To illustrate the method of calculating LDOS that we introduce in the above

contents we calculate LDOS in two cases of the step and smooth potential.

Besides we also compare our results with experimental ones. Figure 4.2 compares the experimental data (a) and the LDOS that we calculated (b). We

point out that the widths of the resonant peaks that are extracted from the cal-

19culated LDOS are almost compatible with the corresponding values obtained

by T -matrix method and describe quantitatively the experimental data. Especially, our result shows that the QBS of the lowest angular momentum, in

accordance with our expectation, is located at a higher energy level compared

to the theoretical calculations present in the experimental paper of Gutierrez

et al

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terms of kx and ky.
the Dirac point. By Taylor’s expansion of the energy E(k) around the Dirac
point with the assumption |q| K we get:
E(q) ≈ ±~vF |q|, (1.2)
where vF = 3ta0/2~ is the Fermi velocity. In the vicinity of the K or K ′ point,
the velocity of electron is equal to the Fermi velocity and does not depend
on the electron’s energy and momentum. In graphene, the Fermi velocity is
about 106 m/s or 1300c with c is the speed of light.
The existence of two sublattices A and B in graphene gives rise to a chiral-
ity in the kinetics of the carrier in graphene therefore two linear branches of
the graphene’s carrier dispersion near the Dirac points become independent
of each other. According to the first quantization language, the carriers in
graphene comply with the Dirac equation:
~vFσ · ∇Ψ(r) = EΨ(r) , (1.3)
where σ = (σx, σy) is the 2D vector formed by Pauli matrices. σ is called
quasi-spin. The wave functions of the carriers in the vicinity of the Dirac
points can be described in terms of two-component spinors that correspond
to the probability of finding carriers in one of two corresponding sublattices.
4
1.2 Graphene bipolar junction
Several experimental studies have developed graphene heterosjunctions in-
duced by local electrodes, also called local gates. In general, to create a GBJ,
one needs to create a design with two electrostatic gates. By changing the
gate voltage, Vb and Vt, independently of each other, one can create graphene
bipolar heterosjunctions in all possible charge density regimes: p-n-p, n-p-n,
p-p′-p, or n-n′-n.
O¨zyilmaz and coworkers studied the quantum Hall transport in graphene
n-p-n junctions and observed a series of fractional quantum Hall plateaus as
the local charge density varies in the p and n regions at large magnetic field
value. Huard et al. measured the resistance (R) across GBJs and reported
a noticeable asymmetry of the R versus Vt curves with respect to the max-
ima. Using a suspended ‘air-bridge’ top gate to avoid a decrease of the carrier
mobility in the region under this gate, Gorbachev et al. were able to fabri-
cate ballistic graphene p-n-p junctions. The Coulomb blockade in graphene
nanoribbon based bipolar junctions is found out in the single electron tran-
sistors.
1.3 Graphene quantum dot
Experimentally, the methods of fabricating QD using electrostatic potentials
to confine carriers have been studied. Zhao and coworkers created nanoscale
confinement areas in graphene in the form of the circular p-n junction using
a scaling tunneling microscope (STM). In the research of Lee et al., GQDs
are created using a technique related to the generation of local charge defects
inside the insulating substrate that is beneath the graphene monolayer film.
In the experiment that was carried out by Gutierrez et al., the CGQD was
created by a piece of copper substrate of the size of several tens of nanometers
causing a potential difference between the inside and the outside of the QD.
Theoretically, CGQDs are usually modeled by a radial confinement poten-
tial depending on the distance in the form of a step or an exponential form.
For pristine graphene with no energy gap, when there is no magnetic field,
the states of the carriers in the GQDs induced by an electrostatic potential
are generally quasi-bound states (QBS) with a finite trapping time instead of
bound states (BS), except for a few special cases in which BSs are observed.
In the case of graphene with an energy gap, researchers have shown that the
energy gap makes the life-time of the carriers longer.
