Quantum Ring in Gapped Graphene Layer with Wedge Disclination in the Presence of an Uniform Magnetic Field

0.43 MB·

by  Jose Amaro Neto

Quantum Ring in Gapped Graphene Layer with Wedge
Disclination in the Presence of an Uniform Magnetic Field
Jos´e Amaro Neto, J. R. de S. Oliveira and Claudio Furtado∗
Departamento de F´ısica, Universidade Federal da Para´ıba,
Caixa Postal 5008, 58051-970, Jo˜ao Pessoa, PB, Brazil.
7
1
0 Abstract
2

n In this paper we investigate the relativistic quantum dynamics of massive excitations in a
a
J
graphenelayerwithawedgedisclinationinthepresenceofanuniformmagneticfieldinz-direction.

9
We use a Dirac oscillator type coupling to introduce the confining potential for massive fermions
]
l
l in this system. We obtain the spectrum and eigenfunctions of energy for this quantum ring pierced
a
h
- by Aharonov-Bohm flux. We also find the persistent current and magnetization for this two-
s
e
m dimensional quantum ring in disclinated graphene layer. The influence of a wedge disclination on
.
t physical quantities is analyzed.
a
m
- PACS numbers: 73.22.-f,71.55.-i,03.65.Ge
d
n
Keywords: Topological defects, Dirac oscillator, graphene, quantum ring, Magnetization
o
c
[

