Bound states in 2D fermion systems of graphen
Analytical solutions for the zero-energy modes of two-dimensional massless Dirac fermions confined within the one-dimensional Lorentz-like potential, which provides а reasonable fit for potential profiles of existing top-gated graphene structures is performed. On the basis of obtained hypergeomet...
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Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України
2009
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| Zitieren: | Bound states in 2D fermion systems of graphen / A.M. Korostil // Моделювання та інформаційні технології: Зб. наук. пр. — К.: ІПМЕ ім. Г.Є. Пухова НАН України, 2009. — Вип. 52. — Бібліогр.: 13 назв. — англ. |
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| author | Korostil, A.М. |
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| citation_txt | Bound states in 2D fermion systems of graphen / A.M. Korostil // Моделювання та інформаційні технології: Зб. наук. пр. — К.: ІПМЕ ім. Г.Є. Пухова НАН України, 2009. — Вип. 52. — Бібліогр.: 13 назв. — англ. |
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| container_title | Моделювання та інформаційні технології |
| description | Analytical solutions for the zero-energy modes of two-dimensional massless Dirac fermions
confined within the one-dimensional Lorentz-like potential, which provides а reasonable
fit for potential profiles of existing top-gated graphene structures is performed. On the basis
of obtained hypergeometrical equations we have studied the conditions for formation of
quantum bound states providing an one-dimensional fermion localization. А simple
relations between the potential parameters and number of modes within the potential are
established. Possibility of realization of the external controlled charge transport in the
studied 2D system is considered.
|
| first_indexed | 2025-11-30T21:27:14Z |
| format | Article |
| fulltext |
УДК 535.3+621.37
A.M. Korostil, IMag, Kyiv
BOUND STATES IN 2D FERMION SYSTEMS OF GRAPHEN
Analytical solutions for the zero-energy modes of two-dimensional massless Dirac fermions
confined within the one-dimensional Lorentz-like potential, which provides а reasonable
fit for potential profiles of existing top-gated graphene structures is performed. On the basis
of obtained hypergeometrical equations we have studied the conditions for formation of
quantum bound states providing an one-dimensional fermion localization. А simple
relations between the potential parameters and number of modes within the potential are
established. Possibility of realization of the external controlled charge transport in the
studied 2D system is considered.
1. Introduction
Carriers within graphene behave as two-dimensional (2D) massless Di1rac
fermions, exhibiting relativistic behavior at sub-light speed owing to their linear
dispersion, which leads to many optical analogies (see [1]). For them as for
relativistic particles there is the known problem of the spatial localization (see [1-
5]). The relativistic particles do not experience exponential damping within а
barrier like their non relativistic counterparts and that as the barrier height tends
towards infinity, the transmission coefficient approaches unity. This inherent
property of relativistic particles makes confinement non-trivial. Carriers within
graphene behave as two-dimensional (2D) massless Dirac fermions, exhibiting
relativistic behavior at sub-light speed owing to their linear dispersion, which leads
to many optical analogies. Features of particle tunneling through р-n junction
structures in graphene has been studied both theoretically and experimentally (see
[1,5,6]). Quasi-bound states were considered in order to study resonant tunneling
through various sharply terminated barriers (see [1,5,7]).
In the presented paper the changed geometry of the problem is considered in
order to study the propagation of fully confined modes along а smooth electrostatic
potential, much like photons moving along an optical fiber (see [1,8,9]). The
Lorentz-like potential is used for confinement of carriers within graphene. The
bound modes within such а channel are analyzed on the basis of the
hypergeometrical equations describing wave states of fermions.
Recently quasi-one-dimensional channels have been achieved within
graphene nanoribbon (see [4]), however the control of their transport properties
requires precise tailoring of edge terminations, currently unachievable. The
solution of this problem can be with the help of truly bound modes creating within
bulk grapheme by top gated structures (see [1]).
