Electron tunneling in a multibarrier potential
The electron and spin transport in nanoscopical heterostructures taking is considered taking into account features of electron spectra. It is shown efficiency of effective mass methods for an quantum-mechanical description of electron tunneling through potential barriers of the system. It is show...
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Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
2009
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irk-123456789-208772013-02-13T02:06:39Z Electron tunneling in a multibarrier potential Korostil, A. Korostil, Yu. The electron and spin transport in nanoscopical heterostructures taking is considered taking into account features of electron spectra. It is shown efficiency of effective mass methods for an quantum-mechanical description of electron tunneling through potential barriers of the system. It is shown the description of coherent electron transport in multibarrier electron potentials. 2009 Article Electron tunneling in a multibarrier potential / A. Koroslit, Yu. Korostil // Моделювання та інформаційні технології: Зб. наук. пр. — К.: ІПМЕ ім. Г.Є.Пухова НАН України, 2009. — Вип. 50. — С. 106-120. — Бібліогр.: 11 назв. — анг. XXXX-0068 http://dspace.nbuv.gov.ua/handle/123456789/20877 72.25.,72.25 en Моделювання та інформаційні технології Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
The electron and spin transport in nanoscopical heterostructures taking is
considered taking into account features of electron spectra. It is shown efficiency of
effective mass methods for an quantum-mechanical description of electron tunneling
through potential barriers of the system. It is shown the description of coherent
electron transport in multibarrier electron potentials. |
| format |
Article |
| author |
Korostil, A. Korostil, Yu. |
| spellingShingle |
Korostil, A. Korostil, Yu. Electron tunneling in a multibarrier potential Моделювання та інформаційні технології |
| author_facet |
Korostil, A. Korostil, Yu. |
| author_sort |
Korostil, A. |
| title |
Electron tunneling in a multibarrier potential |
| title_short |
Electron tunneling in a multibarrier potential |
| title_full |
Electron tunneling in a multibarrier potential |
| title_fullStr |
Electron tunneling in a multibarrier potential |
| title_full_unstemmed |
Electron tunneling in a multibarrier potential |
| title_sort |
electron tunneling in a multibarrier potential |
| publisher |
Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
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2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/20877 |
| citation_txt |
Electron tunneling in a multibarrier potential / A. Koroslit, Yu. Korostil // Моделювання та інформаційні технології: Зб. наук. пр. — К.: ІПМЕ ім. Г.Є.Пухова НАН України, 2009. — Вип. 50. — С. 106-120. — Бібліогр.: 11 назв. — анг. |
| series |
Моделювання та інформаційні технології |
| work_keys_str_mv |
AT korostila electrontunnelinginamultibarrierpotential AT korostilyu electrontunnelinginamultibarrierpotential |
| first_indexed |
2025-07-02T21:26:39Z |
| last_indexed |
2025-07-02T21:26:39Z |
| _version_ |
1836572056305532928 |
| fulltext |
106 © А. Korostil, Yu. Korostil
обобщенная схема ее жизненного цикла, обозначены некоторые проблемы.
В дальнейших своих работах мы намерены рассмотреть и определить
различия между гипотезами, предвидением и прогнозами. В частности,
значительный интерес, в этом отношении, представляет анализ модели
«ПАМЯТЬ - ПРЕДСКАЗАНИЕ» Джеффа Хокинса [5], т. к. он, с одной
стороны, претендует на создание модели мозга, с другой стороны, его
исследования проводятся с целью разработки систем искусственного
интеллекта.
1. Валькман Ю. Р., Быков В. С. Дедуктивные и недедуктивные аспекты в
моделировании образного мышления. // Моделювання та інформаційні технології. Зб.
наук. пр. ІПМЕ, Київ, 2006, Вип. 35, - с. 87 - 96.
2. Валькман Ю. Р., Дембовский О. Ю. Процессы порождения и обоснования гипотез:
индукция, аналогия, абдукция и дедукция. // Моделювання та інформаційні
технології. Зб. наук. пр. ІПМЕ, Київ, 2008, Вип. 46, - с. 12 - 22.
3. Роpper K.R. (1963) Conjectures and Refutations: The Growth of Scientific Knowledge. -
Routledge, London (переводы: Поппер К.Р. (2004) Предположения и опровержения.
Рост научного знания. - М.: АСТ, Ермак. - 638с.; Поппер К.Р. (1983) Логика и рост
научного знания. - М.: Прогресс)
4. Непейвода Н. Н. Прикладная логика. Новосибирск: Изд. Новосиб. ун-та – 2000 –
521 с.
5. Блейксли С., Хокинс Д. Об интеллекте. Москва: Изд. дом «Вильямс» – 2007- 240 с.
Поступила 5.02.2009р.
УДК 72.25., 72.25.
А. Korostil, Yu. Korostil, Kyiv
ELECTRON TUNNELING IN A MULTIBARRIER POTENTIAL
The electron and spin transport in nanoscopical heterostructures taking is
considered taking into account features of electron spectra. It is shown efficiency of
effective mass methods for an quantum-mechanical description of electron tunneling
through potential barriers of the system. It is shown the description of coherent
electron transport in multibarrier electron potentials.
