Kvant-periodiska potentialer med föroreningar: Effekten av föroreningar på den endimensionella Kronig-Penney potentialen

IN

DEGREE PROJECT TECHNOLOGY, FIRST CYCLE, 15 CREDITS

STOCKHOLM SWEDEN 2019 ,

Quantum periodic potentials with impurities

The effect of impurities on the one-dimensional Kronig-Penney potential

ERIK ENGSTEDT

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ENGINEERING SCIENCES

Abstract

In this project a model of a quantum periodic potentials with impurities is implemented and explored. The focus lies on studying the eects that dierent types of impurities have on the energy-band structure found in a one-dimensional Kronig-Penney potential.

The method used is based on a numerical matrix method for solving the Schrödinger

equation for periodic potentials. The results can be summarized in that the impurities

mainly change the band structure in two ways. The rst eect is that new, almost discrete

energy levels are introduced into the previously empty energy gaps which correspond to

the intrinsic energies of the isolated impurities. The other observed eect is that the

previously continuous bands of the periodic potential without impurities split up but

still cover roughly the same range of energies as before, at least when the impurity

density is suciently low. The results reect some important feature found in real doped

materials and looking forward, the work done here can hopefully be applied to analysis

of more realistic three-dimensional models.

Sammanfattning

I det här projektet är en modell för kvantperiodiska potentialer med föroreningar im- plementerad och utforskad. Fokus ligger på att studera eekten som olika typer av för- oreningar har på energibandstrukturen hos en endimensionell Kronig-Penney-potential.

Metoden som används är baserad på en numerisk matrismetod för att lösa Schröding-

erekvationen för periodiska potentialer. Resultatet kan summeras med att föroreningen

huvudsakligen förändrar energibanden på två sätt. Det första är att nya nästan diskreta

energinivåer introduceras i de tidigare tomma energigapen som motsvarar de isolerade

föroreningarnas inneboende energier. Den andra observerade eekten är att de tidiga-

re kontinuerliga banden hos den periodiska potentialen utan föroreningar splittras men

täcker ungefär samma omfång av energier som tidigare, åtminstone när densiteten av för-

oreningar är tillräckligt låg. Resultaten reekterar några viktiga eekter som kan hittas i

riktiga dopade material och i framtiden kan förhoppningsvis arbetet från denna rapport

användas i analys av mer realistiska tredimensionella modeller.

Contents

1 Introduction and background 2

1.1 Periodic potentials . . . . 2

1.2 Energy bands . . . . 2

1.3 Single-electron approximation . . . . 3

1.4 Three-dimensional methods . . . . 3

1.5 One-dimensional methods . . . . 4

1.6 Impurities . . . . 5

1.7 This project . . . . 6

2 Method 7 2.1 The matrix method . . . . 7

2.2 Bloch wave basis . . . . 8

3 Investigation 10 3.1 Problem . . . . 10

3.2 Model . . . . 10

3.3 Analytical calculations . . . . 11

3.4 Numerical analysis . . . . 12

3.4.1 Code implementation . . . . 12

3.4.2 Energy-band representation . . . . 13

3.4.3 The number of basis functions . . . . 14

3.5 Results . . . . 15

3.5.1 Introducing an impurity . . . . 15

3.5.2 Width and strength dependence . . . . 17

3.5.3 Density dependence . . . . 20

3.5.4 Other impurity types . . . . 21

3.6 Discussion . . . . 23

3.6.1 Eects of impurities . . . . 23

3.6.2 Method evaluation . . . . 25

3.6.3 The next step . . . . 26

4 Summary and conclusions 27

5 Appendix 29

Bibliography 32

Chapter 1

Introduction and background

1.1 Periodic potentials

Quantum mechanics is often introduced by solving the Schrödinger equation for some simple one-dimensional potentials such as the innite square well and the harmonic os- cillator. Then, when those are familiar, the theory is often expanded to three dimensions in the form of the hydrogen atom, see for example Gasiorowicz's book in Ref. [1] or any other introduction to quantum mechanics. These examples of potentials are useful since they can be solved analytically and can therefore be used to introduce mathematical aspects of quantum theory in an approachable way. These systems do however only rep- resent an electron interacting with a single atom placed freely in space, which are not very common in Nature. The next step is therefore to study larger systems containing multiple potentials the electrons can interact with. An example of such a system consist- ing of many atoms is a solid and quantum theory of systems containing many potentials are often introduced in books on solid state physics, see for example Kittel's book in Ref. [2]. Solids are often dened in a region of space and then assumed to be repeated under certain translations. This makes it possible to mathematically model a solid by imposing periodic boundary conditions around the repeated region. Thus models the solid as a periodic potential. The simplest example of such a potential is the Kronig- Penney (KP) model introduced in Ref. [3], which consists of a nite square well which is repeated periodically in one dimension. An example where four periods of a KP potential is shown is found in gure 1.1. This potential is of interest since it is one of few that can be solved analytically, see Refs. [3, 4], and therefore is suitable for introducing some aspects of solid state physics.

1.2 Energy bands

The KP model is a very simple model but it captures one important feature that is

found in real solids and molecules, which is the band structure that the electrons allowed

energies form. This is the eect found that when a potential with discrete energies is

repeated periodically, the allowed energies change to form continuous bands separated

by gaps of energies which the electrons cannot have. A theoretical discussion of this

phenomenon can be found in Ref. [5] and a good experimental demonstration of the

formation of bands is given in Ref. [6] where they look at alkane chains, i.e. molecules of

Figure 1.1: Example of four periods of a KP potential.

the form C

H

, of dierent length and observe band formation. The band structure of solids is often of interest when studying a certain material since it determines some of its properties. For example, the conductivity of a material depends on the band gap in between the valence and conduction bands [2].

1.3 Single-electron approximation

Real systems, such as solids, that can be studied as periodic potentials are often very complicated and therefore require some approximations. A solid is a system of multiple electrons with interactions in between them that are generally very complex and therefore hard to model correctly. Consequently one approximation that often has to be made is the single-electron approximation. This is performed by assuming that the interactions between the electrons can be accounted for in an eective potential and thus the problem can be handled by only looking at the wave function of a single electron. This approxi- mation is used by, for example, Bloch in Ref. [7] where he introduced what came to be known as Bloch's theorem and Bloch waves for dealing with periodic potentials which are the basis in many approaches to problems with periodic potentials. There exist models that account for some multiple electron aspects of solids. The simplest such model is to model the electrons in a metal as a non-interacting gas that follows the Fermi-Dirac distribution [2] or a more advanced approach is density functional theory [8], however in this project only single-electron approximated systems will be considered.