The key to the realization of truly bound modes within а graphene waveguide
as zero-energy modes is related to possibility to control of the Fermi level using the
back gate. Then as in an ideal graphene sheet at half-filling, the Fermi level is at
the Dirac point and the density of states for a linear 2D dispersion vanishes. Cannot
escape into the bulk as there are no states to tunnel into.
The model Lorentz potential allows leads an exact analytical solution of
bound modes within а smooth electrostatic potential at half-filling, count the
number of modes and calculate the conductivity of the channel. The conductivity
carried by each of these modes is comparable to the minimal conductivity of a
realistic disordered grapheme system (see [2]). For the considered model potential
there are threshold potential parameters for which bound modes appear.
Thus we present an exact analytic solution for the fully confined zero-energy
modes of massless 2D Dirac fermions in a model smooth potential vanishing at
infinity and then describe the experimental geometry required for the observation
of confined modes within such grapheme waveguides.
2. The Hamiltonian and dispersion
The two-dimensional system of the graphene is consisted of carbon atoms
arranged in hexagonal structure via atomic sp2 bonds (see [1,10]). The structure
can be seen as a lattice with a basis of two atoms per unit cell, and two lattice
vectors 1 23 / 2, / 2 , 0,a a a a , as shown in Fig.1.
Fig.1. A plane structure of the graphene. The circles represent carbon
atoms, 1e and 2e represent lattice vectors.
The reciprocal lattice vectors are given by 1 2 / ( 3),0b a and
2 1/ ( 3),1/b a a . The tight-binding Hamiltonian for electrons in graphene
considering that electrons can hop to both nearest- and next-neighbor atoms can be
represented in the form
3 3
( , ). 1 ( , ). 1
) 'i j i j i j
i j j i j j
H t a b t a a b b
H.c. H.c. , (1)
where ( )i ia a
annihilates (creates) an electron with spin ( , ) on site iR on
sublattice A (an equivalent definition is used for sublattice B ); ( 2.8 )t eV is the
nearest-neighbor hopping energy (hopping between different sublattices), 't is the
next nearest -neighbor hopping energy (hopping in the same lattice). The Fourier
expansion of the mentioned second quantization operators:
2 2
( ), ( )
2 2
i iikR ikR
i i
Bz Bz
d k d ka e a k b e b k
,
where wave vector integration operators is executed within the first Brillouin zone
(Bz) and introduction of the two-component operators
( ) ( ( ), ( )) , ( ) ( ( ), ( ))Tk a k b k k a k b k
allow rewrite the Hamiltonian (1) in the form
2
2 ( ) ' ( )
(2 )Bz
d kH k H k
, (2)
where 'H is a two-dimensional matrix dependent on a wave vector k .
The eigenvalue of the Hamiltonian (2) determines the dispersion of the
studied system
3 3( ) 3 2cos( ) 4cos cos
2 2
3 3' 3 2cos( ) 4cos cos ,
2 2
y y y
y y y
E k k a k a k a
t k a k a k a
(3)
where the plus sign applies to the upper ( * ) and the minus sign the lower ( )
band corresponding to electrons and holes.
The dispersion (3) is characterized by the linear dependence close of six
vertex points of the type 4(0, )
3 3
K
a
(so-called by Dirac points) in the first
Brillouin zone of the hexagonal reciprocal lattice. Such the linear dispersion is
described by the expression [w]
2( ) | | [( / ) ],FE q v q O q K q k K , (4)
where | | | |q K , 3 / 2Fv ta is the Fermi velocity [w].
The most striking difference between this result and the usual dispersion,
2( ) / (2 )q q m , is that Fermi velocity in (4) does not depend on the energy or
momentum. The expansion of the spectrum around the Dirac point including 't up
to second order in /q K is given by
2 2( ) 3 ' 9 ' / 4 (3 / 8)sin(3arctan( / ))F x yE q v t t a ta q q
Whence it follows that presence of 't shifts in energy the position of the Dirac
point and breaks electron-hole symmetry.
