ELECTRONIC PROPERTIES OF GaAs

AND HUBBARD INTERACTION IN

By

Somalina swain
410PH2152

Under the guidance of

Department Of Physics
National Institute Of Technology, Rourkela
 CERTIFICATE

This is to certify that the project thesis entitled ”ELECTRONIC PROPER-

TIES OF GaAs & HUBBARD INTERACTION IN TRANSITION METAL

OXIDE” being submitted by Somalina Swain in partial fulfillment for the re-

quirement of the one year project course (PH592) of M.Sc. Degree in Physics

of National Institute of Technology,Rourkela has been carried out under my

guidance. The result incorporated in the thesis has been reproduced by using

Quantum Espresso codes.

Prof.Biplab Ganguli

Dept. of Physics

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 ACKNOWLEDGEMENT

I would like to acknowledge my guide Prof. Biplab Ganguli for his help &

guidance in the completion of my one year project & also for his encourage-

ment. I am also very much thankful to Ph.D. research scholars of computa-

tional physics lab whose encouragement & support helped me to complete

my project.

Somalina Swain

Roll no.-410PH2152

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 ABSTRACT

Electronic properties of GaAs semiconductor is studied using Local Density

Approximation within Quantum Espresso method. GaAs is found to be a

direct band gap semiconductor with band gap 1.22eV. The effect of Hubbard

interaction on the absolute magnetization of FeO & MnO are also studied.

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Contents

1 Introduction 2

2 Computational Technique 4

2.1 Quantum Espresso . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Local Density Approximation . . . . . . . . . . . . . . . . . . 5

2.2.1 Advantage Of LDA . . . . . . . . . . . . . . . . . . . . 7

2.2.2 Disadvantage Of LDA . . . . . . . . . . . . . . . . . . 7

2.3 Local Density Approximation + U . . . . . . . . . . . . . . . 8

2.3.1 Generalization Of The LDA+U functional . . . . . . . 8

2.3.2 Application Of LDA+U . . . . . . . . . . . . . . . . . 9

3 Result & Analysis 9

3.1 Electronic Properties of GaAs . . . . . . . . . . . . . . . . . . 9

3.2 Hubbard interaction in FeO & MnO . . . . . . . . . . . . . . . 11

4 Conclusion 13

5 References 14

1
1 Introduction

Density functional theory (DFT)[1] has become the most widely used tool to

study the electronic structure of atoms, molecules and solids. As it is used

today, it essentially is an ingenious reformulation of the many-body problem.

Instead of trying to solve the Schroedinger equation of interacting electrons

directly, the problem is cast in a way such as to make it tractable in an ap-

proximate, but in many cases surprisingly accurate way. The success of DFT

is largely due to the availability of increasingly accurate approximations to

the central quantity of DFT, the so-called exchange-correlation energy func-

tional.

While the simple local density approximation (LDA) proved to be surpris-

ingly accurate especially in solid state physics, only the advent of the so-called

generalized gradient approximations (GGAs) with their increased accuracy

led to an explosion of applications of DFT in quantum chemistry[1].

The development of new, improved functionals is an ongoing effort. In this

contribution we are dealing with a particular class of approximations which

are explicit functionals of the Kohn-Sham orbitals rather than explicit func-

tionals of the density (such as LDA or GGA). Treatment of orbital functionals

in the DFT framework requires the use of the so called optimized effective po-

tential (OEP) method to compute the corresponding effective single-particle

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potentials[1].

Due to the simplification in their exchange correlation functional form &

other unphysical features like self-interactions, LDA & GGA fail to describe

the system with strong coulomb interactions such as transition metal oxides

& rare earth compounds[2].

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2 Computational Technique

2.1 Quantum Espresso

Quantum Espresso is a variety of numerical methods & algorithms aimed at a

chemically realistic modeling of materials from the nanoscale upwards,based

on the solution of the density functional theory(DFT).