1.4 Three-dimensional methods

Because of the importance of solids and crystals there exist multiple methods that deal

with three-dimensional models of dierent materials using the periodic potential ap-

proach. One common approximation that is made for this is to assume that the poten-

tial in the regions outside some eective radius of the atoms is constant and that the

potentials close to the atoms have some kind of symmetry. An example is the Green func-

tion method explained in Ref. [9] which has been used to study for example aluminum

[10]. Two other methods which also model the regions outside the atoms as constant are

the augmented plane-wave method [11] and the orthogonalized plane-wave method [12]

which use dierent techniques to piece together the plane waves between the atoms with the bound states of the atoms. They have both been used on dierent models of real materials, for example of solid argon [13] and a diamond crystal [14]. These methods deal with the atom structure dierently and have dierent advantages when it comes to convergence, complexity and accuracy. They are all suitable for studies of dierent specic solids but do require knowledge on the chemical and structural properties of the solid in question to be useful and hence are harder to use for general analysis of periodic potentials.

1.5 One-dimensional methods

Even though there exist three-dimensional methods for specic models of periodic po- tentials, the general one-dimensional treatment may still be of interest, especially as a pedagogical tool. As seen with the original treatment of the KP model, which reveals the band structure of solids, there may exist interesting aspects of periodic potentials that can be found in a one-dimensional analysis that may be of use in further three- dimensional studies. One recent development in solutions of one-dimensional periodic potentials is the use of the method that Marsiglio introduces for solving the harmonic oscillator numerically in Ref. [15]. This method was extended to periodic potentials by him and Pavelich in Ref. [16] by using Bloch's theorem and by Le Vot et al. in Ref.

[17] by embedding a number of cells in an innite square well. The method used here is appealing since it is fairly simple and reduces the Schrödinger equation to a simple algebraic eigenvalue problem that can easily be solved on a computer. The method has been tested against the solution of the KP model in Ref. [3] and is found to be accurate and thus has been used for various other problems in Refs. [16, 17]. There exist other numerical methods for the one-dimensional potential, see for example Ref. [18] where a shooting method for dierential equations is used. However a reason to prefer the matrix method used in Ref. [16] is that it solves the Schrödinger equation for an innite peri- odic potential while the dierential equation approach has to use a model with a limited number of cells of the potential.

The matrix method as mentioned is useful for one-dimensional periodic potentials but can also be extended to multiple dimensions, see Ref. [19] for the extension on the method to two dimensions and where it is applied to a simple model of a graphene sheet. From the extension in Ref. [19] it would also be fairly straight forward to go further to three dimensions, however the time complexity of the method scales poorly with the dimension of the problem.

Therefore it may be more useful to use one of the methods mentioned in the previous section when dealing with a three-dimensional problem instead of the matrix method.

1

The demanding part is to calculate the matrix representation of the Hamiltonian and it is an N × N

matrix where N is the number of basis used. There are roughly needed N

to achieve the same accuracy

for the problem in d dimensions as in one dimension so the time complexity scales roughly as N

.

1.6 Impurities

In the references multiple dierent models of one-dimensional periodic potentials covering dierent physical aspects added to the lattice have been studied. One interesting type of model is one which adds some kind of irregularity to the potential, for example as in Ref.

[17, 20] where the energy bands were calculated for a KP model with alternating distances between the wells and in Ref. [21] where the wells are randomly spaced. Irregularities are of importance because some features in Nature originate from the imperfections. One such type of irregularity which is of special interest is impurities in materials. That are materials where one of the atoms in the original structure is replaced by another type which often is referred to as an impurity. A material containing intentional additions of impurities which aect its properties is often called doped. In gure 1.2 an example of a doped KP potential is shown. Here one well is replaced by one of a dierent size.

A version of this type of model is introduced and studied with the dierential equation approach in Ref. [18] and is what will be focused on in this project.

Figure 1.2: Example of a doped KP potential with an impurity at the fth cell.

The eect of impurities in materials is important as it aects some of its properties

which can be utilized not least in for example semiconductors which are of great interest

in modern technology. There exist a lot of research on semiconductors and an overview

of theoretical studies of impurities in them can be found in Ref. [22] and some experi-

mental data of how impurities change the properties in gallium arsenide can be found in

Ref. [23]. There also exist examples such as in Refs. [24, 25] where similar methods as

those mentioned for three-dimensional periodic potentials were used for studying semi-

conductors, however without impurities. Why impurities aect the properties can partly

be explained by new electron states with energies in between the bands of the pure ma-

terial [22]. These reduce the gap between the valence and conduction bands and allow

electrons to easier access the conduction band which increases the conductivity of the

material [22]. This is however a simplication of the whole eect of impurities, which in

reality depends on that the impurities increase or reduce the number of electrons in the

structure [2]. The single-electron approximation will thus not capture the whole eect of

introducing impurities into periodic potentials but as the KP model gives a simple access

to energy bands in solids may the approach still show some general aspects of impurities

that may be of interest.

1.7 This project

In this project the method used in Ref. [16] on periodic potentials will be applied to

periodic potentials with impurities. The general method will rst be described in chapter

2 and later expanded to be able to handle periodic potentials which are dened over

multiple cells instead of just one. This is done in the beginning of chapter 3 so that

potentials containing impurities can be studied. Chapter 3 then goes on to present how

the energy band structure in a KP potential with an impurity is aected by varying the

strength, width and general shape of the impurity as well as having dierent densities of

impurities. It will be studied how the energy bands change and if new energy levels can

be found and if they correspond to the energy levels found in a system consisting of only

the impurity. Overall, the goal is to gain insight in how dierent impurities aect the

energy levels of the periodic potential and see if it can be compared to real eects found

in Nature. The report ends with a summary and conclusion in chapter 4.