Dirac fermions can be obtained from the Hamiltonian (1) with ' 0t using
expanding the Fourier sum around K and 'K . Then
' '
1 2 1 2,n n n nKR K R KR K R
n n n n n na e a e a b e b e b
Where the index 1i ( 2i ) refers to the K ( 'K ’) point. The new operators, ina
and inb are assumed to vary slowly over the unite cell. Transition in (2) to
coordinate representation in the considered case result in effective Hamiltonian of
the form
1 1 2 2( ) ( ) ( )FH iv dxdy r r r ,
with Pauli matrices ( , )x y , and ( , )i i ia b . In first quantized language the
two-component electron function ( )r close to the Dirac point K obeys 2D Dirac
equation
( ) ( )Fiv r E r ,
which corresponds to the Dirac-like Hamiltonian
FH v .
Note that wave function, in momentum space, for the momentum around K has
the form
exp( / 2)
( )
exp( / 2)
k
K
k
i
k
i
,
where /k x yq q and the signs correspond to the eigenenergies FE v k ,
that is for the * and bands, respectively.
3. Equation and bond quantum states
The Hamiltonian of graphene for the two-component Dirac wavefunctions in the
presence of a one dimensional potential ( )U x can be write in the form
( )F x x y yH v i i U x
where the sign denotes the two nonequivalent Dirac points, 61 10Fv m/s is
the Fermi velocity in graphene. An usage of representation of the Dirac function in
the form exp( )( ( ), ( ))T
y A Biq x x , where A and B correspond two sublattice,
leads to coupled first-order differential equations for the wave functions, which can
be represented in the form
( ) ( ) ( ) 0,
( ) ( ) ( ).
A y B
y A A
dV x x i q x
dx
di q x V x x
dx
Here ( ) ( ) / FV x U x v and / FE v .
Carrying out symmetrization of the wave functions with the help the
substitutions: 1 ( ) ( )A Bx i x and 2 ( ) ( )A Bx i x we can represent the
last system to the system of equations
1 2
1 2
( ) ( ) ( ) ( ) 0,
( ) ( ) ( ) ( ) 0.
y
y
dV x q x x
dx
d x V x q x
dx
(5)
This system describes the potential dependent and free wave propagation along the
x and free directions, respectively, i.e. the propagation in a waveguide.
Differentiation of the second equation of the system (5) and substitution of the
expressions
2 1
2 1
( ) ( ) ( ) ( ),
1( ) ( ),
( ) ( )
y
y
d x V x q x
dx
dx x
dxV x q
allow to reduce (5) to the second order differential equation
" ' 2 2
1 1 1
'( )( ) ( ) 0
( ) y
y
V xx V q
V q
. (6)
If the Fermi energy is at the Dirac Point there no charge carriers within the system
so graphene is insulator. Nonzero conductivity of the graphene waveguide can be
caused by coupled states within the potential well that is controlled via change of
the potential parameters (see [8]). It is shown [pro] possibility of an experimental
fixation of the zero Fermi level. So in the equation (6) we take the value 0 .
As known (see [8]) smooth potentials with controlled parameters which admit
the exact solution of considered problem represent the special physical interest. To
such a type of potential can belong of the Loretz-like potential of the form
2 2( )V x
x
(7)
where 0, 0 ; the negative value reflects a potential well for electron (see
[Pos]. )
Substituting the expression (7) into the equation (6) result in the equation
which at the condition yq takes the form of the differential equation of the
hypergeometrical type (see [11-13])
" '
1 1 12
( ) ( ) 0
( ) ( )
x x
x x
, (8)
Where 2 2( ) ( )x x , ( ) 2x x and 2 2 2( )x x .