1. Introduction
The quantum phenomena of tunneling refers to the possibility that quantum
particles can traverse regions, which are from a classical point of view
energetically forbidden. Tunneling is an intimate consequence of the wave
properties of matter and the probabilistic interpretation of the wave function.
Quantum tunneling was already considered from the early days of quantum
107
mechanics in connection with the problem of field ionization of atoms and the
nuclear decay of alpha particles. The concept of tunneling was firstly applied in
solid state physics to explain the field emission of electrons from metals into
vacuum [1,2]. Single barrier tunneling has found widespread applications and one
of the most prominent is the invention of the scanning tunneling microscope
(STM), in which particles tunnel through a controllable vacuum barrier and which
made it possible to make images on an atomic scale.
In the case of tunneling through a single barrier of height 0V , the energy-
dependent transmission probability ( )T E , which is defined as ratio of the
transmitted to the incident flux, decreases exponentially with the barrier width W :
( )0( ) exp 2 2 ( ) /T E W m V E∝ − − h , where m denotes the particle mass. When a
second barrier of same width is added one might intuitively suggest that, following
Ohm’s law, the total resistance of the structure is just doubled. This is indeed true
if the region between the barriers is much larger than the de Broglie wavelength of
the electrons, which in semiconductors is typically of the order of 10-100
nanometers.98 However, if the middle region is only a few nanometers in width the
carrier transport remains phase-coherent and for some incident energies nE within
a small energy range of width, the particle is transmitted with a high probability,
eventually up to one. This enhancement of the transmission probability is known as
resonant tunneling. The physical explanation is that the resonant energies
correspond to the energies of the quasibound eigenstates of the electrons, which are
localized in such a state, can leak out through the barriers with a finite probability.
Due to the uncertainty principle, the finite lifetime τ of the electron causes an
uncertainty in the energy EτΔ ≈ h , which effectively leads to the broadening of
the resonance /Eγ τ= Δ ≈ h . The whole process of resonant tunneling can be
understood as a constructive interference between the waves leaking through the
first barrier and the reflected waves of the second barrier, similar to what happens
to electromagnetic waves. In a more particle-like picture corresponding to wave
packets an incident electron at resonant energy tunnels through the first barrier,
bounces then several times back and forth in the quantum well in a way that adds
up coherently, and finally tunnels out through the second barrier.
Such double barrier structures were realized by an epitaxial growth of
alternating ultrathin films of two semiconductor materials with different band gaps.
Using GaAs as smaller band gap material and Ga1−xAlxAs as barrier with the
barrier height controlled by the molecular fraction x of Al, the conduction band
profile of the layered structure exhibits sharp discontinuities at the heterointerfaces,
effectively realizing a double barrier structure. The double barrier structure is
usually surrounded by heavily doped layers, which provide low-resistance emitter
and collector contacts. To prevent diffusion of the dopants from the high doped
regions into the inner double-barrier structure usually also thin undoped buffer
layers are included in experiments. By attaching ohmic contacts to the whole
108
structure an external bias can be applied to the resonant tunneling diode (RTD).
The N-shaped current-voltage (IV) characteristic exhibits a region of negative
differential resistance (NDR).
This NDR-behavior of a RTD can be qualitatively easily understood if we
recognize that the electrons which are trapped between the two barriers exhibit a
discrete energy spectrum whose spacing increases if the confinement gets stronger,
i.e., the quantum well width becomes smaller. Let us assume, for simplicity, that
the quantum well is thin enough that there is only one quasibound state in the
energy range of interest. With a positive bias aV applied to the right (collector)
lead the resonant energy level is lowered relative to the energy of the incident
electrons from the left (emitter) lead. In a first approximation one can assume that
the voltage drops linearly from the emitter to the collector side. The electrons in
the left (emitter) and right (collector) contact are considered to be always in
thermal equilibrium which allows to introduce chemical potentials μL, μR for both
reservoirs and to describe the electrons distribution by the Fermi-Dirac function.
