Band Structure Modelling of Strained Bulk and Quantum Dot III-Nitrides to Determine the Linear Polarization for Interband Recombinations

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Datum

Date 21/6/2018

Avdelning, institution

Division, Department

Department of Physics, Chemistry and Biology Linköping University

URL för elektronisk version

ISBN

ISRN: LITH-IFM-A-EX—18/3568—SE

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Språk Language Svenska/Swedish Engelska/English ________________ Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport _____________ Titel Title

Band Structure Modelling of Strained Bulk and Quantum Dot III-Nitrides to Determine the Linear Polarization for Interband Recombinations

Författare

Author

Joakim Andersson

Nyckelord

Keyword

k.p theory, strain, III-nitrides, GaN, InN, bulk, quantum dot, nanostructures, linear polarization

Sammanfattning

Abstract

8-band k. p theory was applied to bulk GaN and InN. The optical transition intensity was computed and results show $>80-90\%$ degree of polarization in the direction of compression. Polarization switching is observed when strain was reversed from compressive to tensile. 6 band k.p theory was used to study InGaN quantum dot/GaN elliptical pyramid structures. The optical transition intensity was calculated for different elongations of the pyramid. Elongation of the pyramid gives rise to a small polarization in the direction of the pyramid elongation. The optical transition intensity was calculated for elongated quantum dots and was strongly influencing the polarization in the direction of the quantum dot elongation, with a degree of polarization of $>90\%$.

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Linköping University | Department of Physics, Chemistry and Biology Master thesis, 30 hp | Physics and Nanoscience, Master's Programme Spring 2018 | LITH-IFM-A-EX—18/3568—SE

Band Structure Modelling of

Strained Bulk and Quantum Dot

III-Nitrides to Determine the

Linear Polarization for Interband

Recombinations

Joakim Andersson

Examiner, Per-Olof Holtz Supervisor, Fredrik Karlsson

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Abstract

8-band k · p theory was applied to bulk GaN and InN. The optical transition intensity was computed and results show > 80−90% degree of polarization in the direction of compression. Polarization switching is observed when strain was reversed from compressive to tensile. 6 band k · p theory was used to study InGaN quantum dot/GaN elliptical pyramid structures. The optical transition intensity was calculated for different elongations of the pyramid. Elongation of the pyramid gives rise to a small polarization in the direction of the pyramid elongation. The optical transition intensity was calculated for elongated quantum dots and was strongly influencing the polarization in the direction of the quantum dot elongation, with a degree of polarization of > 90%.

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0.1 Preface

The report was written as one of the examining parts for the final diploma project of a masters degree. The report was written from work done at IFM, Link¨oping University in 2017-2018.

I want to give special thanks to my supervisor Fredrik Karlsson who provided me with lectures on k · p theory, plentiful discussions regarding topics in ma-terial and semiconductor science in general and supervision of my work along with providing MATLAB scripts for the quantum dot studies. I also want to give a special thanks to Per-Olof Holtz for lending his time for examination of my work.

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6 Circular and elliptical pyramid 47 6.1 Pyramid model . . . 48 6.2 Computational details . . . 49 6.2.1 Computational procedure . . . 49 6.2.2 Parameters . . . 51 6.3 Strain calculations . . . 52 6.4 Piezoelectric calculations . . . 54 6.4.1 Charge distribution . . . 54 6.4.2 Piezoelectric field . . . 55

6.5 Local band edge calculations . . . 56

6.6 Wave functions . . . 57

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6.7.1 Comparison of the optical transition intensity between the circular and elliptical pyramid . . . 60

7 Results and discussion 63

7.1 Influence of quantum dot edge distance on polarization . . . . 63 7.2 Influence of pyramid elongation on polarization . . . 65 7.3 Elongated quantum dot . . . 67

V

Conclusion

71

8 Concluding remarks 73

VI

Appendix

77

9 Band structure 79 9.1 Calculating A’ . . . 79 9.2 Details on transformation . . . 79 10 Polarization 81

10.1 Calculating optical transition coefficients for zincblende . . . . 81 10.1.1 unstrained at k=0 . . . 81 10.1.2 Adding strain . . . 83 10.2 Calculating optical transition coefficients in wurtzite . . . 84 11 Quantum Dot convergence calculations 87 11.1 Equilibrium forces in strain calculations . . . 87 11.2 Removing surface charges . . . 88 11.3 Impact of boundary conditions . . . 90

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List of Figures

3.1 InN (zb) band structure for an unstrained bulk crystal. A and B bands are degenrate at the Γ-point. k-values are along the [111]-direction. a) Full band structure. b) A, B (dashed) and C valence bands . . . 21 3.2 InN (zb) band structure for an unstrained bulk crystal with

A0 = 0 (solid) and A0 = 0.609[¯h2/2m0] (dashed). Bands

di-verge around k = 0.5 [1/nm] with A0 = 0.609[¯h2/2m0]

conduc-tion band having slightly higher energy. a) Full band struc-ture. b) Conduction band highlighting the diverging region between the different values. . . 22 3.3 Strained InN (zb). Comparison of the valence band between a)

-1 GPa and b) 1 GPa uniaxial stress in [¯110]. A and B-bands are not degenerate unlike the unstrained case. . . 24 3.4 ∆E between the first and second band at the Γ-point as a

function of strain, uniaxial stress in [¯110]. Right axes is the corresponding temperature to the thermal energy. . . 25 3.5 InN (ZB) intensity for unstrained transitions between hole

states and conduction band at k=0 as a function of polar-ization angle, polarized in [¯110],[¯1¯12]. a) Ground state (A-and B-b(A-ands) b) C-b(A-and . . . 28 3.6 Energy diagram of the transitions between the conduction

band (CB) and the a) Ground state (A and B bands) b) C band for unstrained zincblende InN. . . 28 3.7 InN (ZB) intensity for 1 GPa uniaxial stress in [¯110] transitions

between hole states and conduction band at k=0 as a function of polarization angle, polarized in [¯110],[¯1¯12]. a) A-band b) B-band c) C-band . . . 29 3.8 InN (ZB) intensity for -1 GPa uniaxial stress in [¯110]

tran-sitions between hole states and conduction band at k=0 as a function of polarization angle, polarized in [¯110],[¯1¯12]. a) A-band b) B-band c) C-band . . . 29

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3.9 Energy diagram of the transitions between the conduction band (CB) and the a) A band, b) B band and c) C band for strained zincblende InN. . . 30 4.1 InN (WZ) band structure for an unstrained bulk crystal.

Un-like zincblende, the hole ground state is not degenerate in the Γ-point. k values are along the z [0001]-direction. a) Full band structure. b) A, B (dashed) and C valence bands. . . 32 4.2 Strained InN (wz) band structure. Comparison of the valence

band between a) -1 GPa and b) 1 GPa uniaxial strain in the x [10¯10]-direction. . . 34 4.3 ∆E between the first and second band at the Γ-point as a

function of strain, uniaxial stress in the x [10¯10]-direction. Right axes is the corresponding temperature to the thermal energy. . . 35 4.4 InN (WZ) intensity for unstrained transitions between hole

states and conduction band at k=0 as a function of polariza-tion angle, polarized in x [10¯10], y [1¯210]. a) Ground state b) B-band c) C-band . . . 37 4.5 Energy diagram of the transitions between the conduction

band (CB) and the a) A band, b) B band and c) C band for unstrained and strained wurtzite InN. . . 37 4.6 InN (WZ) intensity for tensile strained (1 GPa uniaxial stress

in [10¯10]) transitions between hole states and conduction band at k=0 as a function of polarization angle, polarized in x [10¯10], y [1¯210]. a) A-band b) B-band c) C-band . . . 38 4.7 InN (WZ) intensity for compressive strained (-1 GPa uniaxial

stress in [10¯10]) transitions between hole states and conduction band at k=0 as a function of polarization angle, polarized in x [10¯10], y [1¯210]. a) A-band b) B-band c) C-band . . . 38 5.1 Energy difference between hole ground state and first excited

state as a function of strain for InN (red), GaN (blue) and GaAs (pink). InN and GaN in zincblende (straight lines) and wurtzite (dashed lines). GaAs in zincblende. Corresponding temperature to the thermal energy kBT is shown to the right. 40