It is an integrated suite of computer codes for electronic structure calcula-

tions and materials modeling based on DFT,plane waves and pseudopoten-

tials(norm conserving, ultrasoft and projector augmented wave) to represent

the electron-ion interactions[4].

The ESPRESSO stands for opEn Source Package for Research in Electronic

Structure,Simulation & Optimization. The codes are constructed around the

use of periodic boundary conditions, which allows for a straightforward treat-

ment of infinite crystalline systems [4].

Quantum espresso can do the following basic computations[4].

1. Calculation of the Kohn-Sham(KS) orbitals and energies for isolated

systems, and of their ground state energies.

2. Complete structural optimizations of the microscopic(atomic coordi-

nates) & macroscopic degrees of freedom.

3. Ground state of magnetic or spin polarized systems.

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 4. Spin orbit coupling & non-collinear magnetism.

5. Ab-initio molecular dynamics(MD) in a variety of thermodynamical

ensembles.

6. Density functional perturbation theory(DFPT),to calculate second and

third derivatives of the total energy at any orbitary wavelength.

7. Calculation of phonon dispersion, nudged elastic band.

For our electronic structure calculation we used the Quantum Espresso

existing code. Electron corrections are taken into account within the local

density approximation(LDA) & plus advanced functionals like Hubbard U

corrections (LDA+U). Many different exchange-correlation functionals are

available in the frame work of the LDA or generalized gradient approxima-

tions(GGA) plus advanced functionals like Hubbard U corrections.The later

is an area of very active development & more details is based on hybrid func-

tionals & related Fock exchange techniques[2].

2.2 Local Density Approximation

Local Density Approximation are a class of approximations to the exchange-

correlation (XC) energy functional in density functional theory (DFT) that

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depend solely upon the value of the electronic density at each point in space

(and not, for example, derivatives of the density or the Kohn-Sham orbitals).

Many approaches can yield local approximations to the XC energy[6].

However, overwhelmingly successful local approximations are those that have

been derived from the homogeneous electron gas (HEG) model. In this

regard, LDA is generally synonymous with functionals based on the HEG

molecules, which are then applied to realistic molecules [6].

In general, for a spin-unpolarized system, a local-density approximation for

the exchange-correlation energy is written as,

Where ρ is the electron density & ǫxc the exchange-correlation energy den-

sity, is a function of the density alone. The exchange-correlation energy is

decomposed into exchange and correlation terms linearly[6].

Exc = Ex + Ec (2)

so that separate expressions for Ex &Ec are sought. The exchange term takes

on a simple analytic form for the HEG. Only limiting expressions for the

correlation density are known exactly, leading to numerous different approx-

imations for εc .

Local Density Approximation are important in the construction of more so-

phisticated approximations to the exchange-correlation energy, such as gener-

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alized gradient approximations or hybrid functionals, as a desirable property

of any approximate exchange-correlation functional is that it reproduce the

exact results of the HEG for non-varying densities. As such, LDA’s are often

an explicit component of such functionals.

2.2.1 Advantage Of LDA

1. It works well in cases of energetics matter.

2. Errors in the approximation Ex &Ec cancels each other.

3. LDA satisfied the rule for the exchange correlation hole.

2.2.2 Disadvantage Of LDA

1. Poor eigenvalues.

2. Lack of derivative discontinuity at integer.

3. Gaps too small or no gap.

4. Spin & orbital momentum is too small.

5. For transition metal oxides it fails.

6. When it deals with excited state properties it fails.

7. Self interaction problems.

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 8. Mirror image charges.

9. Two body correlations.

2.3 Local Density Approximation + U

Then we use LDA+U method for advanced result.i.e.Hubbard U correction.

LDA+U can be considered as one of the most efficient approaches as it is

successful in understanding strongly correlated systems due to the on-site

Coulomb correlation effect & the modest computational time. It requires

same computational cost as LDA because of the local correction in real space

[6].

It is also due to for unrestricted Hartee-Fock treatments for the localized

orbital states.