Chapter 2 Method

In this project the matrix method as used in Ref. [16] will be somewhat extended to be applied to periodic potentials with impurities. The fundamental idea of the general matrix method is to reduce the Schrödinger equation to an algebraic eigenvalue problem by using a known basis to represent the Hamiltonian in matrix form. The matrix equation can then be truncated, since the matrix generally is innite dimensional, and solved on a computer using matlab, numpy or other software capable of solving eigenvalue equations.

An important step in this method is choosing a relevant set of basis functions and in the references mainly two sets have been used for periodic potentials. The rst is the innite square well basis used in Ref. [17] by embedding some periods of the potential in a innite square well. The second is the basis found by applying periodic boundary conditions and utilizing Bloch's theorem as done in Ref. [16]. In this project the latter one will be used and here follows an explanation of the general matrix method followed by an explanation of the basis functions.

2.1 The matrix method

The method is used for solving the time-independent Schrödinger equation,

− ~

2m ∂

|Ψi + V (x) |Ψi = E |Ψi , (2.1) for some potential V (x) with some boundary condition that restricts the problem to a

nite region of one-dimensional space. The fundamental part of the method is to utilize that the eigenfunctions |φ

i , which are the solutions to the Schrödinger equation without the potential,

− ~

2m ∂

|φ

i = E

|φ

i , (2.2) form a complete and orthogonal basis set [1]. Since the region studied is nite the set of |φ

i will be countable and the solutions to eq. (2.1) can be expressed as a linear combination of them,

|Ψi = X

n

c

|φ

i . (2.3)

Inserting the linear combination (2.3) into eq. (2.1) and then taking the inner product with |φ

i the following expression is found

X

n

hφ

|



− ~

2m ∂

|φ

i + V (x) |φ

i



c

= E X

n

c

hφ

|φ

i . (2.4) Now utilizing the fact that |φ

i is an ortogonal set and satises eq. (2.2) the previous expression can be simplied to

X

n

(E

δ

+ hφ

| V (x) |φ

i)c

= Ec

. (2.5) This is now the Schrödinger equation in matrix form expressed in the basis |φ

i . Dening the matrix elements H

= E

δ

+hφ

| V (x) |φ

i the initial problem of solving eq. (2.1) has been reduced to solving the matrix equation given by

X

n

H

c

= Ec

, (2.6)

which can be solved with basic linear algebra.

To be able to implement this method on a computer a couple of things have to be dealt with rst. Firstly, the matrix (H

) is innite dimensional and has to be truncated to

nite dimensions to be able to solve eq. (2.6) numerically. This is done in Refs. [16, 17]

where they found that their desired accuracy could be achieved by only using the rst

60 basis functions making (H

) a 60 × 60 matrix. Secondly when implementing the method it is desirable to do it in relevant and usable units. In this project it is useful to have the energies in terms of the ground-state energy of the innite square well, E

=

~

π

/(2ma

) , where a is the width of the region in question, so the implemented matrices and potentials can be assumed to be divided by this quantity. The notation h

= H

/E

and u

= V

/E

will be used, where V

is a strength dening a potential.

The matrices were found to be independent of the periodicity, a, of the potentials in most cases so in practice a = 1 could be used without losing any information.

2.2 Bloch wave basis

To be able to use the matrix method the basis functions, |φ

i , have to be chosen. In this project the same approach as in Ref. [16] will be used which is to assume that the potential satises the periodicity V (x + a) = V (x) and then using Bloch's theorem to obtain the basis functions. Bloch's theorem [7] states that the eigenfunctions in a periodic potential will have the following form

φ(x) = e

u

(x), (2.7)

where K is a new quantum number that for an innite periodic lattice provides unique wave functions for all K ∈ [−π/a, π/a] and where u

(x) satises the periodicity of the

1

The order of the Bloch basis functions used in the accordance and here is 0, 1, −1, 2, −2 . . . so when

referring to the rst basis functions it is in accordance to this order.

problem, u

(x) = u

(x + a) . Using eq. (2.7) in the Schrödinger equation without the potential, (2.2), the following equation for u

(x) is found

u

(x) + 2iKu

(x) +  2mE

~

− K



u

(x) = 0, (2.8)

which has the solutions

u

(x) = e

√ 2mE

~ x

. (2.9)

Now for this to satisfy the periodic boundary condition the following must be true

− iKa ±

√ 2mE

~ a = 2πn, n = 0, ±1, ±2 . . . , (2.10) this gives the following allowed energies

E = ~

π

2ma



2n + Ka π



, n = 0, ±1, ±2 . . . . (2.11) With these energies the eigenfunctions are found to be

φ

(x) = r 1

a e

u

(x) = r 1

a e

. (2.12) The inner product in this periodic space is given by

hφ

|φ

i = Z

0

φ

(x)φ

(x)dx, (2.13)

which yields the normalizing factor p1/a in the basis functions. In summary, the periodic boundary condition together with Bloch's theorem gives the following basis set that will be used in the matrix method throughout this project:

φ

(x) = r 1

a e

, E

= ~

π

2ma



2n + Ka π



, n = 0, ±1, ±2 . . . . (2.14)

Now the eigenvalue problem (2.6) can be solved by rst calculating the matrix, (H

) , for

a nite number of eigenfunctions either analytically or numerically and then using some

software that is able to nd the eigenvalues and eigenvectors of the matrix, numpy.linalg

[26] will be used in this project. In the case where the Bloch basis is used the additional

quantum number K has to be accounted for. This is done by repeating the procedure

and solving the eigenvalue equation for multiple values of K in the calculation of (H

) .

Luckily this is easily implemented since only the diagonal of (H

) depends on K so the

matrix does not have to be fully recalculated for each value of K [16].

Chapter 3 Investigation

3.1 Problem

In this chapter the eect of introducing an impurity into the KP model will be studied using the matrix method. The focus lies on establishing how the energy-band structure is aected by the addition of an impurity. Furthermore, it will be investigated how the shape of the impurity, that is how varying the width and strength of a KP impurity and changing the potential form altogether, aects the result. In addition dierent densities of impurities will be tested to both answer how the frequency of impurities aect the results but also to distinguish if the model really can correspond to a system of impurities. The goal is then to try to relate the result with real eects found in materials doped with impurities.