5.2 Degree of polarization for III-nitrides and GaAs for 300K . . . 41 5.3 Degree of polarization for III-nitrides and GaAs for 77K . . . 42 5.4 Degree of polarization for III-nitrides and GaAs for 4.22K . . 42

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6.1 Dimensions of the elliptical pyramid. a) Isosurface model of the GaN elliptical pyramid b) Cross section of the GaN/InGaN pyramid/quantum dot structure with the quantum dot in blue. 48 6.2 Isosurface of the InGaN quantum dot. . . 48 6.3 Computational procedure. . . 50 6.4 Strain components at the base of the QD, a/b = 1 . . . 52 6.5 Strain components taken at the red line in figures 6.4a and

6.4b, at the base of the QD, circular pyramid (a/b = 1) . . . . 53 6.6 Strain components taken at the red line in figures 6.4a and

6.4b, at the base of the QD, elongated pyramid (a/b = 2.4) . . 53 6.7 Cross-section of charge distribution at the centre of the

pyra-mid for a/b = 1 . . . 54 6.8 Charge distribution ρ taken at the red line in figures 6.4a and

6.4b, at the base of the QD, elongated pyramid (a/b = 1) . . . 55 6.9 Charge distribution ρ taken at the red line in figures 6.4a and

6.4b, at the base of the QD, elongated pyramid (a/b = 2.4) . . 55 6.10 Band edge at the centre of the pyramid. Inset shows a

mag-nified region around the quantum dot displaying the local va-lence band edges. The quantum dot is located between 1.5 nm and 3 nm. . . 56 6.11 Isosurface of the wave functions for the circular pyramid. a/b =

1 . . . 57 6.12 Isosurface of the wave functions for the elongated pyramid.

a/b = 2.4 . . . 58 6.13 Optical transition intensity for a circular pyramid between the

conduction band and the a) A-like b) B-like and c) C-like valence bands, a/b = 1. Ptot(A) ≈ 0 . . . 60

6.14 Optical transition intensity for an elongated pyramid between the conduction band and the a) A-like b) B-like and c) C-like valence bands, a/b = 2.4. Ptot(A) = 0.1848 . . . 60

7.1 Polarization for a/b = 2.4, ∆r = 1.5553 nm, Ptot(A) = 0.4330 . 64

7.2 Polarization for a/b = 2.4, ∆r = 3.1 nm, Ptot(A) = 0.1848 . . . 64

7.3 Polarization for a/b = 2.4, Ptot(A) = 0.4330 . . . 65

7.4 Polarization for a/b = 3.4, Ptot(A) = 0.4459 . . . 65

7.5 Strain components taken at the red line (see figures 6.4a and 6.4b in section 6.3), at the base of the QD, elongated pyramid (a/b = 2.4), elongated QD (aQD/bQD = 1.4) . . . 67

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7.6 Optical intensity of the ground state transition for a) non-elongated quantum dot and b) non-elongated quantum dot. De-gree of polarization is a) Ptot = 0.18% and b) Ptot = 0.77%.

Parameters other than the aQD/bQD ratio are the same as in

chapter 6 . . . 68 7.7 Optical intensity for excited state strainsitions. a) B-like

tran-sition b) C-like trantran-sition. aQD/bQD = 1.4 . . . 68

11.1 xx around the quantum dot shown for increasing precisions of

computation on full grid. a/b = 1 . . . 87 11.2 Comparison between the calculated charge distribution a)

be-fore and b) after removal of surface charges. a/b = 1 . . . 88 11.3 Piezoelectric field potential calculated for a pyramid with

sur-face charges a) by fixing the x-axes in centre of the pyramid b) along the z-direction with x and y fixed in the centre c) y and z fixed, along x. a/b = 1 . . . 89 11.4 Piezoelectric field potential calculated for a pyramid

with-out surface charges, minor changes compared with fig 11.3a through 11.3c, a/b = 1 . . . 89 11.5 Piezoelectric field potential seen in a) by fixing the x-axes in

centre of the pyramid b) along the z-direction with x and y fixed in the centre c) y and z fixed, along x. a/b = 1 Compare with 11.4a-11.4c . . . 90 11.6 Piezoelectric field potential seen along the z-direction with x

and y fixed in a corner (x=200px, y=200px), showing expected differences in the edges most affected by the expansion. a/b = 1 90

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List of Tables

2.1 Differences in k·p Hamiltonian parameters between zincblende

and wurtzite. . . 15

3.1 Table of the band parameters for InN (ZB) [6] . . . 20

3.2 Table of the band parameters for InN (ZB) [6]. Bahder and Vurgaftman use different notations in some cases. . . 23

3.3 Basis functions. Source Bahder [5]. . . 27

4.1 Table of the band parameters for InN (WZ) [6] . . . 32

4.2 Table of the band parameters for InN (WZ) [6] . . . 33

5.1 Table comparing the degree of polarization P(xx = 0.23%) for InN, GaN and GaAs. . . 43

6.1 Parameters . . . 51

10.1 Basis functions. Source T. B. Bahder [5]. . . 81

10.2 Calculation of the transition coefficients. . . 82

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Part I

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Chapter 1 Introduction

1.1 Background

Control of linearly polarized light is crucial in novel applications, including quantum computing and cryptography. Linearly polarized light in current technologies is commonly generated by passage through filters, wave guides and polarizers yielding efficiency losses at every stage. Polarized light gener-ated from semiconductor nanostructures can improve the efficiency and pave the way for new technologies.

Linear polarization has previously been seen in nanowires [7, 8, 9] but control-ling polarization direction doesn’t seem to be viable. In this report the strain effects on linear polarization in bulk III-nitrides is investigated. A method of obtaining polarization switching is explored by switching the strain from compressive to tensile.

One demonstrated approach [11] of generating linear polarization is to grow InGaN quantum dots (QD) on top of GaN elongated hexagonal pyramids. The elongation direction has been shown to give linearly polarized light in the same direction as the elongation. Strain and shape of the QD is known to affect polarization [10]. So far it is not clearly understood what the dominat-ing factors givdominat-ing rise to the polarization is. InGaN/GaN QD are modelled and the strain field giving rise to linear polarization is calculated.

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1.2 III-nitrides

III-nitrides are composed of the group III elements Ga, In and Al with the group V element N. GaN, InN and AlN are direct bandgap semiconductors ranging from 0.78 eV (InN) to 6.25 eV (AlN).

III-nitrides have attracted alot of attention in variuos fields. Alloying AlGaN or InGaN gives a wide range of colours in LED from red to the ultra violet. Cheap semiconductor blue lasers based on InGaN quantum wells awarded Isamu Akasaki, Hiroshi Amano and Shuji Nakamura the 2014 Nobel Prize in physics.

1.3 k · p theory for band structure

k·p-theory is derived from quantum mechanical time-independent Schr¨odinger equation p2 2m0 + V (r) ψ(x) = Eψ(x) (1.1)

Bloch theorem states that the wave function in a periodic crystal can be described as a plane wave modulated by a periodic Bloch function.

ψ_kn(r) = u(n)_k (r)eikr (1.2) Combing Shr¨odinger equation with the Bloch theorem gives an equation for the Bloch waves.

p2 2m0 + V (r) + h m0 k · p + h 2_k2 2m0 u(n)_k (r) = (n)_k u(n)_k (r) (1.3)

Assuming that a solution for the Bloch functions and energies is known at k=0, an approximation can be made where only a subset of the complete so-lution is considered. In the 8x8 case the subset consist of 3 valence band and 1 conduction band doubly degenerate states. The minimal subset alone do not provide an accurate enough description, the influence of higher states are commonly included as a perturbation. In this study the influence of higher states, spin-orbit interaction and strain are included as a perturbation. k · p theory provides a good approximation near the Γ-point but is less accu-rate at higher |k| values [1, 2]. k · p theory is used since optical transitions in direct band gap semiconductors such as the III-nitrides takes place at the Γ-point.

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1.4 Influence of strain

Strain is the measure of material deformation when put under pressure. It is typically treated as a tensor. The tensor has components measuring the normal strain in x, y, and z directions and components measuring the shear strain in xy, xz and yz planes.