2.3.1 Generalization Of The LDA+U functional

It can be generalized in the following issues [7].

1. Rotational invariant formulation of the LDA+U functional.

2. LDA average of the on-site coulomb interactions.

3. Choice of the basis,a construction of the local projection operator.

4. Self -consistent determination of the parameters.

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2.3.2 Application Of LDA+U

1. It is applied for the study of Oxide engineering.

2. For Ultra small electronic devices.

3. Large scale electronic structure is calculated.

4. Also applied with the local pseudo atomic orbital basis.

5. Also applied to Quantum phase transitions in the following cases[7].

(a) Metal-Insulator transitions.

(b) Quantum critical point.

(c) Spin/orbital/charge ordering phenomena.

3 Result & Analysis

3.1 Electronic Properties of GaAs

GaAs has equal number of Ga & As ions distributed on the diamond lattice

so that each has four of the opposite kind as nearest neighbors. It is a face

centered cubic lattice with two point basis. It has a higher saturated electron

velocity and higher electron mobility, allowing gallium arsenide transistors

to function at frequencies in excess of 250 GHz. It is a III/V semiconduc-

tor, and is used in the manufacture of devices such as microwave frequency

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integrated circuits, monolithic microwave integrated circuits, infrared light-

emitting diodes, laser diodes, solar cells and optical windows[8] .

From Fig.1 we found the kinetic energy cutoff 25Ry for infinite plane wave

Figure 1: Kinetic energy cutoff of GaAs

Figure 2: Charge density plot of GaAs

basis set. From Fig.2 it is found that the GaAs is covalently bonded. From

figure 3, the band structure is calculated to be 1.22eV for GaAs which is

good agreement with experimental result [8]. The band structure of GaAs

was calculated using empirical pseudopotential method(EPM) with Quan-

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 Figure 3: Band structure plot of GaAs

Figure 4: Density of state (DOS) of GaAs

tum Espresso existing code.

3.2 Hubbard interaction in FeO & MnO

These are transition metal oxides. It is one of the most interesting classes in

solids. These are the examples of strongly correlated electron systems. It has

core level binding energies. Spin orbit splittings & exchange splittings are

found to exhibit interesting variations with the oxidation state of the metal

& nuclear charge. It is used for the study of satellite structure, final state

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effects in x-ray photo spectra ,catalytic activity & semi-conductive properties

[9].

For FeO,

The absolute magnetization of FeO using local density approximation is

found to be 7.09 Bohr mag/cell. When the hubbard energy of 0.31Ry is

applied then the absolute magnetization is enhanced to 7.25 Bohr mag/cell.

For MnO,

Using LDA we get absoluted magnetization is 9.42 mag/cell but with the

addition of Hubbard interaction energy 0.08Ry, the absolute magnetization

is increased to 9.50 Bohr mag/cell.

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4 Conclusion

The electronic properties like band structure, density of states and charge

density of GaAs are studied using quantum espresso code. The band gap is

found to be 1.22 eV. The study of interaction of Hubbard potential in FeO

and MnO shows the enhancement of absolute magnetization.

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5 References

1. NIC Series, Vol. 31, ISBN 3-00-017350-1, pp. 299-334, 2006.

2. www.scielo.br/scielo.php.

3. Myung Joon Han et.al., Phys. Rev. B 73, 045110 (2006).

4. P.Giannozzi et.al., J.Phys.:Condens. Matter 21(2009) 395502(19pp).

5. S.Mishra,B.Ganguli, Solid State Commun.151(2011) 523-528.

6. W.N.Honeyman, K.H. Wilkinson,J.Phys.D:Appl.Phys.4(1971)1182.

7. Jaejun Yu,school of Physics, Seoul National University, 2nd KIAS work-

shop, 2005/06/13.

8. J.R.Chelikowsky and M.L.Cohen, Phys. Rev. Let. 32, 674-677 (1974).

9. www.jstor.org.

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