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© 2019 IJRAR June 2019, Volume 6, Issue 2 www.ijrar.org (E-ISSN 2348-1269, P- ISSN 2349-5138)

IJRAR1ARP026 International Journal of Research and Analytical Reviews (IJRAR) www.ijrar.org 143

HOLE-HOLE INTERACTION IN QUANTUM

WIRE: EFFECT OF CROSS-SECTIONAL

GEOMETRY AND DIELECTRIC SCREENING R. Vijaya Shanthi1

1Department of Physics, Parvathy’s Arts and Science College,

Begambur, Dindigul624002, Tamil Nadu, India.

Abstract: The effect of various cross-sectional Geometries Gi (Lx, Ly) of the GaAs /Al0.3Ga0.7As Quantum wire like G1: (L, L), G2: (L, L/2), G3: (L,

L/4) on the acceptor binding energy and hole – hole interaction of double acceptor has been investigated through effective mass

approximation using variational principle. The observations were made including the influence of spatial dielectric screening at T=300K

and a comparative study is made between single and double acceptor energy. We observe that the reduction of geometry and effect of

screening enhances interaction energy as well as impurity binding energy effectively for smaller wire size.

Keywords: Geometry, Spatial dielectric Screening, Double acceptor, Coulomb interaction energy, Hole – Hole interaction, Quantum Wire

1. INTRODUCTION

Quantum Wires, owing to their nature of eigen states, eigen energies and superior device applications have created immense interest

in research [1]. Quantum Wires are best suited for laser action and display applications. Though GaAs/AlxGa1-xAs multilayer materials

system has dominated the new physics and devices, their unexplored applications are innumeric.Earlier studies made on donor impurity

states in Quantum Wires [2, 3] unveil the behaviour of the donor impurity in different geometries. Recently studies on acceptor impurities

in GaAs/AlxGa1-xAs heterostructure based nanostructures offer a new openup to researchers due to their hole effective mass (relatively

4.5 to 7.5 times larger than the electron mass) which increases the coulomb interactions in many body problem [4, 5, 6]. Unlike donor

states which are simple, parabolic and non- degenerate, acceptor states are complicated with degeneracy in the valance band edges. This

degeneracy is lifted by quantum confinement but the splitting produced by quantum confinement is usually small [7]. The acceptor bands

are anisotropic and non-parabolic. But in the present work, we pursue our study with parabolic approximation and go in for simple

analytic results. Following the first experimental realization of acceptor impurity states by Miller et.al[8], some investigations were made

on the acceptor binding energy in Quantum Well (QW) and Quantum Dot (QD) systems [9, 10, 11] and effect of spatial dielectric

screening on the acceptor impurity in Low Dimensional Systems [12,13].Ignoring the effect of localization on heavy-light hole mixing in

QW and Quantum Wire (QWW) reported by Semina et.al.[14],coulomb interaction of double acceptors in semi magnetic Quantum Dot

and its geometrical effect have been carried out [15, 16]. Since there are no theoretical studies on double acceptor impurities in

GaAs/AlxGa1-xAs low dimensional systems, in the present work, the effect of various cross-sectional geometries Gi:(x, y) of the

GaAs/Al0.3Ga0.7As Quantum Wire like G1:(L, L), G2:(L, L/2), G3:(L, L/4) on the spatially screened coulomb interaction energy as well as

binding energy of double acceptor has been investigated through effective mass approximation using variational technique. In this

investigation two different spatial dielectric screening functions, (2) [17] and 3 [18] along with static dielectric constant of GaAsare

considered. The results are compared with the results of screened single acceptor binding energy for the above said rectangular cross-

sectional geometries of the Quantum Wire, and are presented.

2. THEORETICAL FRAME WORK

The Hamiltonian of the single acceptor impurity in a GaAs/AlxGa1-xAs Quantum Wire is given in atomic units by

𝐇h =−∇2

2m∗w,b

−1

εw,b(r)r+ VB(x, y)(1)

Where m*w is the effective mass of heavy hole in GaAs well = 0.33m0and barrier m*

b= m*w+ (0.18 x)[19], x being Aluminium

composition (x=0.3). r = (x2+y2+z2)1/2. w,b(r) is the spatial dielectric function for well and barrier material which is given in Table 1.

The confining potential of AlxGa1-xAs rectangular Quantum Wire is given by

VB(x, y) = { 0 |x| ≤

𝐿1

2 , |y| ≤

𝐿2

2

V0 |x| >𝐿1

2 , |y| >

𝐿2

2

(2)where L1, L2 is the length of the wire along x and y axis respectively and V0=0.30Eg; Eg

is the difference in the Energy band gaps of GaAs and AlxGa1-xAs.