Inherent strain is induced in nanostructures by using semiconducting ma-terials with different lattice parameters. Lattice mismatch can be used to further increase confinement in quantum well and dot structures [3].

1.5 Optical polarization

Electromagnetic waves oscillate transversely to the direction of propagation with one magnetic and one electric component perpendicular to each other. The polarization specifies the oscillation direction of the wave. If the x and y electric components are in phase, the oscillation is kept in one plane. The wave is then said to be linearly polarized. If there is a phase shift between the components, the oscillation direction will rotate. The electromagnetic wave is then said to be elliptically polarized, with the special case of circular polarization if the phase shift is 90◦.

Emitted light from semiconductors is usually not perfectly linearly polarized in one direction and is therefore partially polarized. The degree of polariza-tion is then defined as

P = Ix− Iy Ix+ Iy

(1.4) Ix and Iy are the optical transition intensity as measured in the x and y

directions respectively. Optical transitions in direct bandgap semiconductors takes place near the Γ-point.

1.6 Research questions

How does strain affect linear polarization in bulk III-nitrides? Will reversal of strain reverse polarization? This could potentially be achieved experimen-tally by applying the semiconductor to a piezoelectric material.

A Lundskog et al have shown that InGaN QDs on top of elongated hexag-onal pyramids emmits linearly polarized light determined by the predefined elongation of the pyramids. How does the underlying strain relaxation affect polarization?

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1.7 Methods

Questions are investigated with the help of computer simulations using MAT-LAB. The theoretical/computational model used is k ·p theory. Optical tran-sitions in direct band-gap semiconductors takes place around the Γ-point and the k · p theory gives a good approximation around the Γ-point. The simplic-ity of k · p theory and its accuracy near the Γ-point makes it the preferred model to use for the purposes of this report.

1.8 Literature

Literature consists of peer reviewed and preprint articles supplemented by textbooks on strain and k · p theory. Special note that Vurgaftman [6, 14] in some cases uses band-parameters that are estimated from computations rather than experiments, especially true for InN in zincblende where experi-mental band-parameters are not well reported, as discussed by I. Vurgaftman and J.R. Meyer [6].

1.9 Structure

Part I gives an introduction going over the background on subject matters. Part II gives a short overview of k · p-theory.

Part III deals with the first research question of strain effects on the optical polarization in bulk InN and GaN. The process of computing the band-structure is described in detail for the InN Zincblende case. The polarization for bulk is described and calculated, and a quick look at the InN wurtzite case follows. Part III ends with a comparison of the polarization between GaN and InN in respective zincblende and wurtzite phases. The III-nitrides are contrasted with the widely studied GaAs semiconductor material. Part IV addresses the second research question of strain effects and polar-ization in elongated GaN/InGaN pyramid/quantum dot structures. The im-plemented process and models are described and computed for circular and elliptical pyramids. Circular and elongated quantum dots are investigated. The report ends with part V giving some concluding remarks.

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Part II

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Chapter 2 k · p theory

In this chapter the k ·p-theory used in this report is introduced. The material is adapted from S. L. Chuang [1], T. B. Bahder [5] and lectures provided by K. F. Karlsson. A more basic text on energy band theory is provided by J. H. Davies [2].

2.1 Basic k · p theory for zincblende crystal

structure

It was previously seen that combining Bloch’s theorem with the Shr¨odinger equation gives an expression for the Bloch waves

p2 2m0 + V (r) + h m0 k · p + h 2_k2 2m0 u(n)_k (r) = (n)_k u(n)_k (r) (2.1)

Bloch functions and energies are assumed to be solved at k = 0 for all bands. Solutions at k = 0 can be used as a basis to expand the solutions for k 6= 0. An approximation is made where the closest bands around the bandgap are used. These are the s-like conduction band state |S > and p-like valence band states |X >, |Y > and |Z >. k · p-theory for zincblende crystals can begin to be constructed by finding the matrix elements in this basis.

When k = 0 eq (2.1) reduces to p2 2m0 + V (r) u(n)_k=0(r) = (n)_k=0u(n)_k=0(r) (2.2) Calculating < S|_2mp2

0 + V (r)|S >= EC gives the conduction band energy. Taking products between < X|_2mp2

0 + V (r)|X >= EV and similarly with Y and Z gives the valence band energies. Taking the product < S|_2mp2

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V (r)|X >= 0 because of orthogonality between the basis functions. Same for products between S and Y, Z as well as products between mixes of X,Y,Z. For k 6= 0 the second term in eq (2.1) contributes to the matrix elements. (k) = h_2m2k2

0 is constant for any given k. So for any of the S and p-like states < i|h_2m2k2

0|j >= (k)δij.

The last remaining part is the k · p term. k · p = kxpx+ kypy+ kzpzexplicitly.

Calculating the inner product between S and X states ¯ h m0 < S|k · p|X >= ¯h m0 < S|kxpx+ kypy+ kzpz|X > (2.3)

The cross terms in the expression of (2.3) _m¯h

0 < S|ky,zpy,z|X >= 0 while

¯ h m0 < S|kxpx|X >= iP0kx. Therefore ¯ h m0 < S|k · p|X >= iP0kx. Be-cause of the cubic symmetry of zincblende _m¯h

0 < S|k · p|Y >= iP0ky and

¯ h

m0 < S|k · p|Z >= iP0kz. All the terms < S|k · p|S >=< X|k · p|X >=< Y |k · p|Y >=< Z|k · p|Z >= 0. The p-like state cross products < X|k · p|Y > etc. are also zero. Combining these results gives

H0 =     |S > |X > |Y > |Z > < S| EC+ iP0kx iP0ky iP0kz < X| −iP0kx EV + 0 0 < Y | −iP0ky 0 EV + 0 < Z| −iP0kz 0 0 EV +     (2.4)

Solving the secular equation gives expression for the effective electron, light-and heavy hole masses. The model provides good results for effective electron me and light hole masses mlh. Heavy hole mhh effective mass behaves like a

free electron and the model need to be expanded by taking into account the influence of higher states. These effects are added as a perturbation to the base functions.

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2.2 Higher states and L¨

owdin perturbation

States are divided into class A and B states. Class A states are S and p-like states in 8-band theory with the class B states being higher states that are added as a perturbation. For a mathematical treatment see L¨owdins original paper [4]. The following theorem follows

”An eigenvalue problem with respect to a system of states belonging to two classes, (A) and (B), can be reduced only to the class (A), if the matrix elements Hmn are replaced by the elements UmnA , where the influence of the

class (B) is taken into account by expansion” Per-Olov L¨owdin [4].

The matrix elements are calculated using group theory and the results are H0 =     A0k2 _Bk zky Bkxkz Bkykx Bkykz L0k2x+ M ky2+ M k2z N0kxky N0kxkz Bkzkx N0kykx M kx2+ L 0 k_y2+ M k_z2 N0kykz Bkxky N0kzkx N0kzky M kx2+ M k2y+ L 0_k2 z     (2.5) A’, B, N’, L’ and M parameters are related to the influence of higher states. B ≈ 0 in zincblende. A’ is a correction term to the electron effective mass and discussed later in section 3.1. N’, L’ and M parameters are commonly expressed in terms of the Luttinger parameters γ₁(0), γ₂(0) and γ₃(0)

L0 = P 2 0 Eg − ¯h 2 2m0 (1 + γ₁(0)+ 4γ₂(0)) N0 = P 2 0 Eg − 3¯h 2 m0 γ₃(0) M = − ¯h 2 2m0 (1 + γ₁(0)− 2γ₂(0))

2.3 Spin-orbit interaction

So far spin-orbit terms have been neglected. Spin is a relativistic effect derived from the quantum relativistic Dirac equation

HDiracψ = i¯h

∂

dtψ (2.6)

with the Dirac Hamiltonian including an external potential V

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From block diagonalization of the Dirac Hamiltonian (giving Pauli Hamilto-nian) the Hamiltonian for conduction and valence band states is obtained

H = p

2

2m0

+ V (r) + ¯h

4m2_c2σ · (∇V × ˆ¯ p) (2.8)

Combining (2.8) with the Bloch theorem (??) Hk·p= p2 2m0 + V (r) + h m0 k · p + h 2_k2 2m0 + ¯h 4m2_c2σ · (∇V × ˆ¯ p) + ¯ h 4m2_c2¯σ · (∇V × ¯hk) (2.9)

The last term can be neglected since the crystal momentum ¯hk is much smaller than the atomic momentum p.

