electronic structure and optical properties of pby and sny...

Electronic Structure and Optical Properties of PbY and SnY (Y = S, Se, and Te)

Nilton Souza Dantas

Licentiate Thesis

School of Industrial Engineering and Management Department of Materials Science and Engineering

Royal Institute of Technology SE-100 44 Stockholm

Sweden

Akademisk avhandling som med tillstånd av Kungliga tekniska högskolan i Stockholm, framlägges för offentlig granskning för avläggande av teknologie licentiatexamen, torsdagen den 14:e juni 2007, kl. 14.00

i sal K408, Brinellvägen 23, Kungliga tekniska högskolan, Stockholm

ISRN KTH/MSE--07/18--SE+AMFY/AVH ISBN 978-91-7178-654-8

Nilton Souza Dantas. Electronic Structure and Optical Properties of PbY and SnY (Y = S, Se and Te) KTH School of Industrial Engineering and Management Department of Material Science and Engineering Royal Institute of Technology SE-100 44 Stockholm Sweden ISRN KTH/MSE--07/18--SE+ AMFY/AVH ISBN 978-91-7178-654-8 © Nilton Souza Dantas, June 2007 Tryck: Universitetsservice US AB

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ABSTRACT

Lead chalcogenides and tin chalcogenides and their alloys are IV−VI family semiconductors with unique material properties compared with similar semiconductors. For instance, PbY (Y = S, Se, and Te) are narrow-gap semiconductors with anomalous negative pressure coefficient and positive temperature coefficient. It is known that this behavior is related with the symmetry of wave functions in first Brillouin zone L-point, which moves the edges of valence band maximum and conduction band minimum towards each other with pressure increasing. SnTe has opposite behavior since its wavefunction symmetry is different from PbY. Therefore, by alloying PbTe and SnTe one can change and control the band gap energy and its pressure or temperature dependence. These chalcogenides alloys have therefore a huge potential in industrial low-wavelength applications and have been attracted the attention of researchers. This thesis comprises theoretical studies of PbY, SnY (Y = S, Se and Te) and the Pb1−xSnxTe alloys (x = 0.00, 0.25, 0.50, 0.75, and 1.00) by means of a first-principles calculation, using the full-potential linearized augmented plane waves method and the local density approximation. The optical properties of Pb1−xSnxTe alloys are investigated in terms of the dielectric function ε(ω) = ε1(ω) + iε2(ω). We find strong optical response in the 0.5–2.0 eV region arising from optical absorption around the LW-line of the Brillouin zone. The calculated linear optical response functions agree well with measured spectra from ellipsometry spectroscopy performed by the Laboratory of Applied Optics, Linköping University. The calculations of the electronic band-edges of the binary PbY and SnY compounds, show similar electronic structure and density-of-states, but there are differences of the symmetry of the band-edge states at and near the Brillouin zone L-point. PbY have a band gap of Eg 0.15−0.30 eV. However, SnY are zero-gap semiconductors Eg = 0 if the spin-orbit interaction is excluded. The reason for this is that the lowest conduction band and the uppermost valence band cross along the LW line. When including

the spin-orbit interaction a gap Eg ≈ 0.2 eV is created, and hence this gap is induced by the spin-orbit interaction. At the L-point, the conduction-band state is a symmetric state and the valence-band state is antisymmetric thereby the L-point pressure coefficient

+4L −

4LpEg ∂∂ /)L( in SnY is a positive quantity. In

contrast to SnY, the PbY compounds have a band gap both when spin-orbit coupling is excluded and included; this gap is at the L-point, and the conduction-band state has and the valence-band state has symmetry, and thereby this band edge yields the characteristic negative pressure coefficient

+4L −

4LpEg ∂∂ /)L( in

PbY. Although PbY and SnY have different band-edge physics at their respective equilibrium lattice constants, the change of the band-edges with respect to cell volume is qualitatively the same for all six chalcogenides. The calculations show that the symmetry of band edge at the L-point changes when lattice constant varies and this change affects the pressure coefficient.

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To my wife Carla Mendes, and my mother Maria (in memoriam).

PREFACE

The following papers are included in this Licentiate thesis

1 Optical characterization of rock salt Pb1–xSnx Te alloys, N. Souza Dantas, A. Ferreira da Silva, and C. Persson Submitted to the 4th International Conference on Spectroscopic Ellipsometry, June 11-15, 2007, Stockholm, Sweden. To appear in Phys. Stat. Solidi.

2 Electronic band-edge properties of rock salt PbY and SnY (Y = S, Se, and Te), N. Souza Dantas, A. Ferreira da Silva, and C. Persson, Submitted to Optical Materials.

The author has contributed to the following research, which however is not discussed in this Licentiate thesis.

3 Electronic structures and optical properties of SbI3 and BiI3 crystals, N. Souza Dantas, C.Y. An, I. Pepe, J. S. de Almeida, A. Ferreira da Silva, and C. Persson, (to be published 2007-2008).

4 Electronic and optical properties of wurtzite and zinc-blende TlN and AlN, A. Ferreira da Silva, N. Souza Dantas, J. Souza de Almeida, R. Ahuja, and C. Persson, Proc. of Int. Bulk Nitride Semicond. (Krakow, 2004), J. Cryst. Growth 281, 151 (2005).

5 Linear optical response of Si1−xGex compounds, A. Ferreira da Silva, R. Ahuja, N. Souza Dantas, I. Pepe, E. F. da Silva,Jr., O. Nur, M. Willander, and C. Persson, Proc. of Int. Symposium on Integrated Optoelectronics Dev. V. 5743 p. 556, (San Jose, CA, USA, 2005).

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6 Electronic and optical properties of TiO2, A. Ferreira da Silva, N. Souza Dantas, E. F. da Silva, Jr., I. Pepe, M. O. Torres, C. Persson, T. Lindgren, J. S. de Almeida, and R. Ahuja, in 27th International Conference on the Physics of Semiconductors, edited by J. Menendez and C. van de Walle, AIP Conference Proceedings Volume 772, p175, Melville, New York, (2005).