The equation (8) with the help of the substitution 1( ) ( ) ( )x x y x where the
function ( )x by the relation of the form '/ ( ) /x can be reduced to the
canonical hypergeometrical equation of the form
( ) " ( ) ' 0x y x y y , (9)
where (see [13])
2
' '( ) , '( ),
2 2
( ) ( ) 2 ( ), ( ) ( )[ ( ) '( )],
'( ).
x k k x
x x x x x x x
k x
(10)
Taking into account that ( )x is a first order polynomial, from the first equation of
the system (10) we obtain two solutions: k and 2( / )k . In the first
case ( )x i and in the second case ( ) ( / )x x . From these
solutions ( ) ( / )x x correspond to the localization condition.
In according with the mentioned solution the system (9) yields
2 2 /(2 )
2
( ) 2 1 , ( ) ( ) ,
'( ) .
x x x x
k x
(11)
Polynomial solution of the equation (9) is built on the basis of the equation of
the form
" '( ) 0n n n n nx v v v , (12)
where ( 1)( ) ( ) ', ' ''
2n n
n nx x n n
and nv is the derivative of the
n-the order of the solution of the equation (9) ( ( )n
nv y ). Polynomial solutions of
the equation (9) are described by the formulae of the form
( )
( ) ( ) ,
( )
nnn
n
B
y x z
x
where the function ( )x is determined by the equation
( ) ' (13)
In the considered case the solution of the equation (13) has the form
2 2 /( ) ( )x x .
The bound modes in the system obey the condition
2
2
1' ( 1) "
2
3 2 0, (0,1, 2...)
n n n n
n n n
(14)
which determine the dependence of number of the bound quantum states on the
potential parameters. From the equation (14) we can obtain corresponding relation
2
1 1 33 2 2
2 2 4
n
. (15)
Thus change of the ratio / , characterizing a shape of the one-dimensional
potential ( )V x , leads to the change of the limit number of bounded quantum states
of electron in the waveguide. Appear of the bound states occurs by discrete leading
to increasing of the density of electron states and the increasing of conductivity of
the waveguide/ Appear of the first bound states is accompanied by a transition
from the non- to conductive electron state.
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|
| id | nasplib_isofts_kiev_ua-123456789-29632 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | XXXX-0068 |
| language | English |
| last_indexed | 2025-11-30T21:27:14Z |
| publishDate | 2009 |
| publisher | Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України |
| record_format | dspace |
| spelling | Korostil, A.М. 2011-12-24T23:34:47Z 2011-12-24T23:34:47Z 2009 Bound states in 2D fermion systems of graphen / A.M. Korostil // Моделювання та інформаційні технології: Зб. наук. пр. — К.: ІПМЕ ім. Г.Є. Пухова НАН України, 2009. — Вип. 52. — Бібліогр.: 13 назв. — англ. XXXX-0068 https://nasplib.isofts.kiev.ua/handle/123456789/29632 535.3+621.37 Analytical solutions for the zero-energy modes of two-dimensional massless Dirac fermions confined within the one-dimensional Lorentz-like potential, which provides а reasonable fit for potential profiles of existing top-gated graphene structures is performed. On the basis of obtained hypergeometrical equations we have studied the conditions for formation of quantum bound states providing an one-dimensional fermion localization. А simple relations between the potential parameters and number of modes within the potential are established. Possibility of realization of the external controlled charge transport in the studied 2D system is considered. en Інститут проблем моделювання в енергетиці ім. Г.Є. Пухова НАН України Моделювання та інформаційні технології Bound states in 2D fermion systems of graphen Article published earlier |
| spellingShingle | Bound states in 2D fermion systems of graphen Korostil, A.М. |
| title | Bound states in 2D fermion systems of graphen |
| title_full | Bound states in 2D fermion systems of graphen |
| title_fullStr | Bound states in 2D fermion systems of graphen |
| title_full_unstemmed | Bound states in 2D fermion systems of graphen |
| title_short | Bound states in 2D fermion systems of graphen |
| title_sort | bound states in 2d fermion systems of graphen |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/29632 |
| work_keys_str_mv | AT korostilam boundstatesin2dfermionsystemsofgraphen |