Hk·p= p2 2m0 + V (r) + h m0 k · p + h 2_k2 2m0 + ¯h 4m2_c2σ · (∇V × ˆ¯ p) (2.10)

This is the k · p Hamiltonian with an extra term. Spin can therefore be added as a correction to the previously derived results. Account for the up and down spin base gives an 8x8 Hamiltonian, 2 conduction bands and 6 valence bands, from the previous 4x4 Hamiltonian.

H0 =             EC+ iP0kx iP0ky iP0kz 0 0 0 0 −iP0kx EV + 0 0 0 0 0 0 −iP0ky 0 EV + 0 0 0 0 0 −iP0kz 0 0 EV + 0 0 0 0 0 0 0 0 EC + iP0kx iP0ky iP0kz 0 0 0 0 −iP0kx EV + 0 0 0 0 0 0 −iP0ky 0 EV + 0 0 0 0 0 −iP0kz 0 0 EV +             (2.11) H0 Hamiltonian correction term describing higher states written in the spin bases follows in a similar way to H0. Spin-orbit correction term is

HSO = ∆0 3             0 0 0 0 0 0 0 0 0 0 −i 0 0 0 0 +1 0 +i 0 0 0 0 0 −i 0 0 0 0 0 −1 +i 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 +i 0 0 0 0 −i 0 −i 0 0 0 +1 +i 0 −0 0 0 0             (2.12)

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2.4 Strain

Next is to include the effects of strain. Consider the original, unstrained position r =   rx ry rz   (2.13)

The corresponding strained position is then

r0 =   r0_x r_y0 r_z0  =   1 0 0 0 1 0 0 0 1  r +   xx xy xz yx yy yz zx zy zz  r = (1 + ˜)r (2.14)

The momentum operator p is affected by strain. Assuming that the strain is small, the strained momentum and wave vector can be written as

p0 = (1 − )p (2.15)

k0 = (1 − )k (2.16)

The strain effects from p is added as a correction term

H₀ =     0 −iP0 P jxjkxj −iP0 P jyjkyj −iP0 P jzjkzj iP0P_jxjkxj 0 0 0 iP0 P jyjkyj 0 0 0 iP0 P jzjkzj 0 0 0     (2.17) Corrections to the Hamiltonian due to the effects of higher states are assumed to be small and is neglected. Spin-orbit effects are also neglected.

The effective potential is affected by strain as well and these effects are included as a perturbation. Calculated matrix elements are

< S|H_V|S >= ac(xx+ yy+ zz) (2.18)

< X|H_V|X >= lxx+ m(yy+ zz) (2.19)

< X|H_V|Y, Z >= nxy,z (2.20)

and analogously for Y,Z. ac is the hydrostatic conduction band deformation

potential. The products are given as matrix correction term to the Hamilto-nian H_V =     ac(xx+ yy+ zz) 0 0 0 0 lxx+ myy + mzz nxy nxz 0 nxy mxx+ lyy+ mzz nyz 0 nxz nyz mxx+ myy+ lzz     (2.21)

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The m, l and n parameters are related to another set of parameters av

(hydro-static valence band deformation potential), bv (uniaxial valence band

defor-mation potential [100]) and dv (uniaxial valence band deformation potential

[111]) commonly used in the literature, with the relation

l = av− bv (2.22)

m = av+ 2bv (2.23)

n =√3dv (2.24)

(2.25)

2.5 k · p theory for wurtzite crystal structure

So far k · p theory for the zincblende crystal has been discussed. However, wurtzite is a more common crystal geometry for the III-nitrides. Wurtzite differs in being a hexagonal rather than a cubic structure and thus has a lattice constant in the z direction differing in length from the x and y di-rections. This gives different effective electron masses in z m||e and x,y m⊥_e.

Furthermore ¯ h m0 < S|k · p|X, Y >6= ¯h m0 < S|k · p|Z > (2.26) with ¯ h m0 < S|k · p|X, Y >= iP2kx,y (2.27) ¯ h m0 < S|k · p|Z >= iP1kz (2.28)

Wurtzite has two parameters P1,2 compared to the one P0 in zincblende. The

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In general it follows that for any parameter in zincblende the lower sym-metry of wurtzite gives extra parameters. The differences are summed up in table 2.1.

Table 2.1: Differences in k · p Hamiltonian parameters between zincblende and wurtzite.

Property Zincblende Wurtzite Effective mass P0 → P1,2

Crystal field splitting Ev → Ev, ∆1

Spin-orbit interaction ∆0 → ∆2,∆3

Higher states γ₁₋₃(0) → A1−7

Strain av, bv, dv → D1−6

Two commonly used approximations, also used in this report, are ∆1 = ∆CR

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Part III

Computational study on bulk

III-nitrides

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Chapter 3 Zincblende band structure

3.1 Unstrained zincblende band structure

The goal of the first part of this project is to understand how strain af-fects the optical polarization. In order to calculate the polarization we need to know the state of the electron, or equivalently, to know the eigenvectors corrsponding to an energy band. k · p theory gives the description of the energy bands as a Hamiltonian, see chapter 2. Eigenvectors are obtained by solving the eigenvalue problem for the k · p Hamiltonian.

Hk·p(k)|ψn>= En(k)|ψn > (3.1)

Solving equation (3.1) gives the eigenvectors |ψn > for the nth state. We also

get the band structure En(k).

The calculations are exemplified using zincblende InN. Zincblende InN is used since it has previously been shown to give highest degree of polariza-tion when an external electric field is applied to a nanostructure [10]. The calculations and the structure of the Hamiltonian are the same for other zincblende materials, however the values for the parameters are different. The model includes 8 bands, one doubly degenerate conduction band and three doubly degenerate valence bands giving an 8x8 Hamiltonian, see eq (15) and (20) from Bahder [5]1_{. E}

g is the band gap between conduction and

valence bands. ∆so is related to spin-orbit interaction, see section 2.3. γ1,

γ2 and γ3 are Luttinger parameters describing the influence of higher states,

see section 2.2. Ep describes mixing of conduction and valence band states.

Ep is commonly used in the literature and is related to the P0 parameter, see

section 2.1, by Ep = 2m0P02

¯

h2 . The values are taken from Vurgaftman [6] and

1_{Equation (16) has an error for R and should read R = −}√3 2 ¯ h2 m0[γ2(k 2 x− ky2) − 2iγ3kxky]

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summed up in table 3.1. A’ is a correction term to account for the effect of higher bands in the conduction band, see section 2.2. They are assumed to not have any significant impact on the band structure, so A0 = 0. B param-eter relates to inversion symmetry. For common zincblende crystals this is close to zero, so B = 0. All parameters are summed up in table 3.1.

Table 3.1: Table of the band parameters for InN (ZB) [6] Parameter Bahder Value

Eg(eV) 0.78 ∆so(eV) 0.005 γ1 γ1L 3.72 γ2 γ2L 1.26 γ3 γ3L 1.63 Ep(eV) 17.2 A0(_2m¯h2 0) 0 B 0

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(a) (b)

Figure 3.1: InN (zb) band structure for an unstrained bulk crystal. A and B bands are degenrate at the Γ-point. k-values are along the [111]-direction. a) Full band structure. b) A, B (dashed) and C valence bands

When the eigenvalue problem is solved for a range of k-values the con-duction band and A, B and C valence bands are reproduced with a band gap of Eg = 0.78 eV. A, B and C bands are here defined by the energy order,

A being the band with highest energy, as seen in figure 3.1a. The A and B bands are degenerate at the Γ-point (k = 0) with a split-off energy of 5 meV as seen in figure 3.1b.

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(a) (b)

Figure 3.2: InN (zb) band structure for an unstrained bulk crystal with A0 = 0 (solid) and A0 = 0.609[¯h2/2m0] (dashed). Bands diverge around

k = 0.5 [1/nm] with A0 = 0.609[¯h2/2m0] conduction band having slightly

higher energy. a) Full band structure. b) Conduction band highlighting the diverging region between the different values.