7 Electronic band structure, effective masses, optical absorption, and dielectric functions of rutile titanium dioxide, C. Persson, J. S. de Almeida, N. Souza Dantas, R. Ahuja, E. F. da Silva Jr., I. Pepe, and A. Ferreira da Silva, Internal report at Instituto Nacional de Pesquisas Espaciais, São José dos Campos, SP, Brazil, INPE-11639-PRE/7019 (2004).

8 Optical properties of large-bandgap PbI2 and SbBiI3 compounds, M. C. Andrade, C. S. S. Brasil, I. Pepe, A. Ferreira da Silva, N. S. Dantas, D. G. F. David, C. Y. An, N. Veissid, L. N. Christensen, E. Veje, H. Arwin, C. Persson, and R. Ahuja, Annals of Optics 5, 100 (2003).

9 Optical absorption of large band gap SbBiI3 alloys, C. Persson, R. Ahuja, J. Souza de Almeida, B. Johansson, C. Y. An, F. A. Ferreira, N. Souza Dantas, I. Pepe, and A. Ferreira da Silva, Mat. Res. Soc. Symp. Proc. Vol.744 , M5.35.1 (2003).

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CONTENTS 1 Introduction 1

2 First principles calculations 5 2.1. The many-body problem 5 2.2. Density functional theory 6 2.3. The exchange-correlation energy 9 2.4. The FPLAPW method 9 2.5. The equation of state 12

2.5.1. The Birch-Murnaghan (EOS) 12

3 Group theory and band structure 15 3.1. Definitions 16 3.2. Irreducible representations (IRs) 17

3.2.1. Properties of the IRs 17 3.3 Symmetry group 19 3.4 Space group 19 3.5 Crystal structure and the space group of rock salt 20

4 Optical properties 25 4.1. Interband transitions and dielectric function 25

5 Summary 29

6 Acknowledgments 31

References 33

Paper 1 37

Paper 2 47

1

Chapter 1

INTRODUCTION

Lead chalcogenides PbY and tin chalcogenides SnY (Y = S, Se and Te) are narrow band-gap (Eg ≤ 300 meV) IV–VI semiconductors used in a wide range of technological applications. They have interesting material properties, which together are unique compared with other semiconductors. For instance, PbY and SnY have direct gap at or near first Brillouin zone L-point, small effective mass of free carries, direct gap and high static dielectric constant [1] – which resulting in very high mobility of free carries. This makes the IV-chalcogenides very promising materials for used in applications as photovoltaic detectors [2] and thermophotovoltaic energy converters [3]. The fundamental band-gap Eg of these materials can be tuned between 0 and 0.3 eV by pressure, temperature, and alloy composition; this property is used in many devices such as far and mid-infrared lasers and solar cells.

PbY (NaCl) SnY S Se Te S (orth.) Se (orth.) Te (NaCl)

Eg [eV] 0.286(4.2) 0.145(300) 0.187(4.2) 1.42(300) 0.9(300) 0.19(300)

[meV/GPap

Eg

∂

∂ ](3) −91 −91 −74 88.5

[ ]meV/KTEg

∂

∂ (3) 0.34 0.24 0.18

mc [m0] 0.229(1) 0.141(1) 0.179(1) 0.45 0.5 mv [m0] 0.229(1) 0.285(1) 0.088(1) 0.95 0.15 0.066 ε(0) 169 210 414 37(2) 50(2) 1200

µ [cm2/Vs] 700 300 1730 90 <7000(77) 840 Table 1. Some properties of PbY and SnY compiled from Ref. 1: Eg is the fundamental band-gap (values in bracket show the temperature of measurements, in units of K), ∂Eg/∂p is its pressure coefficient, and ∂Eg/∂T is its temperature coefficient. mc is the effective mass of the conduction band (m0 is the free electron mass), mv is effective mass of the valence band, ε(0) is the static dielectric constant, and µ is the mobility of the carriers. (1)Geometric mean value. (2)Arithmetic average. (3) From Ref. 4.

Chapter 1 Introduction

2

The PbY and SnY exist almost never in exact stoichiometric composition [5]. An excess of cation acts as electron donors, and an excess of anion acts as acceptors, i.e. the n-type or p-type character depends of the stoichiometric composition of the material [6].

Lead-tin chalcogenides alloys Pb1−xSnxTe have been attracted great interesting since those materials are applied in the fabrication of mid infrared (3-14 µm) photodetectors [2, 7]. Pb1−xSnxTe alloy is a pseudo binary compound with NaCl-like structure in which PbTe and SnTe exist on several proportions in the range of (0.0 x 1.0) [4]. The PbTe has a valence-band maximum at the Brillouin zone L-point with symmetry and a conduction-band minimum with symmetry, whereas SnTe has the opposite band-edge symmetries, that is, the L-point valence-band state is and the conduction-band state is (see Fig. 1.1). Therefore, when the Pb

≤ ≤+4L −

4L

−4L +

4L1−xSnxTe alloy is formed, starting from PbTe and by

adding Sn, the edge symmetry becomes opposite → and → at some alloy composition 0.3 < x < 0.7 [8, 9].

−4L +

4L +4L −

4L

Fig. 1.1 Band inversion in Pb1−xSnxTe alloy, adapted from Ref. 8.

