.4T in GaTe layered semiconductors.
Electron effective mass and dynamics in the superlattice
Peculiarities in the behaviour of galvanomagnetic phenomena in superlattices are
connected with the effective mass in the direction perpendicular to the layer plane
(which can takes negatives values):
and the electron dynamics in the superlattice (see Fig.). We have the Bloch
oscillations, whose frequency lies in the terahertz range.
When the electric field is applied, an electron begins
to accelerate in the field direction. If the crystal is
ideal (there are no defects), then action of the force,
the quasi-impulse begins to grow until it does not
reach kx=/a. Note that the electron effective mass
becomes negative as it approaches the value of kx=/a
. This means that in the coordinate space an electron,
going from the О point, at first accelerates, then slows
down, when it approaches the А point and finally
again begin accelerating, but only in the opposite
direction (moving to the point В), although the
direction and magnitude of the external force
conserve invariable. At kx=0 the electron again turns to
be at rest. Therefore, under influence of an external
field, the electron executes a jump-like motion along
the kx -axis and vibrate in the confined section of the
Motion of an electron in an electrical field:
Х-axis in the coordinate space with the amplitude
in the k - space (a), in the coordinate space (b). A=/2eE and frequency =eEa/h ( is the energy
band width, a is the lattice period).
ELECTRON TRANSPORT PHENOMENA IN
SUPERLATTICES IN A QUANTIZED MAGNETIC FIELD
TAKING INTO ACCOUNT SPIN SPLITTING
A strong magnetic field, perpendicular to the two dimensional layer, quantizes the
motion of current carriers in the layer plane and leads to the following experimentally
observed effects:
semimetalic-to-semiconductor transition in a superlattice in a quantized magnetic
field [N.J.Kawai, L.L.Chang, G.A.Sai-Halasz, C.A.Chang, L.Esaki, Appl. Phys. Lett., 36,
369, 1980].
conductivity oscillations. The oscillation period is determined by the magnetic length
and superlattice constant perpendicular to the layer plane, [B.Laikhtman, D.Menashe,
Phys. Rev. B, 52, 8974, 1995].
unusually sharp growth of resistance with increasing the magnetic field in alternating
layers of GaAs and AlGaAs [V.Renard, Z.D.Kvon, G.M.Gusev, J.C.Portal,, Phys. Rev. B,
70, 033303, 2004] and in the semiconductor layer of InSb [S.A.Solin, D.R.Hines,
A.C.H.Rowe, J.S.Tsai, Yu.A.Pashkin, S.J.Chung, N.Goel, M.B.Santos, Appl. Phys. Lett.,
80, 4012, 2002].
existence of a region of negative differential conductivity in the superlattice at room
temperature (see, e.g. [Estibals O., Kvon Z.D., Gusev G.M., Arnaud G., Portal J.C.
Physica E, 2004, v.22, 446, 2004-449]). Negative differential conductivity in a solid is
caused by the negative effective mass and Bloch oscillations.
 existence of the vertical magnetoresistance in a magnetic field, directed
perpendicular to the layer [Yu.A.Pusep, G.M.Gusev, A.J.Chiquito, S.S.Sokolov,
A.K.Bakarov, A.A.Toropov, J.R.Leite, Phys. Rev. B, 63, 165307, 2001]
 linear growth of magnetoresistance in a superlattice with the magnetic
field, so-called the Kapitsa effect [P.V. Gorskii, Semiconductors, 38, 830,
2004].
 in a strong magnetic field of the order of 30 Т there are observed maxima
and minima in magnetoresistance in GaAs/AlGaAs structure [M.V. Vakunin,
G.A. Al`shanskii, Yu.G. Arapov, V.N. Neverov, G.L. Kharus, N.G.
Shelushinina, B.N. Zvonkov, E.A. Uskova, A.deVisser, L.Ponomarenko,
Semiconductors, 39, 107, 2005]. This fact is explained by change in the density
of states at the Fermi level owing to spin splitting of energy.
In low-dimensional systems, placed in a magnetic field, spin effects appear.
Their main characteristic is magnetoresistance. Because of this fact, we
theoretically studied resistance in ideal superlattices with the cosinusoidal dispersion
law in a strong magnetic field taking into account spin splitting.
As known, transport phenomena are closely connected with the density of states.
Therefore at first consider the density of states of a quasi-two-dimensional electron
gas taking into account spin splitting in a quantized magnetic field.
Density of states in a quantized magnetic field
A strong magnetic field, parallel to the z-axis, quantizes the electron
motion in the layer plane and removes the spin degeneracy; the energy
spectrum has the form:
where g* is the factor of spin splitting of the electron energy. In the
energy spectrum, an additional term, connected with spin splitting,
appears. Each Landau level is split into two spin sublevels. The
density of states has the form:
where
is the magnetic length.
From the Formula it is seen that the density of states has a singularity. The density of
states significantly depends on the ratio between the Fermi level and miniband width.