A0 describes the influence of higher states and has previously been as-sumed to not have any impact, that is A0 = 0. What is the motivation for A0 = 0? We can see the difference by comparing the electron effective mass to an experimental value. The effective mass is calculated by fitting2

the conduction band curve close to the Γ-point from figure 3.1a. This gives me ≈ 0.0435m0. Vurgaftman [6] reports m∗e = m∗e(Γ) = 0.07m0. The

dis-crepancy comes from A’ being set to zero. This can be accounted for by comparing the value given by fitting the curve to the reported value, see ap-pendix 9.1. This gives A0 = 0.609_2m¯h2

0. Comparing A

0 _{= 0 and A}0 _{= 0.609} ¯h2 2m0 shows no difference in the valence bands, while the conduction band has slightly higher energy for A0 = 0.609[¯h2/2m0], see figure 3.2. While the

dif-ference in me is quite big, this does not affect the valence band which is most

important for the optical polarization. Therefore A0 = 0 will be used.

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3.2 Strained zincblende band structure

3.2.1 Uniaxial stress in [¯

110]

When strain is applied, the state of the electron will change and so the band structure will change. Therefore calculating the eigenvectors and correspond-ing band structure when strain is applied is needed. The [111] plane is of interest since it’s a high symmetry plane and the stress is applied in a direc-tion parallel to that plane, x = [¯110]. Strain is derived from stress through Hooke’s law, = C−1σ, where C is the matrix of the elasticity constants and σ is the applied stress. Strain is included by adding correction terms to the Hamiltonian used in the previous section and is described in section 2.4, given by eq (28) and eq (32) in Bahder [5].

acand av is related to the hydrostatic deformation of the conduction and

va-lence band respectively. b and d related to uniaxial deformation in the [100] and [111] direction respectively, see section 2.4. b0 is an inversion symmetry parameter similar to B from previous section, and close to 0 for common zincblende crystals. c11, c12 and c44 are the elastic constants. The band

structure parameters related to strain are summed up in table 3.2.

Table 3.2: Table of the band parameters for InN (ZB) [6]. Bahder and Vurgaftman use different notations in some cases.

Parameter Bahder Value ac(eV) a0 -2.65 −av(eV) a 0.7 b(eV) -1.2 d(eV) -9.3 b0 0 c11(GPa) 187 c12(GPa) 125 c44(GPa) 86

The Hamiltonians given by Bahder are in the [001] coordinate system, how-ever the stress in this case is given in the [111] coordinate system. The transformation of stress to the [001] system is described in appendix 9.2.

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(a) -1 GPa (xx = −0.64%) (b) 1 GPa (xx = 0.64%)

Figure 3.3: Strained InN (zb). Comparison of the valence band between a) -1 GPa and b) 1 GPa uniaxial stress in [¯110]. A and B-bands are not degenerate unlike the unstrained case.

It was seen in the previous section that the A- and B bands are degenerate at the Γ-point for the unstrained case. This degeneracy is removed when strain is applied. Note that the energy difference between A and B-bands is not symmetric with respect to strength of the applied strain. Figures 3.3a-3.3b show a comparison between a) -1 GPa strain and b) 1 GPa strain applied uniaxially in the [¯110]-direction.

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3.2.2 Energy difference between hole ground state and

first excited state at the Γ-point

In order to resolve the ground state (A-band at the Γ-point) from the first excited hole state (B-band at the Γ-point) the energy difference between these bands need to be higher than the thermal energy for a specific temperature. It was shown in the previous section that the degeneracy between A-band and B-band at the Γ-point is removed when strain is applied. It was also seen that the band structure is not symmetric with respect to positive and negative strain and that higher strain gives a larger energy difference.

Figure 3.4: ∆E between the first and second band at the Γ-point as a function of strain, uniaxial stress in [¯110]. Right axes is the corresponding temperature to the thermal energy.

The figure 3.4 shows a large difference between tensile and compressive strain. At room temperature (300K) kBT ≈ 0.026 eV, This suggest that in order

to resolve the ground state from the first excited state in room temperature xx < −0.22% or xx > 0.92% (corresponding to σ0 = −0.345 GPa and

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σ0 = 1.450 GPa respectively) as seen in figure 3.4. Compressive strain gives a larger energy separation between the ground state and first excited state, which will increase the efficiency of generating linearly polarized light in the compressive case.

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3.3 Polarization

3.3.1 Calculating optical transition intensity

The polarization is determined by the states of the electron. In general the initial and final states are in superposition of the 8 basis functions. |s ↓>, |s ↑> of the conduction band and the mixed valence bands |vn >. The basis functions are summed up in table 3.3.

Table 3.3: Basis functions. Source Bahder [5]. |cj >= |s ↓>, |s ↑> |v1 >= uΓ8 −3/2 = −i √ 6|x ↓> + 1 √ 6|y ↓> +i q 2 3|z ↑> |v2 >= uΓ8 −1/2 = i √ 2|x ↑> + −1 √ 2|y ↑> |v3 >= uΓ8 1/2 = −i √ 2|x ↓> + −1 √ 2|y ↓> |v4 >= uΓ8 3/2 = i √ 6|x ↑> + 1 √ 6|y ↑> +i q 2 3|z ↓> |v5 >= uΓ7 −1/2 = −i √ 3|x ↑> + −1 √ 3|y ↑> + i √ 3|z ↓> |v6 >= uΓ7 1/2 = −i √ 3|x ↓> + 1 √ 3|y ↓> − i √ 3|z ↑>

Explicitly a superposed state is described as

|ψi >= As↓i|s ↓> +As↑i|s ↑> + 6

X

n=1

Ani|vn > (3.2)

The amplitudes Ani of either initial or final state i are given from the k · p

calculations of the band structure. The transition coefficients between the final state < ψj| and initial state |ψi >, < ψj|e · p|ψi >, are explicitly

calculated in appendix 10 and the result is < ψj|e · p|ψi > = X n A∗_s↓jAni < s ↓ |e · p|vn > + X n A∗_s↑jAni < s ↑ |e · p|vn > +X k A∗_kjAs↓i< s ↓ |e · p|vk >∗ + X k A∗_kjAs↑i< s ↑ |e · p|vk >∗ (3.3) The intensity is proportional to the transition coefficients (3.3) and have the same intensity regardless of emission or absorption.

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3.3.2 Comparison between unstrained, compressive and

tensile strained zincblende InN

The intensity will be different depending on the polarization angle. The figures shows the optical transition intensity as a function of polarization angle in the [111] plane (see eq:s (3.3) and (3.4)). x-direction is [¯110] and y-direction is [¯1¯12].

(a) A,B and CB (b) C and CB

Figure 3.5: InN (ZB) intensity for unstrained transitions between hole states and conduction band at k=0 as a function of polarization angle, polarized in [¯110],[¯1¯12]. a) Ground state (A- and B-bands) b) C-band

Figure 3.6: Energy diagram of the transitions between the conduction band (CB) and the a) Ground state (A and B bands) b) C band for unstrained zincblende InN.

Figures 3.5a-3.5b shows the case of no strain. Symmetry remains in the xy-plane and therefore the intensity is, as expected, independent of polariza-tion angle. A-band and B-band are degenerate in k = 0 point and therefore

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have the same energy, both being the ground state, see figure 3.6. The total intensity of the ground states (A- and B-bands) is twice that of the C-band.

(a) A and CB (b) B and CB (c) C and CB

Figure 3.7: InN (ZB) intensity for 1 GPa uniaxial stress in [¯110] transitions between hole states and conduction band at k=0 as a function of polarization angle, polarized in [¯110],[¯1¯12]. a) A-band b) B-band c) C-band

Figure 3.8: InN (ZB) intensity for -1 GPa uniaxial stress in [¯110] transitions between hole states and conduction band at k=0 as a function of polarization angle, polarized in [¯110],[¯1¯12]. a) A-band b) B-band c) C-band

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Figure 3.9: Energy diagram of the transitions between the conduction band (CB) and the a) A band, b) B band and c) C band for strained zincblende InN.