The work of Chen and coworkers suggests that the trapping time of the
carriers in GQD increases with the smoothness of the exponential confinement
5
potential. The research by Lee et al. shows that the charge density of a
GQD measured experimentally varies smoothly at the boundary of the p-n
junction. This suggests that QD is induced by a smooth potential. One of
the objectives of this thesis is studying the GQD with the different shapes of
the confinement potentials such as trapezoidal, Gaussian potential in order to
better understand the dependence of states and the trapping time of electrons
in GQDs on the shape of the confinement potential.
1.4 Trapping carriers in graphene by a mag-
netic field
The effect of a magnetic field on graphene sheets can also give rise to BS
because of eliminating the Klein tunneling. De Martino and coworkers the-
oretically pointed out that a perpendicular heterogeneous magnetic field can
be used to create GQDs with the existence of BSs.
Another way to create a GQD, overcoming the obstacles caused by the
Klein’s paradox, is the use of the combination of the electric and magnetic
fields to confine the carriers in a certain confinement area. Giavaras and
coworkers examined the states of the massless Dirac particles in a GQD cre-
ated simultaneously by electric and magnetic fields. The other solution is to
create a GQD having the confinement states that are in the gap between the
Landau levels when QD is put in a magnetic field. Maksym et al. pointed
out that a CGQD can be induced by an electrostatic confinement potential
along with a plane asymptotic potential outside the QD which is placed in an
uniform perpendicular magnetic field.
6
Chapter 2
Calculation methods
2.1 Transfer matrix method
Transfer matrix (T -matrix) is widely used in problems in which electrons
follow the Schro¨dinger equation. In the case electrons obey the Dirac equation
and are affected by a smooth barrier, one can still apply effectively the T -
matrix method. The reason of this is, in principle, any smooth 1D potential
can be approximately considered as a series of the step potentials in which
the potential at each step can be considered constant. By multiplying T -
matrix elements obtained from the solution for this step potential one can
find an overall T -matrix. On the other hand, for each step potential, T -
matrix elements can be obtained from the solutions of the Hamiltonian on its
left and right sides (at that point, the potential is considered constant) by
satisfying an appropriate condition of continuity at the step interface.
In the next chapters, we will use and develop the T -matrix method for both
the graphene bipolar junctions and graphene circular quantum dots. For the
problem of electronic transport through a potential barrier in graphene, using
the obtained overall T -matrix, the transmission probability T through the
studied junction can be calculated as a function of the incident energy E and
the incident angle θ.
7
2.2 Methods of calculating the current, con-
ductance, shot noise and Fano factor
Consider a simple potential barrier shown in Fig. 2.1. One can calculate
the transmission probability T (E, θ) by T -matrix approach. At equilibrium
condition (V = 0), electrons obey the Fermi distribution, the left and the
right hand sides of the barrier have the same Fermi energy µ0. When the bias
V is applied, each side of the barrier has its own Fermi energy, µL and µR
corresponding to the left and the right, respectively. The difference between
them is |µL − µR| = eV .
Figure 2.1: The barrier and a Fermi sea of electrons in an ideal case when the
1D potential bias V is small.
Consider a graphene system under the effect of a small bias. The current
at the linear region I–V is as follows:
I =
geW
h2vF
|(µ0 − UL)eV |
∫ pi/2
−pi/2
dθT (θ) cos θ. (2.1)
Then one can calculate the conductance G = I/V .
A quantity that is usually considered in the ballistic transport is the Fano
factor F . It is defined as ratio of the actual shot noise power S and the
Poissonian noise SP ,
F = S
SP
=
S
2eI
. (2.2)
The shot noise is a consequence of the charge quantization. The Poissonian
noise is a noise produced when the charge carriers are uncorrelated.
