Theoretical Physics
Electron Propagation in Periodic Potentials
Klas [email protected]
Deborah [email protected]
Jakob Wä[email protected]
SA104X Degree Project in Engineering Physics, First LevelDepartment of Theoretical Physics
Royal Institute of Technology (KTH)Supervisor: Patrik Henelius
May 13, 2011
Abstract
In this bachelor thesis we study the propagation of electrons in crystals. The crystallinestructure is modeled as a one dimensional periodic potential primarily composed of deltafunction potential barriers.
We use two different models to describe how a particle behaves in such a periodicstructure. The first model is the Kronig Penney model described by S. Gasiorowicz in[1]. The second model is described by Olsen and Vignale in the article "The Quantummechanics of electric conduction in crystals" [3]. We show that there exists certain rangesof energy for which electron propagation can occur, namely the allowed energy bands se-pareted by regions of forbidden energies. In addition, we solve the Schrödinger equationnumerically for some simple cases and reproduce some of the results seen from the twomentioned models.
Sammanfattning
I det här kandidatexamensarbetet studerar vi elektronrörelser i kristallstrukturer. Kri-stallen modelleras som en endimensionell periodisk potential som först och främst äruppbyggd av deltafunktionsbarriärer.
Två olika modeller används för att beskriva hur en partikel beter sig i en periodiskstruktur som ett kristallgitter. Den ena modellen är Kronig-Penney-modellen som finnsbeskriven av S. Gasiorowicz i [1]. Den andra modellen är beskriven av Olsen och Vignalei artikeln The Quantum mechanics of electric conduction in crystals" [3]. Vi visar attelektronledning enbart kan inträffa för vissa tillåtna energier, de tillåtna energibandenvilka är separerade av regioner av förbjudna energier. Vi löser också Schrödingerekvatio-nen numeriskt för ett enklare fall och reproducerar somliga resultat från de två nämndamodellerna.
Contents
1 Introduction 2
2 Background Material 32.1 Conduction bands and periodic potentials . . . . . . . . . . . . . . . . . 32.2 The Schrödinger equation and quantum mechanics . . . . . . . . . . . . 42.3 Bloch’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Transmission and reflection coefficients for delta potential . . . . . . . . . 6
3 Investigation 83.1 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Description of Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2.1 The Kronig-Penney model . . . . . . . . . . . . . . . . . . . . . . 83.2.2 The Olsen and Vignale model . . . . . . . . . . . . . . . . . . . . 8
3.3 Analytical Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3.1 The Kronig-Penney model . . . . . . . . . . . . . . . . . . . . . . 93.3.2 The Olsen and Vignale model . . . . . . . . . . . . . . . . . . . . 11
3.4 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4.1 The Olsen and Vignale model . . . . . . . . . . . . . . . . . . . . 153.4.2 Numerical solution of the Schrödinger equation . . . . . . . . . . 15
3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.5.1 The Kronig-Penney model . . . . . . . . . . . . . . . . . . . . . . 183.5.2 The Olsen and Vignale model . . . . . . . . . . . . . . . . . . . . 193.5.3 Numerical solution and simulations . . . . . . . . . . . . . . . . . 21
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.6.1 The Kronig-Penney model . . . . . . . . . . . . . . . . . . . . . . 223.6.2 The Olsen and Vignale model . . . . . . . . . . . . . . . . . . . . 233.6.3 Numerical solution and future work . . . . . . . . . . . . . . . . . 23
4 Summary and Conclusions 25
Bibliography 27
1
Chapter 1
Introduction
Quantum mechanics gives a general physical description of our world and is used to studyelectron properties and the mechanisms behind electrical conduction. Various models todescribe how electrons behave in lattice structures have been formulated, and a few havebeen more successful than others. We mention the free electron model and the nearlyfree electron model. In the free electron model electrons are thought of as not affectedby the ion cores of the atoms. In the nearly free electron model electrons are consideredto be perturbed only weakly by the periodic potential of the ion cores.
To begin with, we study a widely used model for periodic potentials, namely theKronig-Penny model, which is formulated in the context of the nearly free electron model.In the Kronig-Penney model square well potentials are often used. However one can alsouse other approximations, such as potentials with the spatial behavior of a delta function.By studying the Kronig-Penney model one can find that electron propagation may onlyoccur for certain energies, the allowed energy bands, which are separeted by forbiddenenergies, named the band gaps.
Another model we study is described by Olsen and Vignale in the article [3]. Thebasic idea of this model is to explain the behavior of electrons in a crystal by studyinginterference of transmitted and reflected electron waves in the crystal. Transmission canonly occur for certain energies that are equal to the allowed energy bands found by usingthe Kronig-Penney model.