Applying strain in the [¯110] direction breaks symmetry in the xy-plane. When strain is applied there is an energy difference between the A-band and B-band in the k = 0 point causing the ground state to consist of only the A-band, see figure 3.9.

Figure 3.7a-3.7c shows tensile elongation along the [¯110]-direction. The ground state transition is polarized along the [¯1¯12]-direction. The B-band is polarized opposite the ground state for very small strains, as strain grows the polarization switches to the [¯1¯12] direction, as seen in figure 3.7b. This will increase efficiency of generating polarized light in the tensile direction. C-band is polarized in the [¯110]-direction. The ground state intensity is less than half of the unstrained ground state intensity. The B-band has an in-tensity comparable to the unstrained ground state while the C-band has a much higher intensity than the unstrained case.

The compressive case is shown in figures 3.8a-3.8c. The transition to the ground state is polarized in the direction of compression, [¯110], and is slightly more intensive than the unstrained case. B-band transition does not change in polarization or intensity compared to the tensile case. C-band polariza-tion is in opposite direcpolariza-tion to the tensile case with same intensity as the unstrained case.

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Chapter 4 Wurtzite band structure

4.1 Unstrained wurtzite band structure

Wurtzite (hexagonal) materials are in a different crystal structure than zin-blende (cubic) materials. Therefore wurtzite is described with a different structure of the Hamiltonian. The calculations for wurtzite follows the same pattern as for zincblende. InN is again used, because of its previously shown good properties1, to exemplify the calculations.

An 8 band k · p model is used with one doubly degenerate conduction band and three doubly degenerate valence bands, same as in the zincblende case. Hamiltonians are given by equations (5),(6), (7), (12) and (14)-(16) from Winkelnkemper [13]. Eg is the bandgap and Ev is the valence band edge.

∆so is the spin-orbit and ∆cr the crystal-field splitting energies. For wurtzite

materials the effective electron mass is different depending on direction, one parallell mek and one perpendicular me⊥ mass. The masses relates to the

Kane parameters P1/2. The Luttinger-like parameters A1−7 describes

in-fluence of higher states. The lower symmetry of wurtzite compared with zincblende leads to an increase in corresponding parameters, as was previ-ously discussed in section 2.5. Band parameters are summed up in table 4.1.

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Table 4.1: Table of the band parameters for InN (WZ) [6] Parameter Value Eg(eV) 0.78 Ev(eV) 0 ∆cr(eV) 0.040 ∆so(eV) 0.005 mek (m0) 0.07 me⊥ (m0) 0.07 A1 -7.21 A2 -0.44 A3 6.68 A4 -3.46 A5 -3.40 A6 -4.90 A7(˚A eV) 0.0937 (a) (b)

Figure 4.1: InN (WZ) band structure for an unstrained bulk crystal. Unlike zincblende, the hole ground state is not degenerate in the Γ-point. k values are along the z [0001]-direction. a) Full band structure. b) A, B (dashed) and C valence bands.

Solving the eigenvalue problem gives the doubly degenerate conduction band and A, B and C valence bands. Vurgaftman [6] uses the same values for the band gap as in the zincblende case which can be seen in figure 4.1a. The A and B bands are not degenerate at the Γ-point in unstrained wurtzite unlike in the zincblende case, as seen in figure 4.1b.

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4.2 Strained wurtzite band structure

4.2.1 Uniaxial stress in [10¯

10]

The approach is the same as in zincblende case, see section 3.2.1. Strain is included by adding correction terms to the Hamiltonian, see chapter 2, given by equation (17) from Winkelnkemper [13].

a1 and a2 are the conduction band deformation potentials while D1-D6 are

the valence band deformation potentials. C11, C12, C13, C33 and C44 are the

elasticity constants. Bandparameters are summed up in table 4.2.

Wurtzite is a hexagonal crystal structure with a rotational symmetry of 60◦. The x direction refers to one of the symmetry points [10¯10] and the y direction is at a 90◦ angle at a corresponding symmetry point to [1¯210].

Table 4.2: Table of the band parameters for InN (WZ) [6] Parameter Value a1(eV) -4.9 a2(eV) -11.3 D1(eV) -3.7 D2(eV) 4.5 D3(eV) 8.2 D4(eV) -4.1 D5(eV) -4.0 D6(eV) -5.5 C11(GPa) 223 C12(GPa) 115 C13(GPa) 92 C33(GPa) 224 C44(GPa) 48

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(a) -1 GPa (xx = −0.65%) (b) 1 GPa (xx = 0.65%)

Figure 4.2: Strained InN (wz) band structure. Comparison of the valence band between a) -1 GPa and b) 1 GPa uniaxial strain in the x [10¯ 10]-direction.

Applying strain either tensile or compressively to the crystal increases the energy difference between the A and B bands, see figure 4.2a and 4.2b.

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4.2.2 Energy difference between hole ground state and

first excited state at the Γ-point

To resolve the hole ground state from the first excited state the energy dif-ference between these bands need to be higher than the thermal energy, see section 3.2.2.

Figure 4.3: ∆E between the first and second band at the Γ-point as a func-tion of strain, uniaxial stress in the x [10¯10]-direction. Right axes is the corresponding temperature to the thermal energy.

There is no significant energy difference between small (< |0.4|%) tensile and compressive strain, unlike what was seen in [111]-plane zincblende, see section 3.2.2. At room temperature (300K) kBT ≈ 0.026 eV, This suggest

that in order to resolve the ground state from the first excited state in room temperature xx < −0.23% or xx > 0.23% (corresponding to σ = −0.35 GPa

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4.3 Polarization

4.3.1 Optical transition intensity

Optical transition intensity was previously investigated for zincblende InN where polarization switching was observed for the ground state, see section 3.3.2. A state i with amplitudes ain can be described as

|ψi > = ai1|S ↓> +ai2|X ↓> +ai3|Y ↓> +ai1|Z ↓>

+ ai5|S ↑> +ai6|X ↑> +ai7|Y ↑> +ai8|Z ↑>

(4.1)

< ψj|e · p|ψi > product between two states are calculated for the wurtzite

case in appendix 10.2. The intensity is given by

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4.3.2 Comparison between unstrained, compressive and

tensile strained wurtzite InN

Figure 4.4: InN (WZ) intensity for unstrained transitions between hole states and conduction band at k=0 as a function of polarization angle, polarized in x [10¯10], y [1¯210]. a) Ground state b) B-band c) C-band

Figure 4.5: Energy diagram of the transitions between the conduction band (CB) and the a) A band, b) B band and c) C band for unstrained and strained wurtzite InN.

The figures 4.4a-4.4c show optical polarization intensity of unstrained bulk InN in wurtzite crystal structure.

When the crystal is unstrained, the xy-plane is symmetric and the A, B and C hole-conduction band transitions are unpolarized. The ground state (A band) is non-degenerate in wurtzite unlike what was previously seen in zincblende, see figure 4.5. The optical transition intensity is the same strength for the A and B bands while C band has a much weaker intensity.

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Figure 4.6: InN (WZ) intensity for tensile strained (1 GPa uniaxial stress in [10¯10]) transitions between hole states and conduction band at k=0 as a function of polarization angle, polarized in x [10¯10], y [1¯210]. a) A-band b) B-band c) C-band

Figure 4.7: InN (WZ) intensity for compressive strained (-1 GPa uniaxial stress in [10¯10]) transitions between hole states and conduction band at k=0 as a function of polarization angle, polarized in x [10¯10], y [1¯210]. a) A-band b) B-band c) C-band

The figures 4.6a-4.6c show the optical transition intensity as a function of polarization angle for tensile strain (1 GPa). Figures 4.7a-4.7c show the intensity for compressive strain (-1 GPa).

The ground state (A-band) is polarized in the direction of compression and opposite the direction of tensile elongation, same as in zincblende (see section 3.3.2).

The B-band is polarized opposite the ground state for compressive strains (figure 4.7b). For tensile strains the B band is polarized opposite the ground state for small to large strains (xx ≈ 0.7%), as seen in figure 4.6b. As strains

grow larger the B band is polarized in the y-direction. The C-band is polar-ized opposite the ground state, as in zincblende.