The general formula of the bias-dependent shot noise power was derived
by Buttiker for a 2D system in the case of a continuous spectral near 0 K:
S = 2
ge2W
h2vF
∫ µL
µR
dE |E − UL|
∫ pi/2
−pi/2
dθT (E, θ)[1− T (E, θ)] cos θ . (2.3)
8
Chapter 3
Graphene n-p-n junction
with the Gaussian
potential
3.1 Model of n-p-n junction
Structures under study are schematically drawn in Fig.1(a), where L is the
top gate length and the x-axis is directed along the graphene stripe with the
origin (x = 0) located at the middle of the top gate (see Fig.1(c)). Along the
(not shown) transverse y-direction the width W of the stripe (and the gates)
is assumed to be large compared to the top gate length. The voltages Vb and
Vt applied respectively to the back gate and the top gate induce in the stripe
a total potential which is considered to be constant along the y-direction and
varied along the x-direction as
U(x) = U21e
−x2/αL2 + U1 , (3.1)
where U21 = U2 − U1.
The massless Dirac-like Hamiltonian takes the form
H = −~vF~σ · ~∇ + U(x)I , (3.2)
where ~σ = (σx, σy) are Pauli matrices, I is the identity matrix. The T -matrix
approach is used to solve the Dirac-like equation with a smooth 1D potential.
9
SiO2
graphene
n++ Si
Ti/Au
Back gate
Top gate
PMMA(a)
Figure 3.1: (a) Scheme of GBJs under study; (b) Diagram of junction charge
density regimes (n for electron and p for hole); (c) Three potential barrier
models are in comparison: rectangular (dash-dotted line), trapezoidal (dotted
line), and Gaussian-type (solid line); and (d) Several potential profiles of
eq.(3): L = 20 nm, Vb = 40 V , and Vt (from top) = −4 V,−3 V,−2 V and
0.1 V .
3.2 Klein tunneling
We present T (θ) dependences in Figure 3.2 for the GBJs modeled by either
rectangular potentials or Gaussian-type potentials. Both solid and dashed
lines show a complete transparency of barriers, T = 1, for the normal inci-
dence θ → 0. This is a typical manifestation of Klein tunneling, regardless of
the barrier shape as well as the barrier size. Nevertheless, the highly transmit-
ted angular region for the Gaussian-type potential (solid lines) is always con-
siderably narrower than that for the corresponding rectangular one (dashed
lines). The root cause of such a difference in the highly transmitted angu-
lar region between the two models is the smoothness of the Gaussian-type
potential.
3.3 Resistance
Figure 3.3(a) presents calculated resistances R = 1/G plotted against the top
gate voltage Vt at different back gate voltages:
For a given Vb, the resistance R strongly oscillates with a slightly increasing
average value as Vt increases in the region of Vt < V
(c)
t , when n1n2 < 0, i.e.
when the studied junction remains in the n-p-n regime. Crossing the last
10
1
30o
60o
0
90o
−90o
−60o
−30o
0o
1 (c)
1.0
30o
60o
0
90o
−90o
−60o
−30o
0o
0.8
0.6
0.4
0.2
0.2
0.4
0.6
(d)
0.8
1.0
1.0
30o
60o
0
90o
−90o
−60o
−30o
0o
1.0
0.4
0.2
0.2
0.4
0.6
0.8
0.6
0.8
(a)
1
30o
60o
0
90o
−90o
−60o
−30o
0o
(b)1
1
30o
60o
0
90o
−90o
−60o
−30o
0o
1 (e)
1
30o
60o
0
90o
−90o
−60o
−30o
0o
1 (f)
Figure 3.2: Polar graphs depicting T (θ) for GBJs in rectangular potential
model (dashed blue curves) and Gaussian-type potential model (solid red
curves): the outmost semicircle corresponds to T = 1 and the center to
T = 0. T (θ)-graphs are shown for various values of parameters of [L (nm),
E (meV), Vb (V), Vt (V)]: (a) [25, 0, 60, −12]; (b) [25, 50, 60, −12]; (c) [50,
50, 60, −12]; (d) [25, 0, 40, −6]; (e) [25, 50, 40, −6]; v (f) [50, 0, 40, −6].