3.2 Model

To be able to model an impurity into a periodic potential the method from the previous chapter has to be adjusted. The model that will be used in this project originates from a normal periodic potential V

(x) which is said to be periodic with periodicity a. The same potential can be represented by looking at a cell containing N + 1 periods of the original potential in a row and dening the periodicity of the problem to be a(N + 1) instead. This does not alter the situation and allows for one of the now called subcells containing the potential V

(x) to be replaced by a subcell containing another potential, V

(x) , instead which now can represent an impurity. The full potential with impurities is now dened in one cell by a total potential function V (x) which satises the periodic condition V (x + a(N + 1)) = V (x). To keep the mathematical denition general the impure subcell will be given an index, l ∈ [0, N], which is the place in the cell where the subcell is placed. This placement should not aect the result but is needed in the mathematical formulation of the potential function of one cell which is now given by

V (x) =

( V

(x), x ∈ [ka, (k + 1)a], k ∈ [0, N ] \ l,

V

(x), x ∈ [la, (l + 1)a]. (3.1)

An example of a cell from a potential of this type is shown in gure 3.1 where N = 5, l = 4 and V

(x) and V

(x) are both KP potentials but with dierent widths and strengths.

The KP potential is denied by its strength, sometimes refered to as depth, V , and the

proportion of the width of the cell the well takes, b, which makes the actuall width of the well ba. The denition that will be used for the potential function of a KP cell is given by

V (x) =

( V, x ∈ [a(

), a(

)]

0, otherwise . (3.2)

This is not the standard way to dene a KP potential but for this project it is more convenient to have the wells centered in each subcell which this denition does. Also note that in this denition the strength is the actual strength and that for the rest of the project this will be replaced by the dimensionless strength u = V/E

.

Figure 3.1: An example of a cell from a doped potential given by eq. (3.1). Here there are N = 5 normal subcells with a KP potential of strength V

and width b

a and one impure subcell with a KP potential of strength V

and width b

a . The impure cell is placed at l = 4.

3.3 Analytical calculations

To be able to use the matrix method on a periodic potential denied by a cell of the type in eq. (3.1) some adjustments have to be done to the method. The basis that will be used is still the Bloch basis, dened in eq. (2.14), but with the periodicity replaced by a → a(N + 1) . The Hamiltonian in matrix form for this type of potential in the Bloch basis is given by the formula

H

= E

δ

+ hφ

| V (x) |φ

i = E

δ

+

Z

V (x) a(N + 1) e

2π(n−m) a(N +1) x

dx. (3.3) The integral can be separated into N + 1 integrals, one over each subcell,

H

= E

δ

+ X

k∈[0,N ]\l

Z

V

(x) a(N + 1) e

2π(n−m) a(N +1) x

dx +

Z

V

(x) a(N + 1) e

2π(n−m) a(N +1) x

dx.

(3.4)

Now since the potential function of each subcell is periodic it does not matter which exact interval they are integrated over so each integral can then be shifted to the interval [0, a] ,

Z

V

(x)

a(N + 1) e

dx = e

Z

0

V

(x)

a(N + 1) e

dx, (3.5) here j = 0, 1 representing both subcell types. Then dening

I

(N ) = Z

0

V

(x)

a(N + 1) e

dx (3.6) as the single subcell integral of each potential the nal Hamiltonian in matrix form for the general potential with impurities can be expressed as

H

= E

δ

+ I

(N )



 X

k∈[0,N ]\l

e



 + I

(N )e

. (3.7) Finally this form is very useful for calculating the matrix form of the Hamiltonian since only two integrals have to be calculated, I

(N ) and I

(N ) , instead of the N+1 integrals in eq. (3.4). So for a given potential V (x) only these two integrals have to be calculated, one for each subcell type, then the matrix can easily be obtained with eq. (3.7). In this project the KP potential, dened in eq. (3.2), will be the most commonly used subcell which in eq. (3.6) gives

I

(N ) = (

N +1

, m = n

u

π(n−m)

e

sin



b



, m 6= n . (3.8)

Note that the dimensionless form of the strength is used here u = V/E

. Later the eect of introducing dierent subcell types will be studied and because of the calculations done here only the expression for the integral I

(N ) has to be calculated to implement each type of potential.

3.4 Numerical analysis

3.4.1 Code implementation

The basic code used for calculating the energy bands in this project can be found in the appendix. It is based on the code found in the appendix of Ref. [16] but extended to be able to handle multiple subcells. The code works by rst letting the user dene the potential in question and provide the integrals from eq. (3.6) for it.

The user also has to specify how many basis functions that should be used to calculate the Hamiltonian and how many values of K in the allowed range [−π/(a(N + 1)), π/(a(N + 1))] the calculations should be done for. How many basis functions to use can be decided by for example plotting the energy bands for an increasing number of basis functions and study the convergence. After all this is given by the user the computer generates the

1

The code support numerical integration but is much slower than if a general expression of eq. (3.6)

is given.

Hamiltonian and calculates the eigenvalues using numpy.linalg.eig [26] and sorts them from lowest to highest. This is then repeated for each K value. A list for each eigenvalue is therefore obtained with dierent values corresponding to dierent values of K. Thus each eigenvalue spans a range of allowed values which form the energy band structure of the potential. Now these bands can be plotted either alone or in comparison to other calculations.

3.4.2 Energy-band representation

As explained in the previous section, the eigenvalues of the Hamiltonian are obtained for dierent values of K and can thus be plotted against K. These plots form curves as seen in the example on the left in gure 3.2 where the ve lowest eigenvalues are plotted for a KP potential with strength u = −10 and width b = 0.5 using 120 basis functions and 201 values of K. All the values these curves cover are the allowed energies of the electrons and as expected they form bands. What happens when the cell consists of multiple subcells is that each band is composed of more eigenvalues as seen on the right in gure 3.2 where the same calculation is done but the cell consists of ten subcells of the same type. Here the same energies are spanned but there are more curves in each band so the plot is harder to read.

Figure 3.2: In the left gure the energy band structure calculated for a cell consisting of only one KP potential with u = −10 and b = 0.5 is shown. In the right it is shown what happens when the calculation is done with ten subcells of the same type. Each red line corresponds to one eigenvalue vs K and the black line is one cell of the potential used. Note that the x-axis for the potentials is not to scale and that the width of each well is equal in both gures.

