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Physics Procedía 54 (2014) 118 - 126

International Conference on Magnetic Materials and Applications, MagMA 2013

Interband Transition Energy of Circular Quantum Dots under

Transverse Magnetic Field

Arpan Deyasi , Swapan Bhattacharyya

aRCC Institute of Information Technology, Beliaghata, Kolkata, INDIA-700015 bCamellia Institute of Technology and Management, INDIA-712134

Abstr act

Eigenstates of two different circular quantum dots (quantum ring and quantum disk) are analytically computed by solving time-independent Schrödinger equation in presence of transverse magnetic field. Intraband transition energy is calculated for the lowest three states. Dimensions and magnitude of applied field are tuned to observe the effect on eigenstates and transition energies. Comparative studies for both eigenvalue and transition energies are made between the devices with similar size and under equal applied field. Results show that transition energy decreases with increasing device diameter, and increases with increase of magnetic field. Change in transition energy indicates the possibility of wavelength tuning by magnetic field for optical emitter/detector applications.

© 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).

Peer-review under responsibility of Department of Physics, Indian Institute of Technology Guwahati

Keywords: Quantum ring; Quantum disk; Transverse magnetic field; Eigenenergy; Intraband transition energy; Wavelength tuning

1. Introduction

Semiconductor nanostructure has already drawn great interest among researchers due to its immense possibility for making novel electronic [1-2] and photonic [3-4] devices. Device dimensions are comparable to de-Broglie wavelength of electrons, which makes quantization of energy levels by restricting carrier motion, and the device is nomenclature as quantum well/wire/ dot depending on number of quantized dimensions. Among them, quantum dot has gained much interest due its quantization along all three spatial directions. Due to the rapid advancements in microelectronics technology [5-6], quantum dots are now practically realizable with various shape, size and materials [7-8]. Circular quantum dots, in recent days has been investigated a lot due to their promising nature of making optical emitter [9], detector [10], transistor [11], modulator [12], switch [13], quantum computer [14]. Quantum ring and disk are the two circular dots which can experimentally demonstrate Ahranrov-Bohm effect [15], one of the best examples of quantum-mechanical phase coherence.

* Corresponding author. Tel.: +91-9831445343; fax: +91-33-2323-2463 . E- mail address: deyasi_arpan@yahoo.co.in

1875-3892 © 2014 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).

Peer-review under responsibility of Department of Physics, Indian Institute of Technology Guwahati doi: 10.1016/j.phpro.2014.10.047

A solid circular-shaped quantum dot may be termed as quantum disk, as shown in Fig 1a. Quantum ring is a modified disk structure where one air-hole is inserted in the centre of the disk, shown in Fig 1b. A tailorable quantum disk/ring having cylindrical geometry has attracted a lot of researchers as its eigenstates can suitably be controlled by varying dimensions as well as by changing external electric/magnetic/crossed field. Since electronic and optoelectronic properties of these structures are dependent on size and material parameters, which, in turn, depends on eigenstates, hence computation of eigenenergy becomes very essential.

Ground state energy of quantum disk is computed without any perturbing potential [16] by effective-index method. Cyclotron transition energies are calculated [17] for the same device in presence of magnetic field. Energy was also calculated when device is subjected to parallel magnetic field [18] for quantum ring and quantum disk [19]. Energy levels of single electron and two-electron quantum rings are calculated in presence of axial magnetic field [20-21]. Quantum disk is already proposed for making DFB [22] with possibility of very accurate control of threshold current. Novel LED can also be designed using heterostructure quantum [23]. Quantum ring is also considered as a promising candidate for designing quantum laser with various geometrical shapes [24-25].

In the present paper, eigenenergies and intersubband transition energies of two different circular quantum dots (quantum ring and disk) are analytically calculated for various structural parameters. Externally applied transverse magnetic field is also varied to observe the effect on eigenstates. Comparisons are made between the devices having similar size and materials. Results are important in designing optical emitter/detector.

2. Mathematical Modeling

We consider the schematic structure of the cylindrical quantum disk shown in Fig. 1a with the assumption that a magnetic field (B) is applied in the plane of the disk.