This means that at low temperatures incident electrons from the emitter with
energies reaching from the bottom of the conduction band up to the Fermi energy
are available. However, since the RTD effectively acts as an energy filter only
electrons with the resonant energy E0 can transmit to the collector side if there are
unoccupied states at that energy; otherwise the electrons are blocked by the Pauli
principle. By applying positive bias to the collector the resonant level passes
through the emitter’s Fermi energy and current starts flowing. Increasing the bias
leads to higher current magnitudes. However, at a certain voltage (the peak
voltage) the resonant level becomes energetically aligned with the bottom of the
emitter’s conduction band. Further bias pushes the resonant level 0E below this
edge, which suddenly cuts off the supply of emitter’s electrons causing a sharp
drop in the current and thereby leading to the phenomenon of NDR
2. Coherent tunneling
For the purpose of obtaining a more quantitative understanding of resonant
tunneling and the related NDR-effect in semiconductor heterostructures we assume
at first that the transport through the structure is fully phase-coherent. This
assumption allows to apply a wave function treatment of the transport similar to
what is done in the description of electromagnetic wave propagation in planar
layers of different permittivity. We restrict our discussion primarily to electrons in
a parabolic conduction band, e.g., one can think of the Γ -valley ( 0k = ) electrons
in GaAs. In the case of coherent transport between two contacts the flowing current
density can be obtained in general from the Landauer-Buttiker formula [1,3,4]
( )2 ( ) ( ) ( )L R
ej dET E f E f E= −∫h
,
where the factor 2 takes into account the spin degeneracy, e is the elementary
charge, 2h π= h is Planck’s constant, and , ( )L Rf E are the electrons distribution
109
function in the left and right reservoir, which are usually assumed to be given by
Fermi-Dirac functions. The single particle transmission function ( )T E describes
physically how likely a single electron of energy E can transmit through the
structure and is more rigorously defined as the sum over all transmission
probabilities ( )n mT E← ) of an electron starting in the input mode m and ending up
in the output mode n of the left and right leads, respectively, which connect the
reservoirs with the structure. In the specific case of planar heterostructures these
lead modes are easily identified with the plain wave electron states of fixed in-
plane momentum q , i.e., of a certain momentum component perpendicular to the
growth direction. If we assume that the in-plane momentum is conserved during
the transport, which means that there is no scattering from one lead mode to
another, the transmission function can be written as
' ',
', ',
( ) ( ) ( ) ( )q q q q q q
q q q q q
T E T E T E T Eδ←= = =∑ ∑ ∑ .
This assumption is reasonable for elastic scatterers, which do not change the
electron’s momentum considerably, and as long as inelastic scattering processes
are not important (which should be actually the case to allow for a phase-coherent
propagation).
The transmission function ( )qT E can be determined from the solution of the
single-particle Schrodinger equation if the electrons can be treated as independent
coherently propagating quasiparticles. This demands that the effect of electron-
electron interactions is describable by an effective single-particle potential, which,
by following the approach of local density functional theory, depends only on the
local electron density. For simplicity we will include here only the self-consistent
Hartree terms and neglect the exchange potentials or other electron-electron
correlations. The influence of the periodic lattice potential of the crystal on the
electrons is treated in the effective mass approximation. Under these assumptions
the steady-state envelope function ( , )r zψ of an single electron in the
heterostructure can be determined from the Schrődinger-like equation
2 2
21 ( ) ( , ) ( , )
2 ( ) 2 r eff
l t
V z r z E r z
z m z z m
ψ ψ
⎛ ⎞∂ ∂
+ ∇ + =⎜ ⎟∂ ∂⎝ ⎠
h h (1)
Here, r is the in-plane or transversal position vector and z denotes the
growth direction or what we call the longitudinal direction, ( )lm z is the
longitudinal effective mass perpendicular to the heterointerface, ( )tm z is the in-
plane effective mass and E denotes the total energy. The kinetic energy operator
for the longitudinal motion takes into account the z -dependence of the
longitudinal effective mass and satisfies the requirement of being Hermitian. The
effective potential ( ) ( ) ( )eff i elV z U z U z= + contains the intrinsic conduction band
discontinuities ( )iU z and the electrostatic potential ( )elU z , which depends on the
fixed ionized impurity density and the electron density profile in the structure.
110
Since the effective potential varies only in the longitudinal direction the in-plane
motion of the electrons, which is of free electrons plane-wave type, can be
separated from the growth-direction dynamics, justifying a product ansatz for the
envelope function: ( , ) exp( ) ( )r z iqr zψ ϕ∝ . With this the lead input and output
modes can be characterized by the plane wave states exp( )iqr Eq. (1) can be
reduced to an effective one-dimensional Schrődinger equation for the growth
direction motion
2 1 ( ) ( , ) ( , )
2 ( ) eff l
l
V z r z E r z
z m z z
ψ ψ
⎛ ⎞∂ ∂
+ =⎜ ⎟∂ ∂⎝ ⎠
h (2)
where we introduce the longitudinal energy 2 2 /(2 )l t tE E q m E E= − = −h , which
we always measure in the following from the bottom of the emitter’s conduction
band. From the definition of the longitudinal energy it is evident that tE l is
conserved during the transport if we assume that the total energy and the in-plane
momentum q are conserved and that mt is independent of z . As we will see, these
assumptions considerably simplify all further calculations, since the transmission
function ( ) ( )q lT E T E= will only depend on the longitudinal energy, having no
explicit dependence on the in-plane momenta q . According to our mean field
approach for the electron-electron interaction the electrostatic potential elU eφ= −
can be obtained from the Poisson equation
( )
0
1( ) ( ) ( ) ( )impz z en z z
z z
ε φ ρ
ε
∂ ∂
= −
∂ ∂
, (3)
where ( )zε denotes the, in general, z -dependent static dielectric constant, 0ε is
the permeability of the vacuum, ( )imp zρ is the fixed impurity charge density of the
structure, and ( )n z is the electron density. The Poisson equation (3) is nonlinearly
coupled to the envelope function equation (2) via the particle density ( )n z , since
the electron density profile of the structure is established by occupying the energy-
dependent scattering states '(z) according to the distribution
functions of the electron reservoirs in the emitter and collector leads. Hence, the
coupled Schrődinger-Poisson system has to be solved in a selfconsistent way,
which can be done iteratively by alternately solving both equations and using the
solution of one equation as input for the other, until convergence is reached.