1
v
1
5
0
2
0
.
1
0
7
1
:
v
i
X
r
a
∗ furtado@fisica.ufpb.br
1
I. INTRODUCTION
In recent years, physical properties of quantum dots in graphene were experimentally
studied in Refs [1, 2], where the eigenvalues of energy were found for the case of the presence
ofanexternalmagneticfield. Othertheoreticalstudieshaveemployedtostudytheproperties
ofaquasiparticleconfinedinnanostructures, i. e. quantumdots[3,4]andquantumrings[5–
7]. It was shown that an electronic structure, transport properties and some other related
physical aspects are significantly altered for graphene with the presence of disclinations.
Disclinated graphene is formed through the removal or addition of an angular sector of the
two-dimensional hexagonal lattice. The angle deficit generates a positive disclination, and
leaves the apex of the cone with a number of ring atoms proportional to the removed section.
Particularly, the low-energy behavior of quasiparticles in graphene has attracted an essential
interest. At low energies, the quantum behavior of excitations in graphene is described by
an equation analogue to a massless Dirac equation. In the absence of a sublattice symmetry,
a gap appears and the corresponding dynamics can be incorporated into a massive term in
the Dirac equation. It is a known fact that a band gap is induced in these samples, and
massive Dirac fermions play a central role in the low-energy limit [8]. In a general way, this
case is called gapped graphene.
It is well known from theoretical and experimental studies of nanostructures of graphene
(cid:112)
that the magnetic length is (cid:96) = (cid:126)/eB ≈ 50nm. This observation allows to show the
importance of investigating the influence of a disclination in a quantum ring in a graphene
layer, noting that the average size of the disclination in this carbon material is of the order
of the interatomic distance between two carbon atoms in this nanostructure. In Ref. [9] the
authors study the influence of external magnetic field for a quantum dot in graphene with
presence of a topological defect. Recently, the influence of a disclination on Landau levels
was investigated in Ref. [10].
Recently the study of soft a confinement in graphene was performed in [11], where it
was observed that the interaction with the substrate of the quantum dots introduces a gap
in graphene. In this contribution we study the two-dimensional quantum ring in a massive
graphenelayer, adoptingamodelofconfiningquasiparticleintheringconfigurationwiththe
potential proposed in Ref. [12], this confining potential is a relativistic version of the Tan-
Inkson confining potential in two-dimensional semiconductor [13]. In this model [12] also
2
two control parameters were used to obtain a harmonic confinement in a two-dimensional
ring, and the quantum point limits are obtained when we make one of the parameters to be
zero, a = 0. In this relativistic model the confining potential is introduced via the coupling
1
with the momentum of quasiparticle similar to Dirac oscillator [14], given by
√
(cid:20) (cid:21)
2Ma (cid:112)
(cid:126)p → (cid:126)p+i 1 + 2Ma r γ0e , (1)
2 (cid:98)r
r
where a and a are the characteristic parameters of the model. Now if we consider the
1 2
limit a → 0 the harmonic confining potential of the quantum dot is recovered. In the
1
case a → 0, the relativistic antidot limit is observed. We use this new coupling in a Dirac
2
equation to describe a quantum ring structure in a massive graphene in the low-energy limit.
The Dirac oscillator coupling was used in several applications for graphene in Refs. [15–20].
Recently, two of us, in Ref. [21] have employed the model proposed in Ref. [12] for the
confining potential and investigated the quantum ring in a graphene layer without/with the
presence of a disclination.
We study, in low energy limit, the system described by a massive Dirac equation, where
a continuous description near Fermi K-points is employed. We use the Dirac oscillator type
coupling to confine harmonically the quasiparticles in a quantum ring pierced by Aharonov-
Bohm flux in a disclinated massive graphene layer submitted to an uniform magnetic field.
We obtain the eigenvalues and eigenfunctions of energy and the persistent current. In the
case of dynamics in the presence of defects, we demonstrate the dependence of these physical
quantitiesontheparametercharacterizingthedisclination, thusdemonstratingtheinfluence
of the defect in the dynamics in a quantum ring.
This paper is organized as follows: in section II, we study the quantum dynamics of
a quasiparticle in a quantum ring in massive graphene layer in the presence of an uniform
magnetic field. In Section III, we obtain the persistent current in this system and investigate
the influence of a topological defect in this physical quantities. In section IV, we obtain the
magnetization at zero temperature and analyze its behavior in this system, and finally, in
section V we present our conclusions.
3
II. QUANTUM RING IN A MASSIVE GRAPHENE LAYER WITH DISCLINA-
TIONS IN THE PRESENCE OF A MAGNETIC FIELD
We can obtain a disclination in a graphene layer by a procedure known as Volterra
process [22]. This transformation can be represented by a cut and glue process where we
are cutting and removing/adding a sector in a glaphene layer, and the resulting disclination
is obtained gluing the the new edges of the lips. Due the symmetry of graphene honeycomb
lattice the removed or added angular sector must be a multiple of π/3. We can introduce a
topological defect in a graphene layer by a fictitious gauge following the approach introduced
in Refs.[23, 25–29], where a gauge field is introduced in Dirac equation in order to reproduce
thealreadyknowneffectof thedisclinationonthe behavior ofthespinor[23,27]. In this way
we can describe the quantum dynamics of a quasiparticle in a gapped graphene layer with a
disclination by Dirac equation in curved space in the presence of a non-Abelian gauge field
a related with K-spin flux. This non-Abelian gauge field is the contribution in the Dirac
µ
equation responsible for mixing of points K [23, 28]. In this way the quantum dynamics of
±
quasiparticles is described by a Dirac equation in a curved background given by following
metric
ds2 = dt2 −dr2 −α2r2dϕ2 (2)
whereαisthestrengthofthedisclinationwhichcanbewrittenintermsoftheangularsector
λwhichweremovedorinsertedinthegraphenelayertoformthedefect, asα = 1± λ , where
2π
in graphene lattice λ ∝ π/3 due to the hexagonal symmetry, so, we have λ = ±Nπ, where
3
N ∈ [0,6] is the number of sectors removed (‘-’) or added (‘+’), that is, positive and negative
disclination. Now we writing the Dirac equation for gapped graphene with a disclination in
the presence of an external magnetic field and a confinement potential, looking like
(cid:18) (cid:19)
∂ a
iγµ −iγµΓ +γµAC −γµeA −γµ µ Ψ = MΨ. (3)
∂xµ µ µ µ r
with γµ being the Dirac matrices defined in a curved space. The Γ term is a spinor
µ
connection which is present due to the curved nature of the geometry of the disclinated
gappedgraphenelatticeinanelasticcontinuouslimit. ThetermscontainedinDiracequation
have the following origin: the term Γ is the spinorial connection arisen due to the change
µ
in the geometry of graphene introduced by disclination, it is responsible for a non-trivial
holonomy due to the parallel transport of spinor in the geometry (2), and for the variation
4
of the local reference frame in this geometry, and produces a geometric phase that acts in
the sublattices A/B of graphene [23, 27]. The second term AC is the confinement potential
µ
given by (1). We note the presence of the term a = ±3(α − 1) which represents a non-
µ 2
Abelian gauge field related to the K spin flux, and contributing for the mixing of the Fermi
points K . The coupling with the potential vector A is responsible for the inclusion of
± µ
the Aharonov-Bohm flux piercing through the centre of the quantum ring and an uniform
magnetic field in the z-direction, given by
(cid:20) (cid:21)
Φ Br
(cid:126)
A = + e .. (4)
(cid:98)ϕ
2πr 2α
The matrices γµ in a curved space can be expressed as functions of triad fields eµ(x). In
a
this curved background frame, the Dirac matrices must be defined by γµ = eµ(x)γa and
a
satisfy the anticommutation relation {γa,γb} = 2ηab ,where ηab the usual Minkowski metric
with a signature ηab = diag(+,−,−). Moreover, the vielbein fields satisfy the relation
gµν = eµeνηab. So that we can write the triad matrix ea and its inverse eµ as
a a µ a
   