Figure 1. Schematic structure of a cylindrical quantum disk under transverse magnetic field; (b) quantum ring

The effect of magnetic field appears through potential operator ( A ). Using the cylindrical co-ordinate system,

vector potential components ( p ,A ,Az ) may be written as

Az = Bp sin e

The time-independent Schrödinger equation for the electron wavefunction ^ can be written as

_L A ( d^l + _L 5 2 y + d2 W pdp I P dp J p2 dO2

dW e 2 B2 +-z— p sin e-— +--p2 sin 2 = E w

dz 2 m

for 0 < p < b and 0<z < d.

This equation is now solved by perturbation technique, assuming that the magnetic field is not very large to have

quantization effect. We can break up the Hamiltonian operator (H) into two part unperturbed operator 0 and perturbing operator H as

H=й + h '

The unperturbed Hamiltonian satisfies the equation

for 0<p<b and z<d

The perturbing Hamiltonian operator is given by

-, ieBñ e2B2 2 ■ 2 „ й =—— psine—+-— p sin2 в

* / -ч _ * /

m dz 2 m

Applying the boundary conditions at z=0 and z=d, Ei™ can be obtained by solving determinant equation

J (-2 C3'2) J (1 Г3'2) J (-2 C3/2) J (-2 C3/2)

where ^1(^2) corresponds to the value of C, at z=0 (z=d).

Using the expressions obtained earlier the p-dependent component of the wavefunction, R, can be determined by solving the differential equation

2 d2R dR

dp2 dp

for a<p<b, where

+ — + (X-p2 - m )R = 0

,2 2m (E0 - Ezn ) ^ =-^^-

Eq. (7) is a Bessel differential equation of order m, the general solution of which is given by

R = AJm (,) + BYm (,)

However, with the application of boundary conditions, R=0 at p=a (p=b), it can be seen that only some discrete values of X and Elmn are allowed. These discrete values can be obtained by solving the following determinants

Jm (X) ^т (X) Jm (X) Ym (X2)

where A,1(A2) corresponds to the value of X at p=a (p=b).

The solution of (10) requires two quantum numbers, n1 and n2 (say), which indicates the n2-th zero (solution) of the n1-th order Bessel function. The solution of Eq. (6), however, gives rise to a single quantum number (n0, say). Thus, the unperturbed energy eigenvalue E0 can be obtained using Eq. (7) and, thus, the energy eigenvalue is quantized with three quantum numbers (l,m,n).

The solution of Eqs. (6) and (10) are obtained using the exact series representation of the Bessel functions. The exact eigenvalue can be approximated by operating H on the wavefunction Y0 and subsequently using the relation

E'=En + Hi,

where,

<p2 >=

nd 2 Jm+1 (^ml )

2 | aml

Using various substitutions and simplifications, the energy eigenvalue of electron in the disk is written as

lmn 2m *

+ —— (nn /d )2

2m *V 1

1 eiBl

2 2m *

In a similar manner, we consider the schematic structure of the cylindrical quantum ring shown in Fig. 1(b). The energy eigenvalue of electron in the quantum ring with approximation of not very large B is given by

w h2 2 h2 I A2 1 e2 B2

E =-Á mi +-(nn/d) +--

2m * 2m * 2 2m *

<P¿ >

^TÍ Jl ttml Pp dP

3. Results and Discussion

Using Eq. (14) and Eq. (16), eigenenergies of quantum disk and quantum ring are computed and plotted as function of outer diameter taking n-GaAs as an example. In Fig 2, lowest three confinement states of the circular dots are plotted in presence and absence of magnetic field. It may be seen from both the plots that energy decreases monotonically with increase of diameter. This is because larger disk size decreases the quantum confinement which makes reduction of energy. Since quantum ring is a heterostructure (composition of air and semiconductor material) device, hence the change of confinement is more for the structure, which makes higher rate of reduction of eigenenergy for ring than that of disk. Also the magnitude of eigenstates becomes higher in presence of magnetic field when it is compared with the value calculated in absence of the same for a particular value of thickness keeping other dimensional parameters constant. This leads to the conclusion that presence of transverse magnetic field enhances the eigenenergy. The effect of magnetic field is clearly distinguishable for larger device dimension, and for higher eigenstates.