In order to solve the Schrődinger equation (2) and to find the transmission
function ( )lT E we introduce here the transfer matrix technique (see (5-7). The
basic idea of the method is to divide the z-axis into a sequence of regions where the
solution can be obtained analytically. These local solutions are then composed to a
global one by using the continuity conditions of the wave function between the
different regions. Let us assume that we have n different layers with different
effective masses im and constant effective potentials iV in each layer. The solution
111
for each individual layer 1i iz z z− ≤ ≤ can then generally be written as the sum of
left and right moving plane wave states
* exp( ) exp( )i i i i i i i iA B A ik z B ik zϕ ϕ ϕ= + = + − (4)
with 02 ( ) /ik m V E= − h and ,i iA B denoting the amplitudes of right and left
moving waves, respectively. The continuity of the wave function and the
conservation of the probability current leads to the system
1
1
( ) ( ),
1 1( ) ( ).
i i i i
i i
i i
z z
d dz z
m dz m dz
ϕ ϕ
ϕ ϕ
+
−
=
=
(5)
These relations between neighboring layers can be rewritten in matrix form,
1
1
1
( ) ( ) , 1,..., 1i i
i i i i
i i
A A
U z U z i n
B B
+
+
+
⎛ ⎞ ⎛ ⎞
= = −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
(6)
with the matrix
( ) ( )( )
'/ '/
i i
i
i i i i
U z
m m
ϕ ϕ
ϕ ϕ
+ −
+ −
⎛ ⎞
⎜ ⎟=
⎜ ⎟
⎝ ⎠
, (7)
where the prime denotes the derivative with respect to z . Starting with 1i = , Eq.
(6) allows to express the transition amplitudes of the second layer as a function of
the amplitudes of the first one, 1
2 2 1 1 1 1( ) ( )C U z U z C−= , using the vector notation
( , )i i iC A B= . The matrix 1
1 2 1 1 1( ) ( )M U z U z−= is called a transfer matrix between
the first and second region since it connects the corresponding amplitudes.
Repeating successively this procedure for 2,..., 1i n= − finally allows to correlate
the amplitudes of the last layer with those of the first one:
1
1
,n n
n n
A A
M
B B
−
−
⎛ ⎞ ⎛ ⎞
=⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
(8)
where we have introduced the composed transfer matrix,
1 1
1 1 1 2 2 2 2
1, 1
( ) ( ) ... ( ) ( )n n n n i
i n
M U z U z U z U z M− −
− − −
= −
= ⋅ ⋅ = ∏ . (8’)
Hence, the total transfer matrix can be composed by the individual transfer
matrices iM just by using conventional matrix multiplications. The amplitudes 1C
are determined by the boundary conditions of the Schrődinger equation. For
instance, if we assume only impinging electrons from the left we can set 1 1A =
and 0nB = . Using the relation 1nC MC= leads to 1 21 22/B M M= with ijM
denoting the matrix elements of M . The knowledge of the first layer
amplitudes 1C allows to successively calculate all other layer amplitudes
( 2 1 1 3 2 2, ,...C M C C M C= = ), constructing in this way the envelope function
throughout the whole structure.
112
The transfer matrix connects the left and right amplitude coefficients of the
structure. This representation is not unique and it is often more convenient to
connect the incoming and outgoing amplitudes by the scattering matrix
1 1 1'
'n n n
B A Ar t
S
A B Bt r
⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞
= =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠
. (9)
The S -matrix is a natural representation for scattering problems, since the
diagonal elements are given by the reflection amplitudes r and 'r for waves
coming from the left and right hand side of the sample, respectively, and the off-
diagonal elements are related to the wave transmission amplitudes t and 't . This
physical interpretation of ijS j becomes immediately evident by recognizing that the
outgoing amplitudes can be always composed by reflected and transmitted parts of
incoming wave amplitudes of electrons impinging from the same and opposite
side, e.g., ( 1 11 1 12 1 'n nB S A S B rA t B= + = + . By using Eqs. (8) and (9) the transfer
matrix can be also expressed in terms of these wave amplitudes
1 1
1 1
'( ') '( ')
( ') ( ')
t r t r r t
M
t r t
− −
− −
⎛ ⎞−
= ⎜ ⎟
−⎝ ⎠
. (10)
It should be noted that for the general case of N incoming channels the
amplitudes 1 1, , ,n nA B A B become complex vectors of length N and the
transmission and reflection amplitudes are replaced by N N× matrices. The
elements of the transfer matrix are not independent due to the flux conservation
and other physical symmetries. For instance, for symmetric structures time reversal
symmetry leads to the relation ( ')Tt t= , where the superscript T denotes
transposition of the matrix. In the simple one-dimensional case, as considered here,
this simplifies to 't t= confirming the intuitive expectation that the transmission
amplitude is the same for left and right incident electrons of equal energy, since the
left-moving electron follows the time-reserved trajectory of the right moving one.