+4L

−4L

−4L

Eg = 0.18 eV +4L

PbTe

and −4L

Pb1−xSnxTe

+4L

SnTe

Eg = 0.30 eV

Chapter 1Introduction

3

Historically, the research on the IV–VI materials began in the 40-ths and gradually increased until the beginning of the 80-ths. But unfortunately, due the main features of those materials such as the difficulties to grow materials with low concentrations of native defects, and the high static dielectric constant, PbY and SnY and their alloys became materials of poor quality compared with II–VI materials. The interest in these kinds of IV–VI materials was therefore low until the beginning of 90-ths when researchers found complex and unexplained physics in new low dimensional structures of the IV–VIs materials [10]. Nowadays the interest in the IV–VI materials is renewed and the main research has been focused on low dimensional structures [11], diluted magnetic semiconductor [12] and n-i-p-i structures [13]. This Licentiate Thesis is divided in two parts. The first part is comprised by four chapters which contain basic theories of the computational method. In the second chapter the main points of first principles calculations are presented beginning with a full Hamiltonian to Schrödinger equation, and then is derived the Kohn-Sham equation within the density functional theorem (DFT) framework. The Birch-Murnaghan equation of state is presented used to optimize the volume of the unit cell. The third chapter discusses the group theory applied to symmetry operations on the wave function in the first Brillouin zone. The fourth chapter presents briefly the linear optical interband transition on the dipole approximation. In the second part two theoretical studies are presented on the optical properties of Pb1−xSnxTe alloys and on the electronic properties of the PbY and SnY binaries. Both studies were completed at Royal Institute of Technology (KTH) in 2006/2007.

The total energy of the Pb1−xSnxTe alloy and of the PbY and SnY compounds was calculated in the framework of the DFT [14, 15] with the local density approximation (LDA) parameterized by Perdew and Wang [16]. In order to calculate the electronic structure and optical properties the fully-relativistic full-potential linearized plane waves (FPLAPW) approach was used, as implemented in the WIEN2k code [17]. In the first work, theoretical calculations of dielectric and absorption functions were performed for Pb1−xSnxTe alloy (x = 0.00, 0.25, 0.50, 0.75, and 1.00). The Pb1−xSnxTe alloy was modeled by an eight-atom super cell face-centered cubic structure. The results were compared with measurements of the dielectric function by spectroscopic ellipsometry, performed by Prof. Hans Arwin’s team

Chapter 1 Introduction

4

at Department of Physics, Chemistry, and Biology, Linköping University. The calculated and measured optical spectra showed good agreement. In the second work, the band-edges in Brillouin zone L-point of the band structure and the pressure coefficient of PbY and SnY were studied. A number of theoretical and experimental works report a negative pressure coefficient to lead chalcogenides [18, 19]. Wei and Zunger [20] pointed that such anomalous behavior in PbY can be explained by Pb 6s band which lies below the valence-band and the symmetry of band-edge at L-point. The Pb 6s band and the valence-band at L-point have same symmetry and the repulsion between them increase when lattice parameter become shorter. In the same way the conduction-band at L-point moves towards valence-band. However, we discuss the pressure dependence in terms of the and symmetries of the wave function at the band-edges of the L-point (see Fig. 1.1 for PbTe). Our results show an inversion in the sign of the pressure coefficient of PbY, compared with experimental finding, but the calculated absolute value agrees with measured data. This failure originates from the band-gap error of the LDA, which generates incorrect band-edge symmetries, that is, the calculated conduction-band minimum of PbY has symmetry and the valence-band maximum has symmetry (cf. Fig. 1.1). The analysis of the band structure show that the sign of the pressure coefficient changes when the symmetries at the valence-band and conduction-band edges are exchanged. Moreover, a small increase of the cell volume of PbY yields the correct sign of the pressure coefficient.

+4L −

4L

+4L −

4L

5

Chapter 2

FIRST PRINCIPLES CALCULATIONS

2.1 The many-body problem The problem of a solid compound as N atoms (~ 10 cm23 −3) each one with Z electrons is easy to describe on a math view but which do not have an exact solution. Therefore it is necessary to make a simplification in order to obtain the solution. The full Hamiltonian can be written as

∑∑∑∑∑−

+∇−−

−−

+∇−=≠

NNNZN

ii

ZN

jiji

i

ZN

i

eZZM

eZem

Hβα

βα

βα

αα

αα

α

,

2

22

,

,

222

0

2

21

221

2 RRRrrrhh 2.1

where Mα is the nucleus mass, ri is electron position, and Rα,β is atom position. In Eq. 2.1,

2

0

2

2 i

ZN

i m∇− ∑

h is the electronic kinetic energy;

∑≠ −ZN

jiji

err

2

21 is the electron-electron interaction;

∑−

−NZN

ii

eZ,

,

2

αα

α

Rr is the electron-nucleus interaction;

∑ ∇−N

Mαα

22

2h is the kinetic energy of the nuclei;

∑−

N eZZβα

βα

βα

,

2

21

RR is the nucleus-nucleus interaction.

Chapter 2 First principles calculations

6

The first important approach is called the Bohr-Oppenheimer approximation [21]. It is based in the fact that the nuclei are heavier (~1840 times) than the electrons. This means that the moves of the nuclei do not change the state of the electrons. In any change in nuclei’s position the electrons react almost instantaneously. This means that the nuclei can be considered to be in fix positions related to the electrons. The important consequence is that the nuclei’s kinetic energy can be discarded in Eq. 2.1 and the wave function Ψ(r,R) can therefore be written as

),..,()...,()( 2121 NNZNe RRRrrrr Θ=Ψ ψ . 2.2

A crystalline solid is a collection of atoms in a periodic position. This means that in a practical view any electron in the solid feels an effective potential which takes into account the electron-electron interactions, electron-nuclei interactions and other electronic correlations. This approach is called the mean-field approximation or the one-electron approximation, and the Schrödinger equation for that solid can be written as

)()(2

2

0

2

rr kkk iiieffi EVm

ψψ =⎥⎦

⎤⎢⎣

⎡+∇−

h . 2.3

Here, m0 is the electron mass and Veff is the effective potential energy. The last one really simplifies the original many-body problem because this change the problem of a solid with N atoms each atoms with Z electrons to an equivalent problem consisting of singles particles under an effective periodic potential. 2.2 Density functional theory There are many approaches to solve Eq. 2.3 which can be grouped in semi-empirical methods, first principles methods and DFT. Essentially, the semi-empirical and first principles methods have as a main goal to find the wave function ψ which contains all information about the material. Once ψ is achieved the next step is to calculate the physical properties as