A-band has the same intensity for ±1 GPa. C-band for compressive strain have weak optical transition intensity.

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Chapter 5 Results and discussion

5.1 Comparing III-nitrides

It was shown in sections 3 and 4 that InN wurtzite and zincblende ground state optical transition became polarized under compressive strain in the direction of compression. When the strain was reversed into tensile the po-larization was also reversed. It was also shown that when either the applied compressive or tensile strain was increased the energy difference between the hole ground state and first excited state increased. GaN quantatively be-haves the same as InN since GaN only differs with different values of the k · p Hamiltonian parameters.

In this chapter the energy difference between the hole ground state and first excited state is extended for GaN in its zincblende (ZB) and wurtzite (WZ) phases and compared with GaAs in zincblende. The degree of polarization is calculated for InN (ZB and WZ), GaN (ZB and WZ) and GaAs (ZB).

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5.2 Energy difference between hole ground

state and first excited state at the Γ-point

Figure 5.1: Energy difference between hole ground state and first excited state as a function of strain for InN (red), GaN (blue) and GaAs (pink). InN and GaN in zincblende (straight lines) and wurtzite (dashed lines). GaAs in zincblende. Corresponding temperature to the thermal energy kBT is shown

to the right.

A strain of = −0.23% has been achieved by placing an InGaAs quantum dot diode integrated on a piezoelectric crystal, J. Zhang et al [15].

The energy difference between the ground state and first excited state are smaller than the thermal energy for strains on the order reported by J. Zhang, as seen in fig 5.1.

Lowering the temperature lowers the thermal energy. At the boiling point of nitrogen, 77K, the thermal energy is low enough that the energy difference between the ground state and first excited state is larger than the thermal energy for strains in the order of |xx| ≈ 0.1%. The ground state could

potentially be resolved from the B-band hole state for these strains under cryogenic conditions.

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5.3 Total polarization

Thermal excitations of excited hole states need to be accounted for as seen in previous section. Done by adding the intensity of excited states multiplied by a boltzman tail.

The degree of polarization is defined as P = Ix− Iy

Ix+ Iy

(5.1) where

Ix,y = Ix,y(A) + Ix,y(B)e

−EA−EB

kB T _{+ I}_x,y_(C)e−EA−ECkB T _(5.2) is the intensity of transition between the A, B or C bands and CB when polarized in x,y. x = [¯110], y = [¯1¯12] for ZB and x = [10¯10], y = [1¯210] for WZ. Band parameters for InN, GaN and GaAs taken from I. Vurgaftman et al [6, 14].

Figure 5.2: Degree of polarization for III-nitrides and GaAs for 300K From previous section it was seen that the energy separation between the ground hole state and first excited hole state for strains on the order of ≈ 0.23% was smaller than the thermal energy. This implies a significant contribution to the total polarisation from the B band - conduction band (CB) optical transition. For compressive (negativ) strains the B band - CB transitions has opposite polarization to the A band - CB transitions. For tensile (positive) strains it is a bit more complicated. The B band transition has an opposite polarization to the ground state transition for small enough strains and then changing polarization for larger strains. See sections on polarization in zincblende, 3.3.2, and wurtzite, 4.3.2, for details. Overall picture is B band - CB optical transition lowering the degree of polarization as seen in fig 5.2.

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Figure 5.3: Degree of polarization for III-nitrides and GaAs for 77K At 77K, the boiling point of nitrogen, the thermal energy is significantly smaller which lowers the intensity contribution from the B band - CB transi-tion. As discussed previously, the B band - CB transition lowering the degree of polarization. Limiting contribution from B band - CB transition gives a higher degree of polarization for all the materials. All of the nitrides has a degree of polarization > 80% for xx = 0.23%.

Figure 5.4: Degree of polarization for III-nitrides and GaAs for 4.22K For 4.22K, boiling point of helium, the thermal energy is lower which gives better degree of polarization for all materials. GaAs still has a low degree of polarization for compressive strains. ZB has a bit better response than WZ, as seen especially when comparing ZB and WZ GaN. WZ exhibits symmetric behaviour with respect to compressive and tensile strains. ZB has asymmet-rical features around the switch between compressive and tensile strain for the III-nitrides. GaAs retains this asymmetry for large |xx|.

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Table 5.1: Table comparing the degree of polarization P(xx = 0.23%) for

InN, GaN and GaAs.

Material xx P(300K) P(77K) P(4.22K) InN (ZB) −0.23% +0.51 +0.97 +0.99 +0.23% −0.52 −0.97 −0.99 InN (WZ) −0.23% +0.46 +0.96 +0.99 +0.23% −0.46 −0.96 −0.99 GaN (ZB) −0.23% +0.35 +0.81 +0.91 +0.23% −0.36 −0.83 −0.96 GaN (WZ) −0.23% +0.43 +0.89 +0.93 +0.23% −0.43 −0.89 −0.96 GaAs (ZB) −0.23% +0.23 +0.56 +0.66 +0.23% −0.25 −0.75 −0.99

Table 5.1 summarize the degree of polarisation for xx = 0.23%, the value

reported by Zhang et al [15], for the III-nitrides and GaAs at room temper-ature, liquid nitrogen and liquid helium. For applications involving polar-ization switching the degree of polarpolar-ization need to be at least 80-90% or better for both compressive and tensile strains. The III-nitrides are within this range for cryogenic temperatures, indicating they could be interesting to further look at experimentally. GaAs has a low degree of polarization in the compressive case for all temperatures.

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Part IV

Computational study on

InGaN/GaN Quantum

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Chapter 6 Circular and elliptical pyramid

The second part of this project is dedicated to understanding how strain ef-fects influence the optical emission of nanoscale heterostructures. An InGaN quantum dot (QD) in a GaN elongated hexagonal pyramids has previously been shown to produce optically polarized light where polarization direction is defined by the elongation of the pyramid [11]. Elongated and symmetric QDs in an electric field have previously been shown to affect polarization strength and direction in computational studies [10]. The chapter investi-gates the influence of elongation of the pyramid itself by comparing a circu-lar pyramid to an elliptical pyramid. The QD is kept circucircu-lar to seperate effects QD elongation and pyramid elongation. The QD is later elongated to investigate the effects of QD elongation.

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6.1 Pyramid model

(a) Isosurface of the pyramid structure (b) Cross section

Figure 6.1: Dimensions of the elliptical pyramid. a) Isosurface model of the GaN elliptical pyramid b) Cross section of the GaN/InGaN pyra-mid/quantum dot structure with the quantum dot in blue.

Figure 6.2: Isosurface of the InGaN quantum dot.

The pyramid structures has an elliptical base as shown in figure 6.1a, defined by parameters a and b, representing the semi-major and semi-minor axes of the ellipse. a = b gives the special case of a pyramid with circular base. The composition of the pyramid is pure GaN. The quantum dot is modelled as a spherical cap sitting at a height h, see figures 6.1b and 6.2, with the composition In0.15Ga0.85N. The quantum dot is covered by a capping

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6.2 Computational details

6.2.1 Computational procedure

The electronic structure is computed using k · p-theory with the model as described in previous section. ”Free space” around the pyramid is computed as strain free GaN. There is a lattice mismatch between the quantum dot structure and the pyramid, and the structure is allowed to relax into an equilibrium position, that is when the internal forces are approaching zero, see appendix 11.1. The resulting strain field is calculated from the relaxed structure.

The strain field gives a charge distribution which is calculated. When cal-culating the charge distribution this leads to formation of surface charges in the interface between free space and the pyramid (see appendix 11.2). Piezo-electric potential arising from the charge distribution is calculated using the Poisson equation by finite differences. A section on the impact of boundary conditions on the piezoelectric field is included in appendix 11.3.

Local band edge is calculated from bulk k · p computations.

InGaN/GaN quantum dot/pyramid is inhomogeneous unlike in bulk k · p cal-culations. Band edges EC and EV depends on composition and strain, both

which differs in the material. Similarly, all other material parameters are position dependent. Replacing band edges, and other material parameters, with position dependent parameters EC,V(r) gives large and sparse matrices.