In this project only the band width and placement will be studied to investigate which

energies can be found in the structure. To make things easier to read will the curvature

of each band be disregarded by plotting them like in gure 3.3. Here the same result is

showed as on the right in gure 3.2 but each energy band is lled between its maximum

and minimum values. Some information is lost by this way of representing the energy

bands, for example the eective mass of the electron can be calculated from the curva-

ture [2, 16], but for this project it conveys the necessary information. These plots will

also be easier to read when impurities is introduces and the now purposeless x-axis can

be used to compare dierent energy-band calculations side by side.

Figure 3.3: The same result as in the right part of gure 3.2 but the overlapping bands are merged and lled for easier reading of the band ranges.

3.4.3 The number of basis functions

In Refs. [16, 17] they use around 60 basis functions for each calculation to gain their

desired accuracy. With the extension to a cell consisting of multiple subcells a greater

number of basis functions is needed to gain good accuracy. To measure how well the

method converges the energy-bands were compared between calculations using dierent

number of basis functions. Each calculation was compared to the calculation which used

the most basis functions by calculating the maximum and average error of the rst 60

eigenvalues between the two runs. These relative errors are shown in gure 3.4 for two

dierent potentials. Both were KP potentials with u = −10 and b = 0.5 where one subcell

was changed to a KP potential with u = −14 and b = 0.8, but one of them had N = 9

normal subcells, plotted in red, to each impurity and the other had N = 19, plotted in

blue. The dashed lines show the maximum error and the full lines shows the average

error. The calculations were done for 100 basis functions up to 400 basis functions in

increments of 10.

Figure 3.4: The maximum and average error between the calculation using dierent numbers of basis functions and compared to the one which used the most basis functions. The blue lines show the result for a potential with ten subcells, nine normal to one impurity, and the red lines show the results for 20 subcells. This shows how well the method converges for dierent number of basis vectors assuming the one using the most basis functions is fairly accurate.

3.5 Results

3.5.1 Introducing an impurity

The eect that an impurity has on a periodic potential was studied by looking at a potential dened by the general formula in eq. (3.1). In all calculations the potential functions could be dened in a similar way as the KP potential in eq. (3.2), with a unitless strength u and a fractional width b. When referring to the strength or width of the normal subcells the subscript 0 was used and the subscript 1 was used in reference to the impurity. For example, u

is the strength of the normal subcell and b

is the width of the impurity.

Now the method could be used to obtain the band structure of a potential containing an

impure subcell. In gure 3.5 the band structure can be seen for a potential containing

N = 9 normal subcells which were KP potentials with u

= −10 and b

= 0.5 and a KP

impurity with u

= −6 and b

= 0.7 . The calculation was done using 200 basis vectors

and 101 values of K.

Figure 3.5: Energy bands for a potential consisting of N = 9 KP potentials with strength u

= −10 and width b

= 0.5 and one KP impurity with u

= −6 and b

= 0.7 .

In gure 3.5 two new bound energy levels can be found and the probability distributions for those two levels can be seen on the left in gure 3.6 in reference to one cell of the potential. This displays where in the potential an electron in each new state is likely to be found. For reference is the same shown for two energy levels found in the original bands in the right plot of gure 3.6. In addition to this change in the band structure are the previously continuous bands broken into smaller pieces, which can be hard to see for the smaller bands and is therefore highlighted in gure 3.7.

Figure 3.6: Probability distributions in one cell of the doped potential from gure 3.5. The left

shows the probability distribution corresponding to the new discrete energy levels and the right

shows the distribution for two energy levels found in the original bands.

Figure 3.7: Demonstration of band splitting in a smaller band when an impurity is introduced to the potential. The left gure shows the band structure without any impurity for a KP potential and the right one shows the band structure for the same potential but one in every ten subcells is replaced by an impurity.

How changing the impurity in dierent ways aect these new features of the band struc- ture could now be studied by plotting the band structure for dierent calculations side by side on the purposeless x-axis in these gures.

3.5.2 Width and strength dependence

To see how the width of the impurity aected the energy band structure of the periodic potential the bands were calculated for multiple values of the width and then plotted side by side. The potentials chosen to be studied for this consisted of N = 9 normal KP subcells with strength u

= −10 and width b

= 0.5 for each impurity with strength u

and width b

. The result is shown in gure 3.8 where the energy bands were calculated for b

= 0.05 to b

= 1 in increments of 0.05. This was done for three dierent values of u

which from top to bottom in gure 3.8 were u

= −6 , −10 and −14.

The same thing was then done for studying the strength dependence and the result is

shown in gure 3.9. The strength was varied between u

= −20 and 0 in increments of 1

and the calculations were done for three values of b

, which were in the gure from top

to bottom b

= 0.2 , 0.5 and 0.8. All calculations used 200 basis functions and 101 values

of K. In addition to the red energy levels in each gure the blue crosses represent the

energies found in a nite square-well potential with the same width and strength as the

impurity. The method for nding these energies can be found in for example Ref. [1].

Figure 3.8: The energy band structure of a KP potential with N = 9 normal subcells with b

= 0.5 and u

= −10 for each impurity vs the impurity's width, b

. In the three gures dierent values of the impurity strength u

were used. From top to bottom were they u

= −6 ,

−10 and −14. The width was varied from b

= 0.05 to b

= 1 in increments of 0.05 and each

calculation was done with 200 basis functions and 101 values of K. In addition to the band

structure which is shown in red the energies for a non-periodic nite square well with the same

shape as the impurity are shown as blue crosses for comparison.

Figure 3.9: The energy band structure of a KP potential with N = 9 normal subcells with

b

= 0.5 and u

= −10 for each impurity vs the impurity's strength, u

. In the three gures

dierent values of the impurity width b

were used. From top to bottom were they b

= 0.2 ,

0.5 and 0.8. The strength was varied from u

= −20 to u

= 0 in increments of 1 and each

calculation was done with 200 basis functions and 101 values of K. In addition to the band

structure which is shown in red the energies for a non-periodic nite square-well with the same

shape as the impurity are shown as blue crosses for comparison.