If the transmission matrix is known, the single particle transmission function
( )lT E can be easily obtained as follows. Physically the transmission function is
defined as the ratio of the transmitted to the incident probability flux of a particle:
/trans incT f f= . Similarly, the reflection coefficient is defined by /refl incR f f= with
frefl denoting the reflected probability flux. Conservation of the total particles flux
demands that 1T R+ = . The incident probability flux is given by the squared wave
amplitude times the group velocity of the incident electron, which we assume here
to impinge from the left, 2
1 1 1| | /incf A k m= h , and the reflected and transmitted
fluxes are accordingly determined by 2
1 1 1| | /reflf B k m= h and 2
1| | /trans n nf A k m= h .
With these definitions the transmission function reads as
2
1 1
2
1 1
| |
( )
| |
n
l
n
k m A
T E
k m A
= .
113
By applying the corresponding boundary conditions of left incident electrons
( 1 1, 0nA B= = ) and by using the relation 1nC MC= we obtain
1
22
det
n
MA A
M
=
The determinant of the transfer matrix results in 1 1det /n nM k m k m= , since
one easily finds
1 1
1 1 1 1 1 1det[ ( ) ( )] 1, det[ ( ) ( )] /i i i i n n n nU z U z U z U z k m k m− −
− −= =
which can be easily verified by using the explicit expressions for M and iU
stated in Eq. (8) and (8’), respectively. With this the transmission function can
finally be written as
1
2
1 22
1( )
| |
n
l
m
k m
T E
k m M
= . (11)
An important point to note here is that the transmission function can also be
defined as the squared current transmission amplitude 2| |T t= % . The current
transmission amplitude t% is related to the wave transmission amplitude t , which
we have introduced in the definition of the S -matrix in Eq. (9), by /L R L Rt t v v→=% ,
where Lv , Rv are the left and right side group velocities. The renormalized
scattering matrix based on current amplitudes, /ij ij i jS S v v= , has the advantage
of being unitary due to current flux conservation. With this it follows that
2( ) ( / ) | |k LT E v v t= , which is consistent with our previous results Eq. (10) and Eq.
(11) by taking into account that 't t=% and, hence, ' / /L R R Lt v v t v v= according
to time reversal symmetry
In order to investigate the basic physics of resonant tunneling we apply these
general results to the special case of a double barrier structure. The typical
appearance of the transmission function versus the electron’s incident energy show
its strongly “spiky” characteristic, the local density of states of the conduction
electrons, in which the forming of quantum well states and their energetic
broadening become clearly apparent. Since such a double-barrier structure consists
of two single barriers in series we can calculate the total transmission matrix M
by using the composition law 1 2M M M= , where 1M and 2M are the transfer
matrices of the first and second single barrier. Using the general expression given
in Eq. (10) the composed transmission amplitude t results in 1 2 1 2( ) /(1 ' )t t t r r= − ,
where ,i it r and 'ir denote the amplitudes of the single barriers 1,2i = i = 1, 2. The
transmission function is given by the squared current transition amplitude
2 1 2
1 2 1 2
( ) | |
1 2 cos
R
l
L
v T T
T E t
v R R R Rθ
= =
− +
(12)
114
with 2
1 1/ | |LT v v tω= , 2
2 2/ | |RT v v tω= with vω denoting the group velocity in the
well, 2 2| | | ' | , 1,2i i iR r r i= = = and θ is the phase of 1 2'r r+ . The phase shift θ
corresponds to the phase acquired by the electron when it makes one round-trip
between the two barriers, which means that the electron is reflected once from each
barrier before transmitting the structure. The analytical form of the transmission
function 1T , 2T for the single barriers is easily obtained from the transfer matrix
technique showing an exponential dependence on the barrier width W in the limit
of thick and/or high barriers 1bWk , where 02 ( ) /bk M V E= − h . The
expression, Eq. (12), for the composed transmission function of the double barrier
structure can be further simplified if we assume, as is normally the case, that 1T ,
2T , and consequently the reflection coefficients are of the order of unity,
1 2, 1R R ≈ :
( ) ( )
( )
2 1 2
2
1 2 1 2
1 2
2
1 2
( ) | |
1 2 1 cos
.