1 0 0 1 0 0
   
ea = 0 cosϕ −αrsenϕ, eµ = 0 cosϕ senϕ. (5)
µ   a  
   
0 senϕ αrcosϕ 0 −senϕ cosϕ
αr αr
From matrices (5), we can obtain the 1-form of connection ωa through the Maurer-Cartan
b
structure equation dea +ωa ∧eb = 0 (without torsion). Due to the symmetry of the defect,
b
the connections have only two non- zero components, ω2 = −ω1 = −(α − 1)dϕ. In this
1 2
way, the spinorial connection is described in terms of the spin connection, by the equation
Γ = iω Σab such that Σab = i(γaγb − γbγa). Hence, the only one non-zero spinorial
µ 4 µab 2
connection component is:
i
Γ = − (α−1)σ3. (6)
ϕ
2
Now, using the triads (5), we obtain the following equation
√
(cid:20) (cid:18) (cid:20) (cid:21) (cid:19)(cid:21)
∂ ∂ 2Ma (cid:112) (α−1)
iγt +iγr − 1 − 2Ma r γ0 + Ψ
2
∂t ∂r r 2αr
(cid:20) (cid:18) (cid:19)(cid:21)
1 ∂ φ eBr a
+ iγϕ +i AB +i +i ϕ Ψ = MΨ, (7)
αr∂ϕ r 2α r
5
with φ = Φ and Φ = 2π the quantum of the magnetic flux. We use a similarity
AB Φ0 0 e
transformation to eliminate the contribution arisen due to the spinorial connection in the
Dirac equation (7) [30]. We choose the following similarity transformation S(ϕ) = e−iϕσ3 to
2
change the representation of local Dirac matrices, (γr,γϕ) = γ¯j. Thus, we write the ansatz
for Ψ(cid:48) given by
 
ψ(r,ϕ)
Ψ(cid:48) = e−iEt−iϕ2σ3  (8)
χ(r,ϕ)
where, as it was mentioned above, ψ = ψ(r,ϕ) and χ = χ(r,ϕ) represent the sublattices A
and B respectively. From the similarity transformations and ansatz (8), we obtain the set
of equations:
√
(cid:20) (cid:21)
∂ 1 2Ma (cid:112)
(E −M)ψ = −iσ1 + + 1 + 2Ma r χ
2
∂r 2r r
(cid:20) (cid:21)
1 ∂ iφ ieBr ia
−iσ2 + AB + + ϕ χ (9)
αr∂ϕ r 2α r
and
√
(cid:20) (cid:21)
∂ 1 2Ma (cid:112)
(E +M)χ = −iσ1 + − 1 − 2Ma r ψ
2
∂r 2r r
(cid:20) (cid:21)
1 ∂ iφ ieBr ia
−iσ2 + AB + + ϕ ψ. (10)
αr∂ϕ r 2α r
Now,weeliminatetheχin(9)substituting(10)intoit,andobtainthefollowingsecond-order
differential equation:
√
∂2ψ 1∂ψ ψ 2Ma (cid:112) 2Ma
(E2 −M2)ψ = − − + − 1ψ + 2Ma ψ + 1ψ +2Ma r2ψ +
∂r2 r ∂r 4r2 r2 2 r2 2
√ √
√ 1 ∂2ψ σ3 ∂ψ 2Ma ∂ψ 2iσ3 2Ma ∂ψ
+ 4M a a ψ − +i −2iσ3 1 − 2 −
1 2 α2r2 ∂ϕ2 αr2 ∂ϕ αr2 ∂ϕ α ∂ϕ
φ ∂ψ iBe∂ψ 2ia ∂ψ
AB ϕ
− 2i − −
αr2 ∂ϕ α2 ∂ϕ αr2 ∂ϕ
√ √ √
2Ma 2Ma 2Ma (cid:112)
+ 2eBσ3 1ψ +2σ3a 1ψ +2σ3φ 1ψ +2σ3φ 2Ma ψ
2α ϕ r2 AB r2 AB 2
√
2Ma r2 (cid:112) eB a φ
+ 2eBσ3 2 ψ +2σ3 2Ma a ψ +σ3 ψ −σ3 ϕψ −σ3 ABψ
2α 2 ϕ 2α r2 r2
φ 2 φ a 2 2φ a a e2B2r2
AB AB ϕ AB ϕ ϕ
+ ψ +2eB ψ + ψ + ψ +2eB ψ + ψ. (11)
r2 2α r2 r2 2α 4α2
Then, to obtain the solution of the Eq.(11), we make the following choice for the function
ψ:
 