Therefore k · p calculations are solved using the jdqr solver [16, 17, 18] that is especially good for large and sparse matrices. Values from the local band edge calculations are used as a starting guess for the jdqr iteration. The valence band (6x6 k · p) and the conduction band (2x2 k · p) are assumed to be independent and solved separately. The conduction and valence bands are separated in order to save computational resources. When solved, the separately obtained solutions describing the valence and conduction bands are reassembled into an 8x8 solution. The wave functions and the envelope functions are given from the k · p calculations and takes the piezoelectric field into effect. The polarization is calculated from the envelope functions as described in section 6.7.

Strain relaxation, piezoelectric and k · p calculations are implemented in MATLAB scripts by K. F. Karlsson [19]. The polarization calculations are implemented in MATLAB scripts by the author. The computational proce-dure is summed up in figure 6.3.

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Strain relaxation Charge distribution Piezoelectric field

k · p calculations

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6.2.2 Parameters

The computational region is 60x60x60 nm3 _{with a resolution of 243x243x243}

finite elements. rQD is the radius of the quantum dot. hQD is the height of

the quantum dot. h is the z position of the quantum dot base as measured from the bottom of the pyramid. Capping is the thickness of the capping layer covering the dot. a and b are the ellipse parameters, distance from the centre of the pyramid to the edge in x- and y-directions respectively. QD position is the distance from the base of the quantum dot to the edge in y-direction. ∆r = QDpos − rQD, i.e. the distance from the edge of the QD

base to the side of the pyramid. Parameters for the circular (a/b = 1) and an elliptic (a/b = 2.4) pyramid are summed up in table 6.1. The ratio a/b = 2.4 for the elliptical pyramid is chosen as such to fill up the computational region. Experimentally produced elongated hexagonal pyramids have a ratio of a/b = 10 [11] with sizes in the µm range. However, selecting a larger ratio would give a more narrow pyramid, giving a smaller ∆r. Increasing the computational region is not possible because of available resources. Effects of edge distance ∆r and a/b ratio are investigated in section 7.1 and 7.2, respectively. Height and radius of the quantum dot used here are slightly smaller compared to the computational values used in S. Amloy et al [10]. The quantum dot dimensions are smaller to give a larger edge distance ∆r.

Table 6.1: Parameters a/b = 1 a/b = 2.4 Size 60x60x60 60x60x60 Resolution 243x243x243 243x243x243 rQD 2.6 nm 2.6 nm hQD 1.5 nm 1.5 nm h 31.5 nm 31.5 nm Capping 5 nm 5 nm Base a = 24 nm a = 28.8 nm b = 24 nm b = 12 nm QD pos 11.4 nm 5.7 nm ∆r 8.8 nm 3.1 nm

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6.3 Strain calculations

Strain is induced by the lattice mismatch, and the strain field needs to be calculated to compute the piezoelectrical properties. Strain is compressive in the QD x and y components.

(a) xx (b) yy (c) xy

(d) zz (e) xz (f) yz

Figure 6.4: Strain components at the base of the QD, a/b = 1

The strain field is of greatest magnitude around the quantum dot, where lattice deformations are the greatest. Figures 6.4a-6.4f shows the strain field for a circular pyramid, i.e. base ellipse parameters a = b.

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(a) xx (b) yy

Figure 6.5: Strain components taken at the red line in figures 6.4a and 6.4b, at the base of the QD, circular pyramid (a/b = 1)

Strain field is symmetric in the xy-plane because of the geometrical sym-metry of the pyramid and because of the hexagonal structure of the wurtzite material, as seen in figures 6.5a, 6.5b.

(a) xx (b) yy

Figure 6.6: Strain components taken at the red line in figures 6.4a and 6.4b, at the base of the QD, elongated pyramid (a/b = 2.4)

The pyramid is elongated with a ratio a/b = 2.4. In order to fit the pyra-mid within the computational region, the length of the semi-minor axis b is halved compared with the circular case and semi-major axis adjusted to fit the ratio, see details in table 6.1. When calculated for an elongated pyramid the xy-plane symmetry breaks and strain field changes as seen in figures 6.6a and 6.6b. Elongating the pyramid appears to have a minor effect on the strain field near the quantum dot. Elongated equivalents to figures 6.4a-6.4f are visually similar.

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6.4 Piezoelectric calculations

6.4.1 Charge distribution

(a) (b)

Figure 6.7: Cross-section of charge distribution at the centre of the pyramid for a/b = 1

The charge distribution is calculated from the strain components and the piezo-electric constants. Figure 6.7b shows a cross section of the red line through the quantum dot. Figure 6.7a is taken following the red line beside the quantum dot. Charges are 2 orders of magnitude higher through the quantum dot compared to a path beside it. Strain field arising from lattice mismatch is at its greatest around the quantum dot, therefore the accumu-lation of charges is at its highest magnitude near the quantum dot.

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(a) ρ, x direction (b) ρ, y direction

Figure 6.8: Charge distribution ρ taken at the red line in figures 6.4a and 6.4b, at the base of the QD, elongated pyramid (a/b = 1)

The charge distribution is symmetric for x and y directions for the circular pyramid, as seen in figures 6.8a and 6.8b.

(a) ρ, x direction (b) ρ, y direction

Figure 6.9: Charge distribution ρ taken at the red line in figures 6.4a and 6.4b, at the base of the QD, elongated pyramid (a/b = 2.4)

When elongating the pyramid, as described in previous section, the charge distribution is affected similarly to the strain field, as seen when comparing figures 6.9a and 6.9b.

6.4.2 Piezoelectric field

The piezoelectric field is calculated using the Poisson equation from the charge distribution. The piezoelectric field is needed when calculating the quantum states from the k · p calculations.

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6.5 Local band edge calculations

Figure 6.10: Band edge at the centre of the pyramid. Inset shows a magnified region around the quantum dot displaying the local valence band edges. The quantum dot is located between 1.5 nm and 3 nm.

The local band edge is the energy position of the bulk bands at a specific point in space. Figure 6.10 is taken across a path in the z-direction at the centre of the pyramid (red line in figure 6.7b). The inset shows a magnified region around the quantum dot.

The band edges depends on the material composition and piezoelectric field, it therefore varies only over the quantum dot as seen in figure 6.10. The band edge calculations are used as a starting point for the jdqr iterations.

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6.6 Wave functions

Envelope wave functions for the A, B and C-like valence bands as well as the conduction band are calculated from k · p calculations. The wave functions are needed for calculating the optical transition intensity of the quantum dot.

(a) A-like valence band (b) B-like valence band (c) C-like valence band

(d) Conduction band

Figure 6.11: Isosurface of the wave functions for the circular pyramid. a/b = 1

Conduction band and A, B and C-like valence band wave functions are cal-culated for the circular pyramid. The wave functions extends around the quantum dot, with a position of 30-35 nm in z-direction, as seen in figures 6.11a-6.11d. Wave functions are symmetrical in the xy-plane.

Band Structure Modelling of Strained Bulk and Quantum Dot III-Nitrides to Determine the Linear Polarization for Interband Recombinations

Band Structure Modelling of

Strained Bulk and Quantum Dot

III-Nitrides to Determine the

Linear Polarization for Interband

Recombinations

Joakim Andersson

0.1

Preface

Contents

I

Introduction

1

II

Fundamentals of k · p theory

7

III

Computational study on bulk III-nitrides

17

IV

Computational study on InGaN/GaN Quantum

Dot/Pyramid nanostructures

45

V

Conclusion

71

VI

Appendix

77

List of Figures

List of Tables

Part I

Chapter 1

Introduction

1.1

Background

1.2

III-nitrides

1.3

k · p theory for band structure

1.4

Influence of strain

1.5

Optical polarization

1.6

Research questions

1.7

Methods

1.8

Literature

1.9

Structure

Part II

Chapter 2

k · p theory

2.1

Basic k · p theory for zincblende crystal

structure

2.2

Higher states and L¨

owdin perturbation

2.3

Spin-orbit interaction

2.4

Strain

2.5

k · p theory for wurtzite crystal structure

Part III

Computational study on bulk

III-nitrides

Chapter 3

Zincblende band structure

3.1

Unstrained zincblende band structure

3.2

Strained zincblende band structure

3.2.1

Uniaxial stress in [¯

110]

3.2.2