[ / 2] 2[1 cos ]
R
l
L
l
l
v T T
T E t
v R R R R E
T T
T T E
θ
θ
= = ≈
− + −
≈
+ + −
(13)
Resonance occurs when the denominator becomes very small, which means thatθ
is a multiple of 2π . At resonance 2
1 2 1 24 /( )resT T T T T= + , which approaches unity
for the case of symmetric barriers 1 2T T= . In the off-resonant case, 1 2 / 4T T T≈ ,
indicating that the double barrier behaves as two independent barriers. Close to the
resonance, 0lE E= , we can further simplify Eq. (13) by performing a Taylor series
expansion of the cosine function
[ ] ( )
9
2
2
0
1 11 cos[ ( )] ( ) 2
2 2l l l
l E
dE E n E E
dE
θθ θ π
⎛ ⎞
⎜ ⎟− ≈ − ≈ −
⎜ ⎟
⎝ ⎠
This yields the known formula for the transmission function near the
resonance,
0
1 2
2 2
1 0 1 2
( ) , ,
( ) [( ) / 2]
l
l j i
E
dE
T E T
dE E
γ γ
γ
θγ γ
≈ =
− + +
(14)
This formula was first derived in studying the decay of resonant states in nuclear
problems and is often also written in the form,
1 2
0
1 2
( )lA E E
γ γ
γ
γ γ
= −
+
with the Lorentzian function
1 22 2( ) ,
( / 2)
A γξ γ γ γ
ξ γ
= = +
+
115
This analytical expression shows that the transmission function is sharply peaked
around the resonant energy 0E and that its broadening is determined by γ , which
corresponds to the full width at half maximum (FWHM) of ( )A ξ . Physically 1 /γ h
and 2 /γ h represents the rate at which an electron leaks out of the quantum well
through barrier 1 and 2, respectively. To make this more plausible one can roughly
approximate the acquired round-trip phase by 2k aωθ ≈ , where kω is the
longitudinal momentum of the electron in the quantum well and a is the width of
the well. Hence,
0
/ (1/ )( / ) | (1/(2 ))( / ) | ( /(2 ))
ri l E l E i idE d dE dk T v a Tω ωγ θ= ≈ =h h h ,
where vω is the group velocity of the electron in the quantum well at the resonant
energy level. The attempt frequency /(2 )v aω tells us the number of escape
attempts of the electron per second through a single barrier when the electron
bounces forth and back in the quantum well. Multiplying the attempt frequency by
the transmission probability of the single barrier gives us the rate of successful
escapes of the electron per second. Hence, the lifetime of the electron is given by
the inverse of the total escape rate 1 2/ ( ) /γ γ γ= +h h . Since is the FWHM of the
resonant transmission peak this again leads to the general result that the energetic
broadening of the quasibound state 0E E0 is inversely proportional to the lifetime of
the electron in this state. If the transmission function is known the current density
can be calculated by using the Landauer-B¨uttiker formula, represented at the
beginning of this section. In the case that the transmission depends only on the
longitudinal energy, as considered here,
3 3
0 0
4 ( ) [ ( , ) ( , )]
(2 ) l l t L l t R l t
emj dE T E dE f E E f E Eπ
π
∞ ∞
= −∫ ∫h
,
where we have rewritten the summation over the in-plane momentum q q in the
usual integral form,
2( ) ( )
(2 )q x y l
q
ST E T dq dq T E
π
= =∑ ∫
with S denoting the cross sectional area of the structure, and transforming to the
longitudinal and transversal energy as integration variables. Assuming Fermi-Dirac
distributions in the leads
( ) 1
, ,1 exp[( ) / ]L R l t L R Bf E E kμ θ
−
= + + −
where ,L Rμ are the chemical potentials in the left and right lead ( R L aeVμ μ= − μR
= μL − eVa), θ denotes the reservoir temperature to avoid confusion with the
transmission T , and Bk is the Boltzmann constant, the integration over the
transversal energy is easily evaluated to give the Tsu-Esaki formula [6]:
116
2 3
0
1 exp[( ) /( )]
( ) ln
1 exp[( ) /( )]2
L l BB
l l
R l B
E kemk
j dE T E
E k
μ θθ
μ θπ
∞ ⎛ ⎞+ −
= ⎜ ⎟+ −⎝ ⎠
∫h
(15)
The logarithmic term is the so-called supply function which determines the energy
interval of interest. The range of electron energies, which can contribute to the total
current, is restricted to the energy window between the left and right chemical
potentials [ ]L Rμ μ⋅ plus/minus several Bk θ due to the thermal smearing of the
Fermi-Dirac functions in the leads. The dominant contributions to the current
integral are given by the resonant peaks of the transmission function.
If we assume that there is only one single transmission peak in the energy
range of interest and that ( )T E is very sharply peaked around 0E due to thick
and/or high barriers we can approximate its Lorentzian form by a Dirac-Delta
function by using the asymptotic limit 0( )lE Eδ − = 0 0(1/ 2 ) lim ( )lA E Eγπ →= − .