Chapter 2 First Principles Calculations

7

ψψ OO = 2.4

where O is any observable. Different from traditional first principles methods, DFT has been the aim searching for electronic density. This is an advantage because the density depends on three parameters [r = (x, y, z)], whereas the total wave function contains about 3×1023 parameters (r1, r2, … and rN). Moreover, the density, unlike the wave function, can be measured, and the density is expected to be a more smooth varying function in r-space compared with the wave function. DFT is implemented through the Kohn-Sham (KS) approach [15], in whish is based on two theorems by Hohenberg and Kohn (HK) [14]. The first one statement that all properties on the ground-state system of particles can be known by charge density ρ0(x,y,z). In other words if one knows ρ0(x,y,z), any property of the ground-state is calculated as a functional of the charge density

[ ]),,(0 zyxOO ρ= . 2.5

The second theorem states that if the functional exists, then it has an energy minimum at the ground state energy of the system. [ ] [ ]00 ρρ EE ≥ 2.6

where ρ is a trial density. From the HK theorem the mean value of Hamiltonian at ground-state can be written as

[ ]),,(00 zyxEE ρ=ΨΨ= H . 2.7


8

By the variational method, the ground-state energy is found as

[ ]δρ

ρδ ),,(0

zyxEE = . 2.8

In the KS approach the functional in Eq. 2.7 is written as [ ] [ ] [ ] )()(0 ρρρρ xceeext EdVVTE +++= ∫ rr . 2.9

Here, T0[ρ] is the kinetic functional energy of the non-interacting electron gas, the second and third term are respectively, the ion-core-electron and electron-electron functional energy. The last term is exchange-correlation functional energy. Only the last term does not have a known analytical expression. The reason is because all complex electron-electron interactions were put into that functional. The Eq. 2.9 suggests that one can write a KS Hamiltonian: [ ] [ ] [ ] )()(0 ρρρρ xceeextKS EdrVVTEH +++== ∫ r 2.10

One can re-write the two last term in Eq. 2.10, which represents the effective potential energy as xceeexteff VVVV ++= 2.11

Now one can write a Schrödinger-like equation

)()(2

1 2

0

rr kkkKS

iiKS

ieffi Vm

φεφ =⎥⎦

⎤⎢⎣

⎡+∇− .

2.12


9

KSIn the Eq. 2.12 ki )(rφ are the KS’s orbitals, which are not the true single-electron orbitals, but it is believed that the KS-orbitals are reasonable good approximations to the exact orbitals. Hence this is possible to find the charge density from

∑=i

KSi

2)(rkφρ . 2.14

2.3 The exchange-correlation energy The simplest approximation to Exc(ρ) is the local density approximation (LDA) which is based on the homogeneous electron gas and the assumption that crystal density varies smoothly, at least locally. [ ]∫= rrr dE xcxc )()( ρερ , 2.15

where εxc[ρ(r)] is the exchange-correlation energy. The form of εxc[ρ(r)] is not known and, therefore, it has been an object of many improvements. In this work, the local spin density approximation (LSDA) derived by Perdew and Wang [16] was used, which can be written as ∫ ↓↑

= rr dE xcxc ),()( ρρµρ . 2.16

For the lead chalcogenides and tin chalcogenides in the present work, the spin-up density )(r

↑ρ equals spin-down density )(r

↓ρ .

2.4 The FPLAPW method The Eq. 2.12 obeys the symmetry of translation of the perfect crystal and therefore commutes with the translation operator. It means that Eq. 2.12 is a familiar Schrödinger-like non-interacting one-electron equation

)()( rr kkKS

iiKS

iKSH φεφ = . 2.17


10

o solve Eq. (2.12) the wave function can be expanded in an adequate basis set T

∑=n nnc )()( rr kkk ϕφ 2.18

here cnk are unknown expansion coefficients and ϕnk(r) are basis functions. In

he FPLAPW approach is a improvement of the augmented plane waves (APW)

Fig. 2.1 The muffin-tin schem s: MT is spherical region and I is the

wthis work the chosen method to expand the Eq. (2.12) was the linear augmented plane waves (FPLAPW) [22] implemented in WIEN2K package [17]. Tmethod developed by Slater [23]. In the APW approach it is assumed that the electrons far from nuclei in the crystal can be treated like plane waves and the electron closed to nuclei could be treated by atomic functions. Therefore, the unit cell is divided in two regions: one region with spherical symmetry around each atom, called the muffin-tin (MT) region, and the space outside the spheres, called the interstitial region (I). In early works the potential had spherical symmetry in the MT region and it was approximated to be constant in the I region. In this work, the full-potential linearized augmented plane wave (FPLAPW) [22] as implemented in WIEN2k package [17] was used to expand the KS-equation of Eq. 2.12.

e to represent two atom

MT

MT I

interstitial region.


11

he APW basis set used in each region is T

⎪⎩

⎪⎨

⎧

∈

∈=

∑

⋅

MTYEuA

IeVE

ml mlllml

i

l

rrr

rr

rk

k

, ,, )(),(

1),(ϕ

2.19

ere, V is the volume of unit cell, Yl,m(r) is spherical harmonics, Al,m are

rHunknown coefficients, El is the eigenenergy and ul is the solution of the adial part of the Schrödinger equation,

llll uEuVll

ddu

rdd

=⎥⎦

⎤⎢⎣

⎡+

++⎟

⎠

⎞⎜⎝

⎛− )()1(12

2

2r

rrrr

2.20

nfortunately, APW functions could become very unstable because the only

Uconstrain is the matching of the wave functions at r = rMT. One solution, purposed by O. K. Anderson [22] is a linearization of APW (i.e., LAPW). Inside of the MT sphere a linear combination of radial functions ul and its derivatives

lu& is used. In this new approach the Eq. 2.19 becomes

⎪⎩

⎪⎨

⎧

∈+

∈=

∑

⋅

MTYEuBYEuA

IeVE

mllllmlml mlllml

i

l

rrrrr

rr

rk

k

)(),()(),(

1),(

,,, ,, &ϕ

2.21

the full-potential (i.e., FPLAPW) LAPW implemented by WIEN2k package

2.22

Inthe crystal potential is treated without any shape-approximations by the inclusion of an interstitial term and the MT term.