maximum at Vt ≈ V (c)t , the junction enters the n-n′-n regime, where the
structure becomes much more transparent and consequently the resistance
experiences a sharp reduction at Vt > V
(c)
t . In the range of Vb under study
Fig. 3.3(a) also shows that an increase of Vb leads to a decrease of not only
the transition voltage V
(c)
t , but also the average resistance in both regions,
Vt < V
(c)
t v Vt > V
(c)
t . Overall, the calculated R(Vt)- dependence shown
in Fig. 3.3 describe quite well the experimental data reported. Observed
oscillations of R versus Vt can arise from the oscillations of the transmission
probability caused by the inside-barrier interference of chiral waves.
11
ĐIỆN THẾ CỦA TOP GATE Vt (V)
H
Ệ
SỐ
F
AN
O
F
Đ
IỆ
N
T
R
Ở
R
Figure 3.3: Resistances R (a), odd resistance 2Rodd (b), and Zero-bias Fano
factors F (c) are plotted versus Vt for three cases with Vb = 40 V (dash-dotted
red lines), 60 V (solid blue lines), and 80 V (dashed green lines). Arrows
indicate the transition top gate voltages V
(c)
t where the transition between
n-p-n and n-n′-n regimes occurs (V (c)t = −2.59 V,−5.39 V , and −8.19 V for
Vb = 40 V, 60 V , and 80 V , respectively)
3.4 Current shot noise
In order to study the currentvoltage (IV) characteristics, it is assumed that
a symmetric bias [+eVsd/2,−eVsd/2] is applied to the two leads (source and
drain), linked to the structure under measurement. Figure 3.4(a) shows the IV
curves for three different GBJs. In general, by gradually rising the bias voltage
Vsd, starting from Vsd = 0, the current I first increases progressively, then
experiences a slowing down at some bias voltage, which mainly depends on
the back gate voltage Vb. Crossing this bias voltage, currents weakly fluctuate
and even go through a slightly negative differential resistance (NDR) region.
Calculations reveal a close association between the observed NDR and the
bias-dependence of the transmission probability. In any case, examining the
IV curves for a number of GBJs with different values of parameters, including
Vb, Vt, L, and W also, shows that within the model of interest the NDR effect
is always rather weak.
Figure 3.3(c) represents the Fano factor as a function of top gate Vt when
12
D
Ò
N
G
Đ
IỆ
N
I
ĐIỆN THẾ BIAS Vsd (mV)
H
Ệ
SỐ
F
AN
O
F
Figure 3.4: (a) Current-voltage and (b) Fano factor-voltage characteristics for
three junctions with [L(nm), Vb(V ), Vt(V )] = [25, 35,−6] (solid blue lines),
[25, 40,−6] (dashed red lines), and [50, 40,−3.5] (dash-dotted green lines).
The bias voltage Vsd is symmetrically applied to the source and the drain.
the bias is equal 0. Within our potential model, the zero-bias Fano factor is
depicted in Fig. 3.3(c) as a function of the top gate voltage Vt.
While the observed harmony of oscillations in resistance R (Fig. 3.3(a))
and in Fano factor F (Fig. 3.3(c)) of the same GBJ is quite well understood,
the fact that all three curves for various GBJs in Fig. 3.3(c) exhibit practically
the same maxima of 0.36 and the same minima of 0.08 causes a little surprise.
In any case, this value of F = 0.36 is rather close to the experimental value
of 0.38.