With this approximation the current density results in
1 2
02
1 exp[( ) /( )]
ln
1 exp[( ) /( )]
L l B
B
R l B
E kej k D
E k
μ θγ γ
θ
μ θγπ
⎛ ⎞+ −
= ⎜ ⎟+ −⎝ ⎠h
, (16)
where we have introduced the constant density of states of a two-dimensional (2D)
electron gas 2
0 /D m π= h . In the special case of zero temperature, 0θ = , this
further simplifies to the expression
1 2
0 0 02 ( ( )), 0L a
ej D E V E
γ γ
μ μ
γπ
= − < <
h
, (16’)
where the voltage dependence of the current is “hidden” in the voltage-dependent
resonant energy level 0 ( )aE V , which is shifted energetically downwards by the
applied bias aV . If we assume, in a first approximation, that the voltage is equally
divided between the barriers the voltage dependence of the resonant level can be
explicitly written as 0 00( ) / 2a aE V E eV= − with 00 0 ( 0)aE E V= = denoting the
resonant level position when no bias is applied. Equation (V.34) shows that the
current initially increases linearly with the applied voltage, reaching its peak value
1 2
0p L
ej D
γ γ
μ
γ
=
h
corresponding peak voltage of 02peV E= . At higher voltages the quasibound state
becomes off-resonant causing a sudden cutoff of the current, as long as no other
higher lying resonant level is pulled down into the energy window of interest
3. Sequential Tunneling
In our discussion of resonant tunneling so far we assumed that inelastic,
phase-breaking scattering processes are negligible, which enables us to apply a
wave function treatment of the underlying electron transport. However, if
scattering is important the electrons will lose their phase memory during
117
propagation and the transport becomes incoherent. In this case one can use the
sequential tunneling model introduced by [8]. In a sequential tunneling process
electrons tunnel through the first barrier, reside some time in the quantum well
where they lose coherence by phase-randomizing scattering processes and, finally,
tunnel out through the second barrier by a second uncorrelated tunneling process.
The regime of sequential tunneling can be characterized by the condition phτ γ h
saying that the lifetime / γh / of the electrons in the quantum well is much greater
than the phase breaking time phτ .
As argued by [8], NDR generally follows from the reduction of the
dimensionality as the electrons tunnel from a three dimensional Fermi sea in the
emitter to a 2D electron gas in the quantum well. Assuming an energy and in-plane
momentum conserving tunneling process leads to the constraining
condition 2
0/ 2lE k m E= =h , where 0E is the energy of the resonant level in the
well, measured from the bottom of the emitter conduction band. Therefore, only
electrons with the fixed longitudinal momentum 0 02 /zk k mE= = h can tunnel
from the emitter Fermi sea into the quantum well. The maximum current is reached
at the equatorial plane 0 0k = . If 0 0E < no resonant tunneling from the emitter
into the well is possible anymore, which leads to an abrupt drop of the current
giving rise to NDR. This explanation shows that for the occurrence of NDR it does
not matter if the electrons propagation is coherent or not.
To calculate the current in the sequential tunneling regime we can use a
master equation approach, since the in- and out-tunneling processes become
uncorrelated. For this purpose, we introduce a single particle distribution function
fα for the electron states |α > in the quantum well. The states | | ,m qα >= > are
characterized by the in-plane momentum q of the electrons and the subband index
m , which enumerates the well quasibound states starting from the ground state
0m = . In real-space representation the state |α > reads
, | exp( ) ( )mr z iqr zα φ< >= , where ( )m zφ is the quasibound wave function. In the
leads the electrons occupy plane-wave Bloch states, shortly denoted by | k > . With
these definitions the master equation for the quantum well distribution function
reads as
, ,
,
(1 ) (1 )j j j j
k k k k
j k
f
f f f
t
α
α α α
∂
= Γ − −Γ −
∂ ∑ , (17)
where j
kf denotes the electron distribution function in the left and right lead
( 1, 2 ,j L R= = and ,
j
kαΓ denotes the transition rate from state |α > in the lead j
to the state |α > in the quantum well. The physical meaning of the two terms on
the right hand side of Eq. (17) is easily understood. The first term is the gain term
which describes the tunneling of the electrons from the leads into the quantum well
118
state |α > by taking into account the Pauli blocking factor (1 )fα− , where as the
second term describes all loss processes due to tunneling out of the state |α > . The
transition rate ,
j
kαΓ can be calculated by using the transfer Hamiltonian approach
[9,10], where it was first developed for describing single barrier tunneling and has
been extensively used in the context of transport in superconducting tunnel
junctions.
In the case of single barrier systems the basic idea of the method is to
represent the total Hamiltonian of the system by L R TH H H H= + + , where LH
and RH describes the Hamiltonian of the left and right subsystem and TH is the
tunneling Hamiltonian describing the transport between the two subsystems. The
main advantage of the method is that if the coupling between the two subsystems is
weak, TH can be treated as a perturbation term, which allows to use perturbative
techniques developed in many-body theory. In our case of a double barrier
structure the total Hamiltonian consists of three subsystems: the emitter LH , the
well Hω , and the collector RH Hamiltonian, which are connected by two
tunneling Hamiltonians j
TH for the left and right barrier: L RH H H= + +
L R
T TH H Hω+ + + . Assuming a free electron gas in the emitter and collector the
corresponding Hamiltonians read ,
,
L R
L R k k k
k
H E c c+= ∑ and the well Hamiltonian is
given by H E c cω α α α
α
+= ∑ with ,kc cα and ,kc cα
+ + denoting the annihilation 0E and
creation operators of the leads and well states, respectively. The
energies, , 2 2
,( ) / 2 ( ) / 2L R
k t z L RE k m k m U= + +h h , and 2( ) / 2E q mα = +h Uω+
include the electrostatic energies of the reservoirs ,L RU and the wellUω . By
measuring the energy from the conduction band edge of the emitter it follows that
0LU = and RU eVα= − . The electrostatic potential of the well Uω depends on the
space charge density in the structure, and has to be calculated in general in a self-
consistent way. The tunneling Hamiltonians are formulated in the standard form,
,
,
. .j j
T k k
k
H t c c h cα α
α
+= +∑ where . .h c abbreviates the hermitian conjugate of the first
term and ,
j
ktα are the tunneling matrix elements. If we assume that the leads are
weakly coupled to the well, the tunneling Hamiltonian can be treated as a
perturbation term and the transition rates between the well and the lead states
2 2
, ,
2 2| | ( ) ( )j j j
k t k k kk H E E t E Eα α α α
π πα δ δΓ = < > − = −
h h
.