⎪⎩

⎪⎨⎧

∈

∈=

∑

∑ ⋅

.)()(

)()(

, ,,

k k

MTYV

IeVV

ml mlml

i

rrr

rrr

rk


12

.5 Equation of state (EOS)

order to determine the best lattice constant a the equation of state (EOS) was

.5.1 Birch-Murnaghan equation of state

he Birch-Murnaghan third order EOS was derived from Murnaghan EOS [24]

2 Inused. The EOS is a important tool to extract parameters related of bulk properties of the material, like equilibrium volume V0, the bulk modulus B0 and its the first derivative '

0B .

2 Tby F. Birch [25]. In this equation the pressure is given by

( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛−+×⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛= 14

431

23)(

3/2

0'0

3/5

0

3/7

00 V

VBVV

VVBVP ,

2.23

d the bulk modulus is given by an

( )

( ) .421

23

14431

35

37

23)(

3/2

0'0

3/5

0

3/7

00

3/2

0'0

3/5

0

3/7

00

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛+

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡−⎟

⎠⎞

⎜⎝⎛−+×⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−⎟

⎠⎞

⎜⎝⎛=

VVB

VV

VVB

VVB

VV

VVBVB

2.24


13

330 340 350 360 370 380 390 400 410-0.92

-0.90

-0.88

-0.86

-0.84

-0.82

-0.80

-0.78 PbSe

Ener

gy (e

V)

Vol. unit cell (Bohr)3

B = 59.4 (GPa)

Fig. 2.2. Energy minimization vs unit cell volume for PbSe with the fitted bulk modulus B by

Birch-Murnaghan EOS [25]. The energy scale was shifted by 6.3×105 eV.

S Se Te-10

-8

-6

-4

-2

0

Ener

gy [1

05 eV/u

nit c

ell]

Anion

SnY PbY

Fig. 2.3. Energy vs anion of PbY and SnY (Y = S, Se, and Te). The calculation was performed

using experimental lattice constants.


14

15

Chapter 3

GROUP THEORY AND BAND STRUCTURE

Group theory is important in Condensed Matter Physics to understand how crystal symmetry affects the properties of the solids. Here, a brief description of group theory is presented, following Cornwell [26] and Tinkham [27]. A full description of group theory to solids can be found in [28, 29]. Mathematically, a group (G,•) is a set of elements a, b, c, .. with a law of composition “•”

(a,b) → a•b or

G × G → G which must satisfy the axioms:

i) Closure. If a and b belongs to G, then a•b = c also belongs to G. The operation denoted by “•” is called product and its definition is specific to every group. Henceforth, the product a•b will written as a•b ≡ ab.

ii) Associative. For any a, b and c that belongs to G: a(bc) = (ab)c; iii) Identity. There is an element of G, e that commutes with all and keeps

it unchanged: ea = ae = a. iv) Reciprocal. Every element a ∈ G has an inverse a-1, which is also in G,

so that: aa-1 = a-1a = e.

The product is not an ordinary operation multiplication; it is only an abstract rule for combining two group elements and generate a third element. Moreover, the product does not need tos be commutative.

Chapter 3 Group Theory

16

3.1 Definitions

i) If a group G is finite, the number of elements of G is called order of the group and denoted by |G|.

ii) A subset H, of G which is closed under the group operation, whose identity element is in G and for ever a ∈ H, a-1 ∈ H, forms a group.

iii) If a, x and b are elements of G, a similarity transformation is defined as

b = xax-1

which transforms some element a by means of another element x into some other element b. The elements a and b are called conjugated elements and the complete subset of elements which is conjugated is called conjugacy class of elements of G.

iv) A law of composition on a finite group can be described by its multiplication table. A multiplication table is a practical way to verify that the postulates of the group are satisfied.

e a be e a ba a b eb b e a

The multiplication table above shown, as example, the group a,b in which the identity element is e.

v) Two groups G and G’, in which a ∈ G and a’ ∈ G’ are called

isomorphic if there exists a one-to-one correspondence a ↔ a’, G ↔ G’, that mapping one element of G to another element of G’. Both groups must have the same order; it means that the isomorphism preserves the structure of original group.

vi) Two groups G and G’, which a ∈ G and a’∈ G are called homomorphic if a → a’, G → G’; In this case the groups have not the same order and some information about the structure of the original group could be lost.


17

3.2 Irreducible representations (IRs) A symmetry operation can be represented by a matrix Γ . The matrix Γ acts on objects v1, v2, ... vn = v that span an n-dimensional space and assign a new position

V’ = Γ v The matrix Γ is reducible if it can be block-diagonalized into submatrices αΓ . A set of square matrices αΓ which obey the same multiplication table of the elements of the group form a faithful representation of group. The number of row and column of αΓ is called dimension of the representation. If each matrix of a representation, through a similarity transformation, can be block diagonalized and the dimension of its sub matrices are the same as of original matrix, then the sub matrices are themselves representation of the group and one can said that the original representation has been reduced. When the process of reduction goes to a limit where no more such reduction is possible, each sub matrix (for each symmetry operation) forms an irreducible representation (IR) αΓ of the group. 3.2.1 Properties of the IRs The IRs of a group form a basic set in which all group’s representations can be constructed. The construction of an IR can be simplified by uses of the orthogonality theorem, which is based on the Schur’s lemma. Schur’s first lemma. If in an IR Γ1, Γ2,.., Γ|G| of a group G, there exists a non-zero matrix which commutes with all matrices of the representation, then such matrix must be a multiple of unit matrix. The immediate consequence of Schur’s first lemma is: if there exists a matrix which is not a multiple of the unitary matrix and if that matrix commutes with all matrices of a representation, then such representation is reducible. Schur’s second lemma. Let Γ 1, Γ 2,.., Γ|G| and Γ’1, Γ’2,.., Γ’|G| be two IRs of a group G with dimensions d and d’ respectively. If there is a matrix M such

MΓα = Γ’αM.

for all α = 1, 2, .. |G|, and if d = d’ then either M = 0 or the two representations differ by a similarity transformation.