Thus, our model provides the shot noise with F ≈ 0.36 for n-p-n GBJs
in the linear regime. A question might then arise of whether the bias voltage
which modifies the potential barrier and changes the transmission probabil-
ity can enhance the noise or even cause a super-Poissonian noise as it did in
conventional semiconductor/metal nanostructures. To shed light on this ques-
tion, we show in Fig. 3.4(b) the F(Vsd) characteristics for the same GBJs with
I(Vsd) characteristics presented in Fig. 3.4(a). Actually, Fig. 3.4(b) demon-
strates that for a given junction, in accordance with the current fluctuation
in Fig. 3.4(a), the Fano factor, starting from the value F(Vsd = 0), fluctuates
against the bias between the values ∼ 0.18 and ∼ 0.25.
13
Chapter 4
Circular graphene
quantum dot induced by
electrostatic potential
4.1 The T -matrix method for circular graphene
quantum dot
We consider a single-layer CGQD defined by the radial confinement potential
U(r) that is assumed to be smooth on the scale of the graphene lattice spacing.
Using the units such that ~ = 1 and the Fermi velocity ~ = 1, the low-energy
electron dynamics in this structure can be described by the 2D Dirac-like
Hamiltonian
H = ~σ · ~p+ ν∆σz + U(r), (4.1)
where ~σ = (σx, σy) are the Pauli matrices, ~p = −i(∂x, ∂y) is the 2D momentum
operator, ν = ±1 is the valley index for the valleys K and K ′ respectively,
and ∆σz is a constant mass term. This term is due to the interaction between
the graphene sheet and the substrate.
Assume that Ψ(r, φ) is an eigen-function corresponding to the energy E
Ψ(r, φ) = eijφ
(
e−iφ/2χA(r)
e+iφ/2χB(r)
)
, (4.2)
where the total angular momentum j takes half-integer values and the radial
14
spinor χ = (χA, χB)
t satisfies the following equation:(
U(r)− E + ν∆ −i(∂r + j+
1
2
r )
−i(∂r − j−
1
2
r ) U(r)− E − ν∆
)(
χA(r)
χB(r)
)
= 0. (4.3)
Consider a particular case of GQD defined by an electrostatic potential of
the form:
U(r) =
Ui, r ≤ ri,Uf , r ≥ rf ,
bt k, r cn li.
(4.4)
We consider some region ra < r < rb, where the potential U(r) is constant,
U(r) = U¯ . For E 6= U¯ ± ν∆, the general solution to equation (4.3) in this
region can be written in terms of two independent integral constants C =
(C(1), C(2))t
χ(r) = W (U¯ , r)C , (4.5)
W (U¯ , r) =
(
Jj− 12 (qr) Yj− 12 (qr)
iτJj+ 12 (qr) iτYj+
1
2
(qr)
)
, (4.6)
where q =
√
(E − U¯)2 −∆2 and τ = q/(E−U¯+ν∆). In which Jj± 12 and Yj± 12
are the Bessel function of the first and the second kind. In case E → U¯ ± ν∆,
the basic solutions of the equation (4.6) become divergent. For simplicity, we
assume that E 6= U¯ ± ν∆ and consider the case E = U¯ ± ν∆.
The spinors at r1 and r2 should be linearly related by a matrix G:
χ(r2) = G(r2, r1)χ(r1). (4.7)
When we represent the spinors at r1 ≤ ri and r2 ≥ rf by the basic coefficients
Ci and Cf , these coefficients are related:
Cf = TCi , (4.8)
T = W−1(Uf , r2)G(r2, r1)W (Ui, r1) . (4.9)
By replacing the equation (4.7) into the equation (4.3) we get
i
∂G(r2, r1)
∂r2
= H(r2)G(r2, r1) , (4.10)
where the Hamiltonian is assumed by
H(r) =
(
i
j− 12
r U(r)− E − ν∆
U(r)− E + ν∆ −i j+ 12r
)
. (4.11)
15
This equation needs to be solved for G(r2, r1) with the initial condition such
that G(r1, r1) is the (2 × 2) identity matrix. In fact G(r2, r1) can be solved
by a relevant numerical method such as Runge-Kutta method. After getting
G(r2, r1), the equation (4.9) allows us to calculate the T -matrix.