Eg=1.155x+0.37x2[20](3)

The trial wave function of the ground state of the single acceptor in a Quantum Wire is given by

ψ1s

= 𝑁1𝑠 {cos 𝛼 x cos 𝛼𝑦 𝑒−λr|x| ≤

𝐿1

2, |y| ≤

𝐿2

2

𝐵𝑒−𝛽(x+𝑦)e−λr|x| >𝐿1

2, |y| >

𝐿2

2

(4)Where N1s is the normalisation constant, = [(2m*wE)/2]1/2 and = [2m*

b(V0-E)/2]1/2.

λ is the variational parameter. B is obtained by applying the boundary conditions at x= L1/2, y=L2/2. The subband energy E is obtained by

solving the transcendental equation.

[tanL2/2+tan L2/2] = 2(5)

The binding energy of the single acceptor is found by solving the Schrodinger equation variationally, and is given by

EB=E - <H>min(6)The Hamiltonian of two acceptors in a GaAs/Al0.3Ga0.7As Quantum Wire using effective mass approximation is given

by

𝐇hh =−∇1

2

2m∗w,b

−∇2

2

2m∗w,b

−𝟏

εw,b(r)r1−

𝟏

εw,b(r)r2+ VB(r1) + VB(r2) +

1

εw(r)|𝐫𝟏−𝐫𝟐|(7)where the term1/εw(r)|𝐫𝟏 − 𝐫𝟐|is the coulomb hole-hole

interaction energy term with

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r1=√(x1)2 + (y1)2 + (z1)2and r2=√(x2)2 + (y2)2 + (z2)2

The trial wave function of the ground state of the double acceptor in a Quantum Wire is given by

ψ1s

(r1, r2) = 𝑁ℎ {cos 𝛼 |x1| cos 𝛼|y1| cos 𝛼 |x2| cos 𝛼 |y2|𝑒−𝜆ℎℎ|r1+r2||x1|, |x2| ≤

𝐿1

2, |y1|, |y2| ≤

𝐿2

2

𝐵𝑒−𝛽(x1+y1)𝑒−𝛽(x2+y2)𝑒−𝜆ℎℎ|r1+r2||x1|, |x2| >𝐿1

2, |y1|, |y2| >

𝐿2

2

(8)whereλhh is the variational

parameter, Nh is the normalization constant.

The minimum of the Hamiltonian<Hhh>minis evaluated and the variational parameter λhh is fixed in the wave functionψ1s

(r1, r2)and this

wave function is used to evaluate the hole- hole interaction energy.

Ehh=⟨ψ1s

(r1, r2)|1

𝐫𝟏−𝐫𝟐|ψ

1s(r1, r2)⟩(9)Table 1:

Different dielectric-screening functions used for GaAs/Al0.3Ga0.7As Quantum Wire:

(r) Specific form Remarks

1 Static dielectric constant - 0 0(w)=13.18, 0(b)=0(w)-(3.12*x)[21]

2[16] [0-1+(1-0

-1)exp(-r/sr)]-1

sr – Screening radius

sr= 1.15468 a.u., which is one-fourth of nearest

neighbour distance.[22]

3[17] [a1+a2exp(-b1r)+a3exp(b2r)]-1 a1=0.07634, a2=0.87244, a3=0.05122, b1=1.6977,

b2=0.3435 for GaAs.[23]

3. RESULTS AND DISCUSSION

The cross-sectional geometries of the GaAs/Al0.3Ga0.7As Quantum Wire are defined as Gi(x,y): G1:(L1,L2), G2:(L1,L2/2), G3:(L1,L2/4)

where L1, L2= L. The variation of single and double acceptor impurity binding energy with various thicknesses of Wire for geometry G1

and G3 is shown in fig.1.

Fig.1. Variation of acceptor binding energy as a function of Wire size for (A) G1=L,L and (B) G3=L,L/4. Results are obtained including

the effect of spatially dependent dielectric screening.

From fig.1A, it is seen that for both single and double acceptor cases, the maximum binding energy is observed for wire size L

~aB*, (aB

*= 22Å for acceptor). Binding energy is larger for screened acceptor and follows the trend as BE(2)>BE(3)>BE(1) and is

significant for single acceptor impurity in a Quantum Wire. This variation in binding energy of double acceptors with screening is

noticeable in the smaller wire size, only when the geometry is reduced as given in fig.1B. The reason behind this isthe increase in

subband energy, effect of coulomb potential energy and decrease in size of the wire leads to enhancement of binding energy as we go

from geometry G1 to G3. Significant nature of screening on binding energy of double acceptor impurity is clearly shown in inserted

graph.Binding energy of double acceptor is smaller than the binding energy of single acceptor. The coulomb repulsive potential of the

two holes enhances the Hamiltonian of the double acceptors which in turn reduces the binding energy.Moreover the expected turn over in

binding energy occur in Quasi 1D region and increase of wire size leads to decrease of binding energy leading to bulk system. When the

size of the wire is reduced below 20Å, the variation of binding energy of double acceptor is erratic which is very much negligible in the

case of single acceptor. This is due to the large repulsive energy between the holes in the smaller wire size and hence leads to the

tunnelling of the carrier through the barrier. This is manifested in the succeeding discussions on the hole-hole interaction.