18

Orthogonality theorem. Let Γ1, Γ2,.., Γ|G| and Γ’1, Γ’2,.., Γ’|G| kth and k’th irreducible representations of a group G with dimensions d and d’ respectively . For all α, α’ = 1, 2, .. |G| the matrices Γα and Γ’α in that representation has the following rule,

( ) ( ) '''*

'''

kkjjiijik

ijk

dG

δδδα

αα =ΓΓ∑

(3.1)

Consequences of the orthogonality theorem.

i) The character of one representation. Taking the sum over the diagonal of kth and k’th irreducible representations in Eq. (3.1), the character of kth class is defined as k

αχ ( )kTr αΓ . Setting i = i’ and j = j’ in Eq. (3.1) and take the sum over n class,

'1

'kk

nkk Gn δχχ

αααα =∑

=

(3.2)

ii) The orthogonally relation above can be used to shows that the number

of possible IRs is equal the number of the conjugacy classes. iii) If the representation with c classes is irreducible, then

Gnec

=∑=1

2

ααα χ .

(3.3)

If it is not irreducible, then

Gnec

>∑=1

2

ααα χ .

(3.4)

The IR for one group is not unique but its characters are invariant. Therefore, the characters are a central point in applications of group theory in many physical problems.


19

3.3 Symmetry group An important group in solid state physics is the set of symmetry operations of a crystal called symmetry group. With the Schoenflies notation [30], the symmetry operations are defined as: E: The identity transformation. Cn: Rotation (clockwise) through an angle of 2π/n radians, where n is an integer. Cn

k: Rotation (clockwise) through an angle of 2kπ/n radians. Both n and k are integers. Sn: Rotation (clockwise) through an angle of 2π/n radians followed by a reflection in the plane perpendicular to the axis of rotation. This operation is also called an improper rotation. σ: A mirror plane. σh: Horizontal reflection plane - passing through the origin and perpendicular to the axis with the ‘highest’ symmetry. σv: Vertical reflection plane - passing through the origin and the axis with the ‘highest’ symmetry. σd: Diagonal or dihedral reflection in a plane through the origin and the axis with the ‘highest’ symmetry. Alternative, with the Cornwell [26] notation, the symmetry operations are defined as: E: The identity transformation. Cn: Rotation (clockwise) through an angle of 2π/n radians, where n is an integer. -Cn: Rotation (counterclockwise) through 2π/n. I: The inversion transformation. ICn: Rotation (clockwise) through 2π/n, times inversion. -ICn: Rotation (counterclockwise) through 2π/n, times inversion. 3.4 Space group The symmetry operations of a crystal comprise a type of group called space group. An element of this space group, which has standard notation Ti = Ri|ti, consists of rotation and translation operation. In this notation a pure rotation is written as Ti = Ri|0 and a pure translation is written as Ti = E|ti. The inverse operation T-1 is written as T-1

i = R-1i|R-1

i ti.


20

There are 230 different space groups in the three-dimensional space. Each element of the space group can be expressed as the product of an element of a translation group t, and a point group R, element; the latter consist of either a rotation or the product of a rotation and an inversion. There are 32 point group in three dimensions. It can be shown that all operations of the space group commute with the one-electron Hamiltonian of the perfect crystal. It means that the electron energy must have the symmetry of the crystal structure in k-space or, in another words, E(k) keeps invariance under any of the space group operation performed in k-space. The IR Γα of an eigenfunction is obtained from iiii ψψ αΓ=tR | . (3.5)

and the characters Tr(Γα) for all Ri|ti show the symmetries of the eigenfunctions. In Eq. 3.5, the symmetry operator acts on a spin-independent eigenfunction ψi. For spin-dependent systems where spin-orbit coupling is included, the rotation is r-space imply also a rotation in spin-space. The transformation is then, ±U(Ri)Ri|ti )(rkjψ where U(Ri) is a 2×2 matrix acting on the spin states, and )(rkjψ = α )(rk

↑jψ +β )(rk

↓jψ contains both spin-up and spin-

down parts. The IRs and their characters are presented in, so called, character tables. 3.5 Crystal structure and the space group of rock salt This work will focus on the groups of symmetry operations at points and lines of highest symmetry within Brillouin zone for rock salt structures (or NaCl-like structures); specifically, lead chalcogenides PbY and tin chalcogenides SnY, where Y = S, Se, or Te. The PbY compounds and SnTe have rock salt structure as stable crystalline structure [26]. Although SnS and SnSe are orthorhombic structure as most stable geometric arrangement, they can be grown as rock salt epitaxially. Also, the ternary alloy PbSnY2 is rock salt-like [30].

Rock salt structures are among structures derived from close-packed arrangements [in this case face-centered cubic (FCC)] in which also the interstitial sites of a simple FCC are occupied. The most common structure of PbY and SnTe is the rock salt structure. The space group of rock salt is (equivalently,

5hO

mFm3 or mmF /23/4 , #225 [31]).