In a two-dimensional electron gas (>20) there are oscillations of the density of states,
characteristic for two-dimensional electron systems in a strong magnetic field [T.Ando,
A.B. Fowler, F.Stern, Review of Modern Physics, 54, 437 (1982)], which vanish in the
quasi-two-dimensional case (<20). Besides, from Figure1 it is seen that spin splitting
significantly influences the density of states and at large values of the g*- factor, the
density of states linearly depends on the magnetic field.
Density of states as a function of the magnetic field. The solid line corresponds to no spin
splitting, the dashed line does to g*=5, the dotted does to g*=2. a -  >20, b -  <20.
Vertical longitudinal magnetoresistance in a quantized
magnetic field taking into account spin splitting
If the direction of an external magnetic field B and current j coincide
and they are directed along the z-axis, i.e., B II j II oz magnetoresistance is
called vertical longitudinal magnetoresistance. The electron motion
quantization in the magnetic field, leads to longitudinal magnetoresistance,
due to the fact that in a quantized magnetic field the probability of the
current carrier scattering and the Fermi level substantially depends on the
magnetic field.
In a quantized magnetic field, the relaxation time at the scattering by
acoustic phonons is inversely proportional to the density of states of
electrons in a magnetic field:
Taking this fact into account for electrical conductivity zz= II , we have:
From this expression it is seen that magnetoresistance takes an infinitely large
value, if the condition
is satisfied.
Stormer and others in the work [H.L.Störmer, J.P.Eisenstein, A.C.Gossard,
W.Wiegmann, K.Baldwin, Phys. Rev. B, 56, 85 1986] experimentally observed turning
into zero of II(B) at specific values of the magnetic field.
Ratio II (B)/II(0) vs. the magnetic fields when  > 20 and m=0.2m0 (the solid line), m=0.3m0 (the dashed line) taking into
account spin splitting (a), without spin splitting (b) for the following parameters:
From the Figure it is seen that resistance strongly oscillates in the magnetic field if
the Fermi level becomes larger than the miniband width (a two-dimensional electron
gas). These oscillations weaken with decrease in the effective mass along the layer.
Spin splitting leads to the oscillation period decrease. For a quasi-two-dimensional
electron gas the oscillations of resistance become weaker an in an ultra- strong
magnetic field vanish.
Existence of peaks of resistance, to all appearance, is associated
with metal-insulator transitions.
Superlattices are distinct from two-dimensional systems by conductivity along
of the magnetic field II.
In the quantum limit for a two dimensional electron gas (>20, and
also B >20) longitudinal magnetoresistance has two peaks, positions of
which are determined from the transcendental equation:
If B>Bk, where Bk=na/e, the peak-like behaviour of magnetoresistance is
observed. For the used values of the concentration and lattice constant in the
z-direction, the magnetic field induction Bk equal to 2Т.
From dependence of magnetoresistance on the magnetic field (see Fig.), it is seen
that longitudinal magnetoresistance can be positive and negative. In superlattices the
negative magnetoresistance is associated with spin splitting of electron levels.
The negative longitudinal magnetoresistance can be explained as follows:
Spin splitting changes the relation between
the Fermi level, Landau level and
miniband width, and the density of states
decreases, and therefore, resistance
decreases since  --1  g() [S.R.Figarova,
Phys. Stat. Sol. (b), 243, R41, 2006].
Longitudinal magnetoresistance of a quasi-two-dimensional
electron gas vs. the magnetic field taking into account
spin splitting.
For magnetoresistance in the case of
(na2R2<<1), we have:
ultra-strong magnetic fields
It is seen that in the limiting case specific resistance linearly depends on the
magnetic field. Therefore under specific conditions, it is possible to observe
Kapitsa effect.
Thus, experimentally observed effects, namely the oscillations of conductivity,
unusually sharp growth of resistance and metal-insulator transition are
explained by quantum confinement. However the negative resistance and
linearly growth of resistance takes place only at expense of spin-splitting.
CONCLUSIONS
Thus the behaviour of charge carriers in low-dimensional
structures is determined by the following basic phenomena:
quantum confinement, ballistic transport, tunneling and spin
effects. In regard to quantum confinement and spin effects in
superlattices with a cosinusoidal dispersion law in a magnetic
field we draw the next conclusions:






In superlattices, longitudinal and transverse transport take place. In
semiconductor superlattices minima of resistance in the layer plane
are accompanied by similar minima of conductivity perpendicular to
the layer plane.
In a nonquantized magnetic field, directed along the layer plane, Hall
coefficient at electron conductivity in the two-dimensional case is
positive.
Transverse magnetoresistance of superlattices is positive in the layer
plane and is negative across the layer plane in a nonquantized
magnetic field. Changing the magnitude and direction of the
magnetic field, one can essentially vary resistance.
In a quantized magnetic field vertical longitudinal resistance has the
following features: oscillates in a magnetic field . metal-insulator
transition occurs, linearly grows in an ultrastrong magnetic field,
magnetoresistance can be negative and positive. The latter two
features take place if spin splitting is taken into account.
Peculiarities of transport phenomena in superlattices are connected
with dynamics of electrons in the superlattice, Bloch oscillations,
negative effective mass regions in the miniband and spin splitting.