The effect of geometry and importance of screening on the repulsive coulomb interaction of double acceptor impurities in a Quantum

20 40 60 80 100 120 140

20

40

60

80

100

120

140

160

Double Acceptor

Bin

din

g E

nerg

y (

me

V)

Wire size (Å)

G3=L,L/4

x=0.3

Single Acceptor

B

30 35 40 4580

85

90

95

Bin

din

g E

ne

rgy

(m

eV

)

Wire size (Å)

20 40 60 80 100 120 140

20

30

40

50

60

70

80

90

100

110

Double Acceptor

Bin

din

g E

nerg

y (

me

V)

Wire size (Å)

G1=L,L

Single Acceptor

x =0.3A




20 40 60 80 10020

40

60

80

100

120

140

160

Eh

h (

meV

)

Wire size (Å)

G1G2

G3

Ax=0.3

20 40 60 80 100

40

80

120

160

200

G3G2

Eh

h (

me

V)

Wire size (Å)

x=0.3

G1

B _ _ _ _ _

\

Wire is shown in fig.2and 3.

Fig.2. Variation of Ehh as a function of Wire thickness for different Geometry with (A) constant Screening and (B) spatially dependent

dielectric screening.

From fig.2A, it is seen that the interaction energy (Ehh) drastically varies with wire size and increases with increase of carrier

confinement (i.e.) from higher to lower geometry. The variation in interaction energy with constant screening is larger for narrow wire

size and decreases gradually with increase in wire size for all geometries. Fig.2B gives the coulomb interaction energy of spatial screened

double acceptor as a function of various wire size for different geometries. It is observed that the effect of screening on interaction energy

is conspicuous in the smaller geometry of the system.

The important role of geometry and screening of impurities on interaction between two holes in a Quantum Wire is given in

fig.3 by the Ehh profile with wire size for two geometries with constant and spatial dielectric screening.

Fig.3. Variation of Ehh with wire thickness for G1and G3 with constant and spatial dependent dielectric screening.

From fig.3 it is clear that, for all the three cases 1, 2, 3 the Ehh increases appreciably as we go from geometry G1 to G3. The

fact that two major forces influence on the holes, the first being an attractive force due to restricted “Degree of freedom” in a QWW that

brings the holes together, and the second being the repulsive force due to the coulomb interaction between holes themselves which are

confined in a Quantum wire. One can clearly understand from fig.2 and fig.3, G3 being smaller geometry, it brings the two holes closer

thus enhancing the repulsive force, which reflect in increment of Ehh as well as Binding energy particularly for L<40Å. In geometry G1

and G3, the coulomb interaction energy for screened acceptor is larger than that of constant screening which is predominantly seen for

smaller wire size and the coulomb interaction energy for two holes follows the trend as Ehh(2) >Ehh(3) >Ehh(1). The spatial dielectric

screening reduces the attractive Coulomb potential of opposite charges and brings the two holes nearer which increases the repulsive

potential between them.

Figures 4, 5 are self-evident for the degree of localization of impurity wave functions in terms of position probability density

|ψ|2for geometry G1 and G3for Quantum Wire size L=40Å.

20 40 60 80 100

40

80

120

160

200

Eh

h (

me

V)

Wire size (Å)

x=0.3

G1= L,L

G3 = L,L/4




Fig.4.3D plot of |ψ|2for single acceptor in Quantum Wire for L= 40Å.

Fig.5.3D – plot of |ψ|2 for double acceptor in Quantum Wire for L= 40Å.

For smaller wire size (L=40Å),the probability oflocalization of impurity inside the well region is enhanced as we go

from Geometry G1 to G3 and thiseffect is smaller for double acceptor than that of single acceptor due to its nature of repulsive

potential.

4. CONCLUSION To conclude, the effect of reduction of geometry and spatial dielectric screening on the impurity binding energy is obviousfor single

acceptor binding. But in double acceptor, the repulsive force overcome the attractiveforce due to reduction of geometry and affects the

impurity binding energy as well as coulomb interaction energy particularly for lower L limit. The interaction between two holes leads to

strong correlations, and the study on coulomb interaction for many body problems will lead to obtain deep knowledge of impurity states

in compound semiconductor based Quantum heterostructures.

Ly (Å) Ly (Å)

Spatial Dielectric Screening 2

|ψ|^2 |ψ|^2

Constant Screening 1

|ψ|^2 |ψ|^2

Geometry G1 = L, L Geometry G3 = L, L/4

LX(Å)

Ly(Å)

Ly (Å)

Lx (Å)

Lx (Å)

Ly (Å)

Lx (Å)

Ly (Å)

|ψ|^2

|ψ|^2

Spatial Dielectric Screening 2

|ψ|^2

|ψ|^2

Constant Screening1

Geometry G1 = L, L Geometry G3 = L, L/4

Lx (Å)

Ly (Å)

Lx (Å)

Ly (Å)

Lx (Å)

Ly (Å)

Lx (Å)

Ly (Å)




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