21

cation X= Pb, Sn

anion

Y=S,Se,Te

Fig. 3.1(a) shows schematic representation of rock salt structure for PbY and SnY compounds and Fig. 3.1(b) shows the first Brillouin zone of it. The direct net is two interpenetrating FCC sub lattices displaced from each other by a vector (a/2, a/2, a/2), where a is the lattice constant. Each atom is bonded to six neighbors and the environment around it forms a polyhedron called octahedron. The PbY compounds have the fundamental direct band-gap at the L point [that is, at (1,1,1)π/a] in the first Brillouin zone and in the SnY compounds the fundamental band-gap occurs very close to the L point. For the Γ point at the first Brillouin zone of the FCC structure there is a group with 48 operations divided into ten classes. The identity operation E alone forms one class. Other k-points in that structure have fewer symmetry operations in their groups. The symmetry operations of a group can be classified into classes. The table 3.1 shows the characters of IRs at the Γ point of first Brillouin zone for FCC structures, and the table 3.2 shows the characters if IRs at the L point of first Brillouin zone for same structures. The notation is from Koster et al. [28]. The difference between IRs and is its property under spatial inversion. +Γα

−Γα

evenevenevenI ψψψ α +=Γ= +0| ; eveneven tIRtR ψψ || =

oddoddoddI ψψψ α −=Γ= −0| ; oddodd tIRtR ψψ || −=

(3.6)

(a)

a

(b)

Fig 3.1. Rock salt cell (a) and its Brillouin zone (b)


22

E 8C3 3C2 6C4 6C2

’ I 8IC3 3ΙC2 6IC4 6IC2’

+Γ1 1 1 1 1 1 1 1 1 1 1 +Γ2 1 1 1 -1 -1 1 1 1 -1 -1 +Γ3 2 -1 2 0 0 2 -1 2 0 0 +Γ4 3 0 -1 1 -1 3 0 -1 1 -1 +Γ5 3 0 -1 -1 1 3 0 -1 -1 -1 −Γ1 1 1 1 1 1 -1 -1 -1 -1 -1 −Γ2 1 1 1 -1 -1 -1 -1 -1 1 1 −Γ3 2 -1 2 0 0 -2 1 -2 0 0 −Γ4 3 0 -1 1 -1 -3 0 1 -1 1 −Γ5 3 0 -1 -1 1 -3 0 1 1 -1 +Γ6 2 1 0 2 0 2 1 0 2 0 +Γ7 2 1 0 -2 0 2 1 0 -2 0 +Γ8 4 -1 0 0 0 4 -1 0 0 0 −Γ6 2 1 0 2 0 -2 -1 0 -2 0 −Γ7 2 1 0 -2 0 -2 -1 0 2 0 −Γ8 4 -1 0 0 0 -4 1 0 0 0

Table 3.1. Characters table for Γ-point in the FCC structure. The point group is Oh which has 48 symmetry operations in 10 classes. This data is a reproduction of Table 87 on page 103 in Koster et al. [28] and Table 71.4 on page 641 in Altmann et al. [32]. The first row shows the ten classes and the first column shows labels which represents wave functions at Γ-point. The ten conjugacy classes of Oh are: identity E, inversion, 8 rotations by 120o, 3 rotations by 180o, 6 rotations by 90o, 6 rotations by 180o, 8 rotations by 120o followed by inversion, 3 rotations by 180o followed by inversion and 6 rotations by 180o followed by inversion. The sign + indicates even function and the sign – indicates odd function.


23

E 2C3 3C2 I 2IC3 3IC2 +1L 1 1 1 1 1 1 +2L 1 1 -1 1 1 -1 +3L 2 -1 0 2 -1 0 −1L 1 1 1 -1 -1 -1 −2L 1 1 -1 -1 -1 1 −3L 2 -1 0 -2 1 0 +4L 2 1 0 2 1 0 +5L 1 -1 I 1 -1 I +6L 1 -1 -i 1 -1 -i −4L 2 1 0 -2 -1 0 −5L 1 -1 I -1 1 -i −6L 1 -1 -i -1 1 I

Table 3.2. Characters table for the L-point in the FCC structure. The point group is D3d, 12 symmetry operations in 6 classes. This data is a reproduction of Table 55 on page 58 in Koster et al. [28] and Table 42.4 on page 371 in Altmann et al. [32] The six conjugated classes of D3d are: identity E, inversion, 2 rotations by 120o, 3 rotations by 180o, 2 rotations by 120o followed by inversion and 3 rotations by 180o followed by inversion.


24

25

Chapter 4

OPTICAL PROPERTIES

When a sinusoidal electromagnetic plane wave with electric field vector )sin(),(),( tt ωω −⋅= rqqErE traveling through a dielectric medium it induces

a polarization vector P, which is related to the applied field by [33]

∫= dtdtttt ij rrErrrP ),()',,',()','( χ (4.1) where χij is the electric susceptibility tensor. In a regime of weak electric field the relationship between P and E can be assumed linear. In this case the displacement vector can be written as ),(),(),( ωωεω qEqqD = (4.2)

where ε(ω,q) = ε1(ω,q) + iε2(ω,q) is the complex dielectric function of the medium. 4.1 Interband transitions and dielectric function In a dielectric medium the electrons at the valence band perturbed by an electromagnetic field vector can excite to an unoccupied conduction band. The Hamiltonian of such electron is

)()(2

1 2

0

rAp Vem

H ++= . (4.3)

Where p is the momentum operator, e is the electron charge, m0 is its mass, and A is the vector potential which can be written as

Chapter 4 Optical properties

26

constant)(

0 += −⋅ wtieA rkA (4.4)

That transition can occur in two way:

i) in first way, whish is called direct transition, only photon participate in process;

ii) in the second way, the indirect transition, the electronic transition is mediate by photon plus a phonon.

In this work, the imaginary part dielectric function ε2(ω) is related with the direct transitions of the linear optical response [34]. The calculation can be done in momentum representation in the long wave length limit q = (k’ – k) = 0 (where k is the crystal momentum of electron at unperturbed band and k’ is the momentum of electron after jump). The imaginary part of dielectric function can be obtained as [34]

)()1(''4)0,( '

2

22

22

2 ωδω

πωε h−−−Ω

== ∑ nnnnn

EEffnnm

ekkkk

kkpkq

(4.5)

where Ω is the crystal volume, fkn is the Fermi distribution and |kn⟩ is the crystal wave function corresponding to the nth eigenvalue Ekn with crystal momentum k. In the present work the wave functions are LAPW functions (see chapter about LAPW ). The real part of dielectric function is calculated from ε2 by the Kramers-Kronig transformation

∫∞

∞−⎟⎠⎞

⎜⎝⎛

++

−+==

ωωωωωεω

πωε

'1

'1)'('

211)0,( 21 dq .