3.5.3 Density dependence

In the previous section the density of impurities was constant at one in ten. In this section it was instead studied how the band structure was aected by having dierent numbers of normal subcells to each impurity to replicate density dependence. The calculations were done with the same principles as in the previous section. The band structure was calculated for dierent values of N and plotted side by side for comparison. The calculations were done for the same potential as previously, N KP potentials with u

=

−10 and b

= 0.5 to each impurity. The values of N varied between 1 and 19 and the results are shown for four dierent impurities in gure 3.10. From left to right in the

gures, the strength were u

= −6 , −6, −14 and −14, and the width b

= 0.2 , 0.8, 0.2 and 0.8. Each calculation was done with 200 basis functions and 101 values of K and the energy band is shown in red for each density. In addition to these bands the bands for a periodic potential with only the impure cell is shown in green to the left and the energy bands for a potential without impurities is shown to the right in blue.

Figure 3.10: Density dependence for a periodic potential with u

= −10 and b

= 0.5 containing four dierent impurities which from left to right had the strength and width u

, b

= −6, 0.2 ,

−6, 0.8 , −14, 0.2 and −14, 0.8 respectively. In addition to the red energy bands for dierent N, the green energy band corresponds to the energy levels from a potential consisting only of the impurity and the blue is the potential without any impurity. Each band calculation was done with 200 basis functions and 101 values of K.

A clear band splitting eect could still be seen for the lower densities. To investigate this

further a zoomed-in view of the second and third bands from the top left plot in gure 3.10 can be found in gure 3.11.

Figure 3.11: A zoomed-in view of the second and third bands from the top left plot in gure 3.10 to highlight the band splitting.

3.5.4 Other impurity types

In the previous sections the impurities introduced into the periodic potential were of the same type as the regular potentials they replaced, a KP potential but with a dierent size than the rest of the lattice. The next thing that was explored was the eect of having other types of impurities in the lattice. The results of three dierent potentials are presented here. The rst is a harmonic oscillator type potential dened by

V

(x) =

( −u 

2x−a ba

− 1 

, x ∈ [

,

]

0, otherwise . (3.9)

The second is a square-root analog to the harmonic oscillator dened by

V

(x) =







−u q  

  − 1 

, x ∈ [

,

]

0, otherwise . (3.10)

Finally a cosine potential was used dened by

V

(x) = (

2

cos π

− 1
, x ∈ [

,

]

0, otherwise . (3.11)

All three potentials were dened this way so that they would have the width ba and

strength u, exactly like in the denition used for the KP potential in eq. (3.2). This

makes it easier to replace the impurities in the lattice and compare the results. These

potentials were not chosen for any special reason other than the integral from eq. (3.6)

could be calculated analytically for each one of them. The integrals, calculated with

maple, is listed here and they were all tested against numerical integration to verify their

validity. For the harmonic oscillator the integral was calculated to

I

(N ) =







2ub

3N +3

, m = n

−2u(N +1)eⁱ

π(n−m) N +1

π²b(n−m)²

 cos 

π(n−m) N +1

b 

−

sin 

π(n−m) N +1

b 

, m 6= n , (3.12) and the integral for the root potential was

I

(N ) = (

4N +4

, m = n

−iRu

P Q(n−m)π

e

(erf(P )Q − erf(Q)P ) , m 6= n , (3.13) where R = 0.4431134627, P = pibπ(n − m)/(N + 1) and Q = p−ibπ(n − m)/(N + 1), and nally for the cosine potential the integral was found to be

I

(N ) =



 



 



ub

2N +2

, m = n

(N +1)²u

2π(n−m)((N +1)²−b²(n−m)²)

e

sin 

N +1



, m 6= n

ub

4N +4

e

, if b(n − m) = ±(N + 1)

.

(3.14)

These were now placed as impurities in the standard potential with N = 9 KP potentials

with u

= −10 and b

= 0.5 for each impurity. Each impurity was placed with the

strength u

= −10 and width b

= 1 and the energy bands are shown in red for each

impurity type in gure 3.12. In addition to these energy bands the energies for a potential

without the impurity are shown in blue to the right for comparison. In section 3.5.2 the

energies were also compared to those found in the nite square well, the corresponding

problem to these impurities is not as easy to solve. Instead the energies found in a

periodic potential consisting of only the impurity cell type are shown in green to the left

to give some reference.

Figure 3.12: The energy band structure for the three impurities dened in eqs. (3.9-3.11) is shown in order from left to right. Each impurity had b

= 1 and u

= −10 and was located in a lattice of KP potentials with u

= −10 and b

= 0.5 with N = 9 normal subcells to each impurity. Each calculation was done with 200 basis functions and 101 values of K.

3.6 Discussion

3.6.1 Eects of impurities

The goal of this project was to introduce impurities into the KP model and see how the energy band structure was aected. Throughout all the results presented mainly two new features were introduced into the energy band structure of the potential. Firstly new almost discrete energy levels could be found in between the bands from the potential without impurities. These bands were found to correspond to states bound to the impure cell as emphasized in gure 3.6. The second feature was that the previously continuous bands split up into closely packed almost discrete levels, but which still spanned roughly the same range of energies as before. This is highlighted in gure 3.7.

The eect that changing the width and strength of the impurity had on the band structure

was investigated closer in section 3.5.2. No new eects on the energy band structure could

be found other than that the new energy levels depended on the impurity's shape. The

new energy levels were compared to the energies found in the isolated non-periodic nite

square well and it was found in gure 3.8 and 3.9 that these matched very well together

with the exception of when the energies approached 0, which is close to free electron levels, or when the energy fell close to another energy band. This leads to the conclusion that a fairly good approximation to the energies in the doped potential is the union of energies from the potential without impurities with the intrinsic energies from the impurity. This was suggested in Ref. [18] but was now conrmed more thoroughly here.

In section 3.5.4 the eects of having other potential types as impurities, dierent from a KP potential were explored. The results of this did not reveal any new eects in comparison with the KP impurities. No test on how well the new introduced energy levels correlate to the energies found in the isolated impurity, as done with the nite square well in section 3.5.2, could be done either. If it had been done it would probably have yielded the same results as previously found but it would be interesting to see if they possibly deviated more because of them not being of the same type as the rest of the potential which may lead to some kind of interference. A comparison was made with the bands found in a periodic potential with only the impure cell and the new energies were at least found to lay in the ranges of those bands so they do correspond to the isolated impurity levels in some way.