R (r)
ψ(r,ϕ) = eimϕ +  = eimϕRs(r), (12)
R (r)
−
6
where m = l + 1, is semi-integer with l = 0,±1,±2,... and R = (R (r),R (r)). In this
2 s + −
way we obtain the following set of equations
(cid:20) d2 1 d δ 2 ω 2M2 (cid:21)
+ − s − α r2 +(cid:15) R (r) = 0, (13)
dr2 rdr α2r2 4 s s
with parameters δ , ϑ , (cid:15) and ω defined as:
s s s α
(cid:112)
δ = ϑ +αs 2Ma ,
s s 1
(1−αs)
ϑ = (l+αφ )+ +αa ,
s AB ϕ
2
(δ +αs) (cid:16) ω (cid:17)
(cid:15) = E2 −M2 −M s ω s+ c ,
s 0
α α
(cid:114)
2ω ω s ω 2
ω = ω 2 + 0 c + c (14)
α 0 α α2
(cid:113)
with ω = 8a2 being the characteristic frequency and ω = eB is the cyclotron frequency.
0 M c M
(cid:113)
To solve the equation (13) we make the change of the variable ρ = ωα2M2r2 so that
4
afterwards our equation takes the form:
(cid:20) d2 d δ 2 ρ (cid:15) (cid:21)
s s
ρ + − − + R (ρ) = 0. (15)
dρ2 dρ 4α2ρ 4 2Mω s
α
In this way, we do the asymptotic analysis of Eq. (13), for the limits R → 0 and r → ∞,
s
it is possible to present radial equation in the following form:
−ρ |δs|
R (ρ) = e ρ F (ρ) (16)
s 2 2α s
Substituting this into the previous equation we obtain:
d2F (ρ) (cid:20)|δ | (cid:21) dF (ρ) (cid:20) (cid:15) |δ | 1(cid:21)
s s s s s
ρ + +1−ρ + − − F (ρ) = 0, (17)
dρ2 α dρ 2Mω 2α 2 s
α
where Eq. (17) is the hypergeometric equation whose solution is the hypergeometric func-
tion:
(cid:18) (cid:19)
|δ | 1 (cid:15) |δ |
s s s
F (ρ) = F (a,b;z) = F + − , +1,ρ . (18)
s 1 1 1 1
2α 2 2Mω α
α
Now to get a finite solution anywhere, we impose the condition where the solution of hy-
pergeometric series becomes a polynomial of degree n, that is: |δs| + 1 − (cid:15)s = −n from
2α 2 2Mωα
this equation, substituting the parameters (14), we obtain the energy spectrum for the par-
ticle confined in two-dimensional quantum ring pierced by Aharonov-Bohm quantum flux
in gapped graphene in the presence of the uniform magnetic field:
(cid:18) (cid:19)
|δ | (δ +αs) (cid:16) ω (cid:17)
E2 = 2n+ s +1 Mω +M s ω s+ c +M2, (19)
n,l α 0
α α α
7
wheres = +1correspondstosublatticeAands = −1correspondstosublatticeB. Notethat
in the spectrum, there is the dependence of the quantum numbers n and l, the parameters
a and a depend implicitly on frequency ω , also, the uniform magnetic field depends on
1 2 0
the cyclotron frequency ω , and, in addition, on the new frequency related to the topological
c
defect ω through the dependence on the parameter α . Moreover, the Eq. (19) has a
α
contribution of the non-Abelian gauge field a responsible for the Aharonov-Bohm effect of
ϕ
the spinors, due to the presence of the disclination.
(a)
Figure 1. Graphic of the behavior of energy in function of quantum number n for several values of
α. We have adopted the following values for parameters : B = 1T, M = 1, l = 1, φ = 1, s = 1
AB
and a = 9,0122.10−6eV.m2 and a = 2,222.10−17eV.m−2 are Tan -Inkson parameters [13].
1 2
Note that we can see in Fig. 1 that the for positive disclinations the energy eigenvalues
have magnitude greater than in the flat case, and that in the case of negative disclinations
the energy assumes smaller values for the same quantum number n.
The solutions with positive energy describe the dynamics of electrons in the conduction
band while negative energy corresponds to the dynamics of holes in the valence band. In this
paper we address only the first case. Also, as it was argued earlier, components of the spinor
8
in graphene describe the contributions of sub-lattices, with the real spins of the particles
are not considered within this approach. For equation (19), the spinors corresponding to
positive energy for s = +1 and η (r) = 0 are written as:
−
 