Fig 2a: Lowest three energy states with outer diameter for quantum ring in presence and absence of magnetic field

Fig 2 b: Lowest three energy states with outer diameter for quantum disk in presence and absence of magnetic field

In Fig 3, lowest three energy states of quantum ring and disk are plotted as a function of outer diameter in presence of transverse magnetic field. From the plot, it may be seen that for a particular value of diameter, eigenvalue of ring is higher than that of the disk when both the devices are subjected to similar magnetic field and other dimensions are kept same. This is because in case of ring, the air in the core region makes quantum confinement higher, which increases the energy. The difference may safely be neglected when device size is extremely large, and the dimension of hole of the ring is small compared to its outer radius.

Fig 3: Comparative study of eigenenergy with outer diameter of the ring and disk in presence of transverse magnetic field

Keeping device size constant, lowest three energy states of disk and ring are computed and plotted with thickness, as shown in Fig 4. It is observed from the plot that energy monotonically decreases for both the structures with increase of thickness. For any eigenstate, magnitude of energy is higher for ring than that of the disk due to the same reason discussed earlier. However, the notable feature is that the rate of decrease of energy for any confinement state is same for ring and disk. This is due to the fact that both ring and disk are vertically symmetric as far material is concerned. It may also be observed from the plot that as we consider higher energy states for both ring and disk, energy difference between them decreases, i.e., energy gap between E111 for ring and disk is higher than that between En3 due to the less effect of inner core layer (air) on quantum confinement.

Fig 4: Comparative study of lowest three energy states with thickness of the ring and disk in presence of transverse magnetic field

When comparison is performed with increasing the magnitude of applied magnetic field between the devices keeping all the dimensions same, it is seen that energy monotonically increases, as plotted in Fig 5. The rate of

increment is high for ring than that of the disk. It may be seen from the plot that eigenenergy is higher for ring than that of the disk for a given value of magnetic field. The difference of energies for a given energy state Elmn is more distinguishable with larger value of magnetic field. Lowest three energy states are plotted for comparative analysis.

Fig 5: Comparative study of lowest three energy states for quantum ring and disk with magnetic field

Intersubband transition energy AEj,i represents transition between ith state to jth state. For example, AE2,1 signifies a transition from ground energy level (Em in this case) to first excited state (En2). In Fig 6, transition energy is plotted with outer diameter for ring and disk. It may be observed form the graph that with increasing outer diameter, transition energy monotonically decreases. The reason can be explained as follows: increase of diameter decreases the confinement, which reduces energy for both types of circular dots. The separations of energy states are less for ring than the disk, which gives higher intersubband transition energy of quantum disk.

a=10(nm)-QR

d=20(nm)-QD,QR

\ --------Disk ^>-aE21

s Ring AE3,

- s* N [B=2T AE32

•-*—

->---1

--O-O-O-O-o-

■ 1 1 1 I .... I ... .

1 1 ■ ■

30 b(nm)

Fig 6: Comparative study of intersubband transition Energy with outer diameter of the ring and disk in presence of magnetic field

a=10(nm)-QR

b=40(nm)-QD,QR

d=20(nm)-QD,QR

-QDisk _QRing

Fig 7: Comparative study of lowest three transition energies states for ring and disk with magnetic field

Fig. 7 shows the variation of intraband transition energy with transverse magnetic field for quantum disk and ring having same outer dimension and material composition. It is seen for the plot that disk has higher transition energy that is clearly distinguishable with higher field. This is because the inner core layer (air) of ring increases the quantum confinement of the ring than the disk, which results higher eigenenergy of ring with little separation. This reduces the transition energy of ring.

4. Conclusion

Lowest three eigestates and corresponding intersubband transition energies in n-GaAs quantum disk and quantum ring are analytically computed and plotted with various device dimensions and strength of magnetic field. Eigenenergy decreases with increasing radius and thickness, and reduces with increasing the magnitude of applied transverse magnetic field. This results change in subband energy. Results suggest that for similar device dimensions, energy of ring is higher than that of the disk, but transition energy of disk has higher magnitude because of the wider separation between eigenstates. However, transition energy for both the devices remains invariant with thickness. The variation of intraband energy indicates the possibility of wavelength modulation by magnetic field.

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