4.1.1 Bound states
The bound states are the states of the carriers affected by a potential but these
carriers tend to localize in a given region. The general equation to determine
the energy spectrum of all the bound states in the considered energy regions
for a GQD is given by
T11 + iT21 = 0. (4.12)
4.1.2 Quasi-bound states
Electron can be confined temporarily at the quasi-bound states (QBSs) with a
finite lifetime. Each QBS is characterized by a complex energy E = Re(E) +
i Im(E) with the imaginary part Im(E) < 0 relatively small. This part plays a
role of a disturbance. Re(E) defines the position of the QBS (i.e. the resonant
level). |Im(E)| is a measure of the resonant level width and its inverse is a
measure of the carrier lifetime at the QBS, τ0 ∝ 1/(2|Im(E)|).
The general equation to determine the energy spectrum of the QBSs in
CGQD is as follows
T11 + isT21 = 0. (4.13)
4.1.3 Density of states
By using T -matrix we can easily get LDOS. With each angular momentum j,
LDOS can be calculated as
ρ(j)(E) ∝ |E|
pi
1∣∣∣T (j)11 ∣∣∣2 + ∣∣∣T (j)21 ∣∣∣2 . (4.14)
By summing Eq. (4.14) over j, we obtain the total LDOS:
ρ(E) =
+∞∑
j=−∞
ρ(j)(E) . (4.15)
16
4.2 Graphene quantum dot defined by a radial
trapezoidal potential
Consider a CGQD induced by the radial trapezoidal potential
Ui = U0, ri = (1− α)L,
Uf = 0, rf = (1 + α)L,
v U(r) = Ui +
r−ri
rf−ri (Uf − Ui) vi ri < r < rf .
This potential becomes the rectangular potential in the limiting case of α = 0.
l
l
l l
l
l
l
l
l
l
l
l
l
l l
l
l
l
l l
l
l
ll
l
l
l
ll l
l l
l
l
l
l
l
lll
l
l
l
ll
l
ll l
ll
l
l l
l
l
l
l
l
l
l
l
l ll
ll ll
l
l lll
l
l
l
lll
l
l
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l
l
l l
l
l
l
l
l
l
l
l
ll
l
l
l
l
l
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l
l
l
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ll
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l
l
l
l
l
l
l
l
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l
l
l
l
l
l
l
l
l
l
l
ll
l
ll
l
l
l
l
l
l
0 10 20
0.25
0.50
Re[E]
−
Im
[E
]
(a)
0 10 20
E
LD
O
S α = 0.3
α = 0.5
α = 0.7
(b)
Figure 4.1: QBS spectra (a) and LDOS (b) of a GQD induced by the trape-
zoidal radial potential of L = 1 and U0 = 20 are presented for ν = +, j =
3
2
and various α. In (a): 5 curves correspond to 5 QBS levels, each describing
how the QBS energy (Im(E) and Re(E)) changes as α varying regularly from
0.3 (top) to 0.7 (bottom) from larger point-sizes to smaller point-sizes.
Figure 4.1(a) shows how the complex energies of five different QBSs change
as the smoothness α varies from 0.3 to 0.7. With increasing potential smooth-
ness α, while the real part of the energy just changes slightly, the imaginary
part decreases substantially. This result proposes an approach of creating the
states with a long lifetime and can be manipulated such that they can satisfy
the requirements of manufacturing electronic devices. Figure 4.1(b) repre-
sents LDOS (in arbitrary unit) for three spectra with the α values examined
in figure 4.1(a) to compare the correlation between QBSs and LDOS. The
positions of QBSs in (a) and the resonant peaks of LDOS in (b) are in a good
17
agreement. Moreover, the imaginary parts of the QBS energies represent the
widths of the corresponding LDOS peaks quite well.