By assuming that the in-plane momentum is conserved during the tunneling
process this becomes
2
, ,(2 / ) ( ) ( )
t
j j
k m z k q kt k E Eα απ δ δΓ = −h with the zk dependent
119
tunneling matrix element ( )j
m zt k , which physically corresponds to the overlap of
the lead and well wave function in the barriers and is given by [9]
2
* *( ) ( ) ( ) ( ) ( )
2 z z
j j j
m z k m m k
d dt k z z z z
m dz dz
ψ φ φ ψ⎡ ⎤= −⎢ ⎥⎣ ⎦
h .
Here,
z
j
kψ is the longitudinal part of the lead wave function, which is exponentially
decaying in the barrier regions, the superscript * denotes complex conjugation,
and the expression has to be evaluated at some point z0 inside the j th barrier-
region. The total leaking rates from a certain quantum well state | | ,m qα >= >
through the left and right barriers into the leads are defined by 2
2
, 0
/
z
m
k
k k
αγ
>
= Γ∑h
2
2
, 0
/
z
m
k
k k
αγ
>
= Γ∑h which can be readily simplified to 2/( 2 )m j
j mmL Eγ = h ,
2| | /(2 )j j
m m m jE k m E U Uω= = + =h with L denoting the length of the leads. With
these definitions and by exploiting the microscopic reversibility of the tunneling
processes j j
k kα αΓ = Γ , the master equation Eq. (17) can be written in the form
[ ( ) ]
j
m
j
j
f
f E f
t
α
α α
γ∂
= −
∂ ∑
h
.
If we now assume, as before, that jf are given by Fermi-Dirac distributions and
that there is only one resonant level 0E in the energy range of interest we can
obtain a simple rate equation for the quantum well particle density, which is
defined by n fα
α
= ∑ . Summation of the master equation over all well states
| , qα α= > yields the rate equation
1 1 2 2/ ( / ) ( / ) ( / )dn dt n n nγ γ γ= + +h h h . (18)
with
0( ) ln 1 exp[( ) /( )]j j B j Bn f E D k E kα α
α
θ μ θ⎡ ⎤= = + −⎣ ⎦∑
and 0
j jγ γ= , 1 2γ γ γ= + . This expression confirms the naive expectation that at
steady state the quantum well has to establish a “compromise” between the
opposing efforts of equilibrating with both leads at the same time and, hence, the
particle density becomes a balanced sum of the lead particle densities weighted
according to the coupling strengths to the particle reservoirs. The steady state
particle density n0 follows from the condition / / 0dn dt j dj dt= ∇ = = whence
0 1 1 2 2( ) /n n nγ γ γ= + . Therefore, it does not matter at which z -point the current
density is evaluated, and calculating the current at the first barrier yieldswhich by
using Eq. (18) results in
120
1 2
0 0
1 exp[( ) /( )]
ln
1 exp[( ) /( )]
L l B
B
R l B
E kej k D
E k
μ θγ γ
θ
γ μ θ
⎛ ⎞+ −
= ⎜ ⎟+ −⎝ ⎠h
.
which is exactly the same result as we get for the coherent model in Eq. (16) in the
limit of a delta-like resonant level. This limit is physically reasonable, since in
order to apply the transfer Hamiltonian formalism we had to assume that the well is
only very weakly coupled to the reservoirs and accordingly the electrons can stay a
long time in the well before they tunnel out. A long lifetime in the well
corresponds to only a very narrow energetic broadening of the quasibound states
resulting in a delta-like resonance.
Thus sequential and coherent tunneling models give essentially the same
values for resonant currents, although the underlying physical pictures are very
different. In particular the peak current of the IV-characteristic has been shown to
be insensitive to scattering. This conclusion can be justified by using a more
general model that includes both a coherent and sequential part of the tunnel
current (see [11]), showing that scattering processes effectively lead to an
additional broadening of the resonant level. This broadening hardly influences the
total current density, bearing in mind that the current is proportional to the folding
integral of the transmission function with the supply function. In contrast to the
peak current, the off-resonant valley current depends strongly on the presence of
inelastic scattering processes.
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Поступила 12.01.2009р.
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