(4.6)

In the same way another important parameter used to characterization of dielectric material is the absorption coefficient, calculated as

2/1

2)())(Re(2)( ⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=

ωεωεωωαc .

(4.7)

s s
Chapter 4 Optical propertieChapter 4 Optical propertie
27

0 1 2 3 4 50

10

20

30

40

50

60

70

80

90

100

(a) Energy [eV]

ε 2(ω)

PbS

PbSe

PbTe

SnS

SnSe

SnTe

0 1 2 3 4

−40

−20

0

20

40

60

80

(b) Energy [eV]

ε 1(ω)

PbS

PbSe

PbTe

SnS

SnSe

SnTe

Fig. 4.1. The Fig. shows imaginary (a) and real (b) part of dielectric function for PbY and SnY calculated by WIEN2k [17].

0 1 2 3 40

30

60

90

120

150

Energy [eV]

Abs

orpt

ion(

104 /c

m)

PbS

PbSe

PbTe

SnS

SnSe

SnTe

Fig. 4.2. This Fig. shows absorption for lead and tin chalcogenides calculated by WIEN2k [17].

Chapter 4 Optical properties

28

29

Summary The electronic band-edge structure of rock salt PbY and SnY (Y = S, Se, and Te) as well as the optical response of the rock-salt-like alloys Pb1−xSnxTe have been studied using the first-principles and all-electron FPLAPW method and the local density approximation. The optical properties of Pb1−xSnxTe alloys (0 ≤ x ≤ 1) were investigated in terms of the dielectric function ε(ω) = ε1(ω) + iε2(ω). We found strong optical response in the 0.5–2.0 eV region arising from absorption around the LW-line of the Brillouin zone. Moreover, the response peak at EA = 1.6–1.8 eV are shifted towards lower energies for high Sn compositions as a consequence of narrower W-point band-gap Eg(W) for the Sn-rich alloys. The calculated linear optical response functions agree well with measured spectra from ellipsometry spectroscopy by the Laboratory of Applied Optics at Department of Physics, Chemistry and Biology, Linköping University. The electronic band-edges of the binary PbY and SnY compounds have similar electronic structure and density-of-states, but there are differences of the symmetry of the band-edge states at and near the Brillouin zone L-point. We found that: (i) SnY are zero-gap semiconductors Eg = 0 if the spin-orbit interaction is excluded. The reason for this is that the lowest conduction band and the uppermost valence band cross along the Q ≡ LW line. (ii) Including the spin-orbit interaction splits this crossing and creates a gap along the Q-line, thus away from the L symmetry point. Hence, the fundamental band gap Eg is induced by the spin-orbit interaction and the energy gap is rather small Eg ≈ 0.2–0.3 eV. At the L-point, the conduction-band state has symmetric and the VB state is antisymmetric thereby the L-point pressure coefficient

+4L

−4L pEg ∂∂ /)L( is

a positive quantity. (iii) PbY have a band gap at the L-point both when spin-orbit coupling is excluded and included. In contrast to SnY, the spin-orbit interaction decreases the gap energy in PbY. (iv) Including the spin-orbit interaction, the LDA yields incorrect symmetries of the band-edge states at the L-point; the conduction-band state has and the valence-band state has symmetry. However, a small increase of the cell volume corrects this LDA failure, producing an antisymmetric conduction-band state and a symmetric valence band state, and thereby also yields the characteristic negative pressure

+4L −

4L

Summary

30

coefficient in agreement with experimental findings. (iv) Although PbY and SnY have different band-edge physics at their respective equilibrium lattice constants, the change of the band-edges with respect to cell volume is qualitatively the same for all six chalcogenides. (vi) Finally, in the discussion of the symmetry of the band edges, it is important to clearly state the chosen unit cell origin; a shift by (a/2,0,0) changes the labeling of the irreducible representations as ⇔

pEg ∂∂ /)L(

+4L −

4L .

Acknowledgments First of all I would like to thank my wife Carla for supporting me with your love, and your confidence. I know the last eight months were very hard for you! I would like to thank very much my Supervisor Dr. Clas Persson, who invited me to develop this part of my Ph. D. studies at KTH, for your friendship and for the full support to me here at Sweden, for teaching me a lot about scientific research, about quality of work, and not less important, for your big patience with my poor skills in English Language at the begin (I would like to say in Portuguese of the Bahia: Valeu a força!). I would like to thank very much Prof. Börje Johansson for accepting me in your research group. My appreciation also goes to other people at KTH, especially Elizabeth Keller, an English teacher at the Unit of Language and Communication, my colleagues on the Technical English Course, and to people that crossed my way and gave me your attention, sometimes only with a small smile that does a big difference for one person in a foreign country. Keep smiling, your nice people! I want to express my gratitude to Prof. Antônio Ferreira da Silva, my Supervisor in Brazil, for fully incentivizing me, and people of Engenharia e Tecnologia Espaciais Course at Instituto Nacional de Pesquisas Espaciais (INPE) who supported me with my requirements, specially, Dr. Chen Yin An. Special thanks goes also to my Brazilian friends, specially, people at LaPO-IFUFBa where I spend my time working and enjoying when I stayed in Bahia, to Moysés Araújo, his wife Emilene in Uppsala, to Adriano, “The Aborígene”, in BH-Brazil, and to Jailton Almeida for your friendship and attention with me during my time here in Sweden. Finally I am very grateful to Fanny Akritidis and her mother Anette Rudholm who helped me a lot with the apartment; two very nice people.

This work was financially supported by the Swedish Institute (SI), the Swedish Foundation for International Cooperation in Research and Higher Education (STINT), the Swedish Research Council (VR), Brazilian Agencies FAPESB (Bahia), and REMAN/CNPq. The computational resources were provided by Centre for Parallel Computers (PDC) at KTH, and Swedish Infrastructure for Computing (SNIC).

31

Acknowledgments

32

33

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electronic structure and optical properties of pby and sny...

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