4.3 CGQD induced by an arbitrary axially sym-
metric electrostatic potential
4.3.1 Method for calculating LDOS from normalized wave
functions
The eigen function Ψ(E)(r, φ) has the form which is defined by the equation
(4.2). The wave function χ(E,j)(r) = (χ
(E,j)
A (r), χ
(E,j)
B (r))
T obeys the equa-
tion:
i
∂χ(E,j)(r)
∂r
= H(r)χ(E,j)(r), (4.16)
where the Hamiltonian H(r) =
(
i
j− 12
r U(r)− E
U(r)− E −i j+ 12r
)
. Because of the axi-
ally symmetric potential, LDOS only depends on the radial coordinate r:
ρ(E, r) =
+∞∑
j=−∞
ρ(j)(E, r), ρ(j)(E, r) ∝ 1
∆E
‖χ(E,j)(r)‖2, (4.17)
ρ(j)(E, r) ∝ 1
∆E
‖χ(E,j)(r)‖2, (4.18)
where ∆E is the level spacing at the energy E and χ(E,j)(r) is the normalized
wave function.
For r ≥ rf , the wave function can be represented in term of two integral
constants Cf = (C
(1)
f , C
(2)
f )
T : χ(E,j)(r) = Wf (r)Cf , where
Wf (r) =
(
Jj− 12 (qfr) Yj− 12 (qfr)
iτfJj+ 12 (qfr) iτfYj+
1
2
(qfr)
)
, (4.19)
Because the wave function is equal to 0 at r = L, we can find the level spacing:
∆E = piL .
The normalization condition can be found in the form:
4L‖Cf‖2
|E−Uf | = 1. The
eigenfunction of the equation (4.16) near the center of QD has the form
χ(E,j)(r) = N
(
Jj− 12 (qir)
iτiJj+ 12 (qir)
)
, (4.20)
18
N
ăn
g
l
ng
ượ
N
ăn
g
l
ng
ượ
Figure 4.2: (a, b) LDOS of CGQD with R0 = 5.93 nm, V0 = 0.43 eV (a back-
ground correction leads to a shift of ED = −0.347eV in the energy position.):
(a) Experimental data; (b) Calculated results using the present approach; (c)
Two TDOSs calculated from the data in (a) (dashed) and (b) (solid line) [log
scale, arbitrary unit]. The resonances are labelled by their angular momen-
tum. Panels (d−f) compare the partial LDOS for the state of j = 12 : (d) from
the experimental paper; (e) Eq. (4.18) without normalization; (f) Eq. (4.18)
with normalization.
with qi = |E − Ui|, τi = sign(E − Ui) and N is the normalization coefficient.
Then we take the solutions given in the equation (4.20) at ri as the initial
values and solve the equation (4.16) for χ(E,j)(r). With the normalized wave
function, we can calculate LDOS using the Eq. (4.17).
4.3.2 Comparison with experiment
To illustrate the method of calculating LDOS that we introduce in the above
contents we calculate LDOS in two cases of the step and smooth potential.
Besides we also compare our results with experimental ones. Figure 4.2 com-
pares the experimental data (a) and the LDOS that we calculated (b). We
point out that the widths of the resonant peaks that are extracted from the cal-
19
culated LDOS are almost compatible with the corresponding values obtained
by T -matrix method and describe quantitatively the experimental data. Es-
pecially, our result shows that the QBS of the lowest angular momentum, in
accordance with our expectation, is located at a higher energy level compared
to the theoretical calculations present in the experimental paper of Gutierrez
et al.
4.4 CGQD in the magnetic field
In this section we develop the T -matrix approach for CGQD induced by an
axially symmetric electrostatic potential and subjected to a perpendicular
uniform magnetic field.
4.4.1 Wave function form
When an uniform magnetic field is applied on the QD, the Hamiltonian of the
Dirac fermion has the form:
H = vF~σ ·
(
~p+
e
c
~A
)
+ ν∆σz + U(r), (4.21)

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