1
 
 0 
(cid:16) (cid:17) 
Ψ =f F −n, |δ+| +1, ωαMr2  
+ + 1 1 α 2  0 
 
 (cid:20) √ (cid:21)
 i 1[(ω +ω /α)+ω ]r+ (ϑ+−|δ+|+α 2a1) − ωcr 
E 2 0 c 0 αr 2α
 
0
 
(cid:18) (cid:19) (cid:16) (cid:17)0
+ if+ n(ω0+ωc/α)r F −n+1 , |δ+| +2 , ωαMr2  , (20)
E |δα+|+1 1 1 α 2 0
 
1
and for s = −1 and η (r) = 0 as
+
 
0
 
(cid:18) (cid:19) 1 
|δ | ω M  
Ψ = f F −n, − +1, α r2  (cid:20) √ (cid:21)
− − 1 1 α 2  Ei 12[(ω0 +ωc/α)+ω0]r− (ϑ−+|δ−α|−rα 2a1) + ω2cαr 
 
0
 
0
(cid:32) (cid:33)  
if n(ω +ω /α)r (cid:18) |δ | ω M (cid:19)0
+ − 0 c F −n+1 , − +2 , α r2  . (21)
E |δ−| +1 1 1 α 2 1
α  
0
Both in (20) and (21), the factor f = N e−iEt looks like
s s
 (cid:16) (cid:17) 1/2
fs =  ωαΓ |(cid:104)δαs|(cid:16)+n+1(cid:17)(cid:105)  ×e−iEtei(l+1/2)ϕe−ωα4Mr2(cid:18)ωα2M2(cid:19)|4δαs| r|δαs|. (22)
Γ(n+1) Γ |δs| +1 2 4
α
Note that at α = 1, the non-Abelian gauge field α goes to zero and the parameters δ , ϑ ,
φ s s
(cid:15) and ω tend to the values of the parameters λ , ξ , ε and ω respectively. In other words,
s α s s s
we recover the spectrum, persistent current, and spinors of positive energy for the case of
flat sheet without defect, i.e., α = 1 represents the absence of topological defect of the
gapped graphene layer. Moreover, if we switch off the magnetic field, i.e., make B = 0 and
by consequence ω = 0, we recover the case of a two-dimensional ring in a gapped graphene
c
layer, and in the case M = 0 we obtain the results which were considered in Ref. [21].
9
III. THE PERSISTENT CURRENT IN THE PRESENCE OF A DISCLINATION
Now we calculate the persistent current for our model of quantum ring in graphene layer
with a disclination. The persistent current carried by a given electron state is calculated us-
ingtheByers-Yangrelation[31]. Thisrelationexpressesthepersistentcurrentasaderivative
of the energy with respect to the magnetic flux
(cid:88) ∂E
n,l
I = − . (23)
∂Φ
n,l
In this way, the persistent current for this system is written as:
(cid:18) (cid:19) (cid:18) (cid:19)
∓e (cid:88) δ ω
s c
I = M ω + +sω ×
α 0
4π |δ | α
s
n,l
(cid:40) (cid:18) (cid:19) (cid:41)−1/2
(δ +αs) (cid:16) ω (cid:17) |δ |
× M2 +M s ω s+ c + 2n+ s +1 Mω . (24)
0 α
α α α
Where the persistent current is expressed as a function of the control parameters a , a
1 2
implicitly depending on frequency ω , the quantum numbers n and l implicitly depend on
0
the parameter δ , moreover the linear magnetic field is related to the cyclotron frequency
s
ω . In addition, we also observed the emergence of the correction term α . Note that,
c ϕ
the persistent current is a function of the parameter characterizing the curved geometry
introduced by a topological defect. The expression for the persistent current given by (24)
is a function of the non-Abelian magnetic flux and the Aharonov-Bohm flux on which the
parameter δ depends. The current is a periodic function of the Aharonov-Bohm flux Φ ,
s AB
and when the oscillations vanish for the large values of the Aharonov-Bohm flux, we can
observe this behavior in Figure 2.
We can compare our results with those ones obtained for quantum dots in graphene, and
the oscillatory behavior turns out to be similar to that one observed in Ref. [10].
IV. THEMAGNETIZATIONFORQUANTUMRINGINPRESENCEOFDISCLI-
NATION
Now, we investigate the magnetization for a quasiparticle in quantum ring in a graphene
sheet at zero temperature, in order to obtain an analytic expression for a magnetization M
of the form
∂E
n,l
M(B) = , (25)
∂B
10