In the previous results discussed the potential investigated had a constant density of one impurity in ten cells. In section 3.5.3 other densities were tested to see how they would change the energy band structure. It could be seen that the general structure of the bound bands, E < 0, stabilized at ratios around one impure in every seven to ten normal subcells. The band splitting is however still present at these densities but the ranges each band spanned become fairly xed and could probably be approximated as continuous already for these densities. This implies that these densities and lower can be considered to model impurities and that higher densities should probably be regarded as molecule structures instead. However a question arises about how well these really correspond to real doped potentials. In reality the ratios of impurities are in the order of 1 :10

or lower [22]. Those ratios are not possible to study with this method because a huge number of basis functions would be necessary. However from the graphs in gure 3.10 it seems likely that nothing would change for those kinds of densities anyway. The ratio 1 :10

is also measured in three-dimensional materials which cannot be directly translated to the corresponding ratio in one dimension which should be lower.

The result observed throughout this project can be summarized by the introduction of

energy levels in the previous empty regions and the band splitting. The rst of these is

the most important and has the clearest relation to real material. In real materials, the

conductivity changes when impurities are present [23] which can partly be explained by

the impurity's introduced energy states [22]. The conductivity depends on the energy

gap between the valence and conduction bands and by introducing new energy levels

in between them some electrons can occupy these states and be more easily thermally

excited into the conduction band and thus increases the conductivity [2]. In this project

new energy levels corresponding to the impurity's intrinsic energies were found in between

the conduction and valence bands. Here the valence levels are assumed to be the energies

less than 0 and the conduction band is roughly the ones over 0. No close investigation

of the conduction band was done because as seen the structure is more complicated but

throughout the results one can observe that the lower edge of it is fairly constant so the

energy gap does only change because of the new introduced energies. These energies probably correspond to the energies found in real materials and may therefore be of use in further studies to calculate conductivity dependence and other properties in more sophisticated models of realistic materials.

The band splitting observed does not have any corresponding eects found in real mate- rials but can probably be explained as a consequence of the breaking of the translation symmetry of the potential. A periodic potential without impurities that can be assumed to be innite is allowed continuous values of K which lead to continuous bands. However when the potentials are not innite the values of K become discrete [7]. This is probably what the method used captures with the band splitting because the normal cells are only invariant under N translations when an impurity is present and there are N levels in the splitting in the right plot of gure 3.7. The eect is probably realistic because it can be explained by the transition between a molecule structure and impurity structure as seen in gure 3.10. The eect will however not be measurable in real materials because low densities of impurities gives almost continuous bands and other external eects will dominate over the small energy dierences in the bands. Thus can the splitting probably be disregarded in reality. So overall, the success of nding the new discrete energy levels was that of most interest in this project and may be what allows it to be more useful in further studies.

3.6.2 Method evaluation

In Refs. [16, 17] which used the matrix method the results could be compared with the analytic solution to the KP model and the results were found to be accurate. Thus to evaluate the method used in this project the result of a calculation using multiple subcells, without any impurities, was compared to the calculations from those references using only one subcell. The result is shown in gure 3.2 and the ranges are the same which indicates that the method used here also was accurate. However some information is lost, the E(K) plots are harder to read, so some improvements have to be done if those are of interest, but in this project only the ranges of the bands were studied so the method was deemed satisfactory and suitable to be used on the impurity calculations.

For a single-cell calculation only about 60 basis functions were used to gain the desired

accuracy in the references. For the multicell approach used here more functions were

necessary to reach the same desired accuracy. Since no analytic results exist for impurity

calculations only relative errors could be studied to decide how many basis functions

were necessary for each calculation. This was done in gure 3.4 for two potentials, one

with ten subcells which was the standard number used throughout the project and one

with twenty which was the largest potential studied in this project. The relative error

was decided to be small enough at around 200 basis functions for both potentials which

was the number used throughout the rest of the calculations. This is quite an increase

from 60 and the time for a band calculation increased from a couple of seconds to about

10-20 seconds on a standard desktop computer. This is noticeable, especially if multiple

calculations were done, but not terrible for what was gained by being able to use multiple

subcells. However if the multicell approach was to be used in higher dimension this may

cause problems.

3.6.3 The next step

This project may not have yielded any direct useful results for any application but it highlighted that the method of periodic potentials can capture more complicated aspects found in real materials. The next step relating to this project is to rst expand the matrix method into three dimensions using the approach for two dimensions in Ref. [19].

Then use the method to analyze the band structure of some real solids and compare it

to methods such as those in Refs. [9, 11, 12]. When this has been done maybe a similar

expansion as used in this project can be applicable in three dimensions. It will probably

not be feasible to capture the eects of impurities at the densities found in Nature but the

subcell approach can be used for multi-atomic structures such as ionic crystals. However

since the time complexity scales poorly with the dimension a lot of optimization has to

be done in the code for it to be usable in higher dimensions, this was not a priority

in this project. Some possible improvements that could be done is parallelization of

the matrix operations and maybe taking advantage of the symmetry for the K values

better. Another possible approach, which may not require as much improvement, is to

just model a specic symmetry direction of a three-dimensional crystal. This reduces a

more realistic model down to one dimension and with this approach it may be feasible

to try to analyze a more realistic model of impurities.

Chapter 4

Summary and conclusions

The goal of this project was to use the periodic potential model to calculate the energy band structure for potentials with impurities. The matrix method from Ref. [16] was thus expanded to deal with a periodic potential dened over multiple subcells. This was done by simply increasing the periodicity from just one cell to a cell containing multiple cells of the original potential. This approach was compared to the single cell method and was found to replicate the result. The method was thus deemed to be suitable to use to investigate the problem of impurities in periodic potentials.

The method was applied on simple one-dimensional KP potentials containing impurities.

The investigation focused on determining how the energy structure of the electrons in the potential depended on the size and density of the impurities. It was found that the impurity inuenced the energy band structure by introducing new discrete energy levels that corresponded fairly well with the isolated impurity's intrinsic energies and by splitting the previously continuous band found in the pure potential into smaller tightly packed bands spanning roughly the same interval as in the pure potential. The general band structure was found not to depend much on the impurity's shape except for the placement of the new energy levels. The structure was found to be more dependent on the densities of impurities but at around one impurity to eight normal subcells the band structure seemed to stabilize and not change for lower densities. Thus at these densities the band structure, at least for the bound energies, was concluded to be approximately the band structure without impurities plus the discrete energy levels from the impurity.

The results seemed to reect the physical reality fairly well for being such a simple

method. It captured the fact that the impurities introduce new energy levels and that the

rest of the structure was fairly unaected by the impurity. This opens up the possibility

for this approach to be used in the future in three-dimensional analyses of realistic models

of materials. However as discussed there may exist great challenges with reducing the

computing time to be quick enough to be usable. In conclusion, the introduction of

impurities was deemed successful by capturing the new energy levels and hopefully the

approach can be used in the future expansion of the matrix method into three dimensions.

Acknowledgements

I would like to thank my supervisor Tommy Ohlsson for guiding me through this project.

I want to especially thank him for the help he provided regarding the structure and

language of the report.

Chapter 5 Appendix

Here follows the basic code used in this project to calculate the band structure. The function getEnergies is the important one which returns the energies for all K and can be used in dierent ways to create the plots in this report. To use other types of poten- tials a similar class as the KP class in the code has to be implemented. The function plotEnergyBandsDoped is an example which plots the simple energy band structures. If the provided code is copied exactly the result yielded is gure 3.5.

import numpy as np

import numpy.linalg as linalg

import scipy.integrate as integrate import scipy

import matplotlib.pyplot as plt class Potential():

def __init__(self,pot):

self.pot=pot def func(self,x):

return self.pot(x) def integral(self,m,n,N):

if (m==n):

integrand =lambda x: self.pot(x)*1/(N+1) else:

integrand=lambda

x:self.pot(x)*1/(N+1)*np.exp(1j*np.pi*2*(n-m)/(N+1)*x) real =lambda x:scipy.real(integrand(x))

img =lambda x:scipy.imag(integrand(x))

return integrate.quad(img,0,1)[0]*1j+integrate.quad(real,0,1)[0]

class KP(Potential):

def __init__(self,u0,b):

self.u0=u0 self.b=b def func(self,x):

return self.u0*(x>(1-self.b)/2)*(x<(1+self.b)/2) def integral(self,m,n,N):

if (m==n):

return self.u0*self.b/(N+1)

return np.exp(1j*np.pi*(n-m)/(N+1))/(np.pi*(n-m))

*self.u0*np.sin(np.pi*(n-m)*self.b/(N+1)) def createMatrix(NB,N,I0,I1,l):

ns=[]

for n in range(NB):

ns.append(int((1-(2*n+1)*(-1)**n)/4)) h0=np.zeros((NB,NB),dtype=complex)

for i in range(NB):

for j in range(i,NB):

m=ns[i]

n=ns[j]

if (n==m):

h0[i,j]=4*n**2/(N+1)**2+I0(m,n,N)*N+I1(m,n,N) else:

s=0 for k in range(0,N+1):

if(k==l):

s+=I1(m,n,N)*np.exp(1j*2*np.pi*(n-m)/(N+1)*l) else:

s+=I0(m,n,N)*np.exp(1j*2*np.pi*(n-m)/(N+1)*k) h0[i,j]=s

return h0+np.conj(np.transpose(np.triu(h0,1))) def getEnergies(NB,N,I0,I1,l,Ka):

ns=[]

for n in range(NB):

ns.append(int((1-(2*n+1)*(-1)**n)/4)) h0=createMatrix(NB,N,I0,I1,l)

energies=[]

for ka in Ka:

h=np.array(h0) for n in range(NB):

h[n,n]+=4*ns[n]*ka/(np.pi*(N+1))+(ka/np.pi)**2 [E,v]=linalg.eig(h)

E=np.sort(np.real(E)) energies.append(E) return np.array(energies) def getBandInfo(energies,maxE):

maxs=[]

mins=[]

gaps=[]

width=[]

b=0

while b<len(energies[0,:]) and np.min(energies[:,b])<maxE:

maxs.append(np.max(energies[:,b])) mins.append(np.min(energies[:,b])) width.append(maxs[b]-mins[b]) if b>0:

gaps.append(mins[b]-maxs[b-1]) return b+=1 maxs,mins,width,gaps

def mergeBands(maxs,mins,width,gaps,t):

nMax=[]

nMin=[mins[0]]

nGaps=[]

nWidth=[]

for i in range(0,len(maxs)-1):

if(gaps[i]>=t):

nGaps.append(gaps[i]) nMax.append(maxs[i])

nWidth.append(nMax[-1]-nMin[-1]) nMin.append(mins[i+1])

nMax.append(maxs[-1])

nWidth.append(nMax[-1]-nMin[-1]) return nMax,nMin,nWidth,nGaps

def plotEnergyBandsDoped(NB,ittr,N,pot0,pot1,l,maxE, plotFuncs=True,merge=False,merTol=0):

Ka=np.linspace(-np.pi/(N+1),np.pi/(N+1),ittr)

Es=getEnergies(NB,N,pot0.integral,pot1.integral,l,Ka) maxB,minB,widthB,gapsB=getBandInfo(Es,maxE)

if(merge):

maxB,minB,widthB,gapsB=mergeBands(maxB,minB,widthB,gapsB,merTol) i=0 while (i<len(minB) and minB[i]<maxE):

m=np.min([maxB[i],maxE])

plt.fill_between([-1,1],[minB[i],minB[i]],[m,m],color='red') else: i+=1

b=0

while(b<len(minB) and minB[b]<maxE):

plt.plot(Ka*(N+1)/np.pi,Es[:,b],color='red') b+=1

plt.ylabel(r"$E/E^{isw}_0$",rotation=0) if(N==0):

plt.xlabel(r"$Ka/\pi$") else:

plt.xlabel(r"$Ka(N+1)/\pi$") if(plotFuncs):

potential=np.array([]) for k in range(N+1):

if(k==l):

potential=np.append(potential,pot1.func(np.linspace(0,1,200))) else:

potential=np.append(potential,pot0.func(np.linspace(0,1,200))) plt.plot(np.linspace(-1,1,200*(N+1)),potential,color="black",alpha=0.6) plt.ylim(top=maxE)

plt.tight_layout() plt.show()

plotEnergyBandsDoped(200,101,9,KP(-10,0.5),KP(-6,0.7),5,10,merge=True)

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