The Schrödinger equation contains all information about a quantum mechanical sys-tem and quantum mechanical properties can be studied by solving the equation. TheSchrödinger equation may be solved analytically in some cases, but far from all problemsin quantum mechanics can be solved this way. This creates a demand for other means tosolve the problem, where often computers and numerical methods are employed. There-fore we also study a numerical solution of the Schrödinger equation, where a wave packetis used to model a propagating electron.
2
Chapter 2
Background Material
2.1 Conduction bands and periodic potentialsThe inside of a crystal may be described by the free electron model. In the free electronmodel the potentials between the ion cores is supposed to be insignificantly small andthere will only be a great potential at the edges of the material. Inside the materialthe conduction electrons can then be seen as free, but still they will be bounded to thematerial by the large potential at the edges. The free electron model describes a particlein a box problem. This model is sufficiently good when studying some properties but it isnot good enough to explain the difference between insulators, metals and semiconductors.
When improving the free electron model by adding potential barriers between the ioncores one get a model consisting of periodic potentials, the nearly free electron modelwhere the electrons are weakly bounded to the ion cores. When studying the nearly freeelectron model it will be seen that electron propagation only occurs for some energies.The regions of energy where electron propagation can occur are named the allowed energybands and the regions of energy where there is no electron propagation are called bandgaps or forbidden energy bands. [2]
The property of energy bands found by using the model of periodic potentials mayexplain the difference between insulators, metals and semiconductors. The electrons ofthe material will occupy the states in the allowed energy bands. If any allowed energybands are partially filled the material will be a metal, since the electrons in the partiallyfilled band can move to neighboring states that are empty. If an electrical field is applied acurrent will arise. If the allowed energy bands are completely filled or empty the electronswill not be able move and the material is an insulator. For a semiconductor the forbiddenenergy band is narrow, making it possible for electrons to get excited to a higher allowedenergy band and thus the material can behave either as an insulator or a metal. Whenan electron is excited to a higher allowed energy band it will provide an empty state,known as a hole, in the lower band. This hole will also contribute to the conductivity.The highest allowed energy band containing electrons is called the conduction band butit is empty for an insulator. [1]
3
Figure 2.1: Figure showing electron occupancy in allowed energy bands.(a) In an insulatorthe allowed energy bands are either filled or empty and electrons cannot be accelerated by anelectrical field. (b) In a metal an allowed energy band is partially filled and electrons can beaccelerated by an electrical field. (c) In a semiconductor the forbidden energy band is narrowmaking it possible for electrons to be excited to the conduction band. The material can beeither an insulator or a metal. Figure from [1].
2.2 The Schrödinger equation and quantum mechanicsFrom two-slit experiments it is known that electrons produce an interference pattern,e.g. p.13, figure 1-9 [1]. To explain this peculiarity, a wave description seems plausible.The wave function must also allow for self-interference, making linearity of the equation anecessity. When using a wave packet to describe a particle, e.g. an electron, it is not clearexactly where the electron is localized. Instead one does a probabilistic interpretationof the electron position. The Heisenberg uncertainty principle ∆x∆p ≥ ~
2relates how
well one can determine the momentum and the position. The Heisenberg uncertaintyprinciple can be derived as general result from Fourier analysis. When one has a wavepacket description of the electron movement, one can motivate the Schrödinger equationby accounting for momentum, which can be described using the Planck relation ω = E
~and the de Broglie relation k = p
~ , where ω is the frequency, E the energy, ~ is Planck’sconstant, k the wave number and p the momentum. The motivation is performed inchapter 2 in [1]. The name “Quantum mechanics” comes from the fact that describingthe energy relations of certain phenomena was fruitful when using a description involvingquantized levels of energy, instead of a continuous energy description used in classicalmechanics.
2.3 Bloch’s theoremIn the following section we briefly explain how the central equation is used in solid statephysics, and utilize the central equation to prove Bloch’s theorem. The result of Bloch’stheorem is used when describing the solution of the Schrödinger equation in periodicpotentials. The proof is adapted from the solid state physics textbook by Kittel. [2]
In a linear lattice with a lattice constant a, let U(x) denote the potential energy ofan electron that traverses the potential. The potential is invariant under crystal latticetranslation, U(x) = U(x + a). From the theory of Fourier series we have that since thefunction U(x) is periodic, it may be expanded in a Fourier series. With the reciprocallattice vector G the Fourier series of the potential energy is written as
U(x) =∑G
UGeiGx .
4
To maintain a physical interpretation, we demand that the potential U(x) is real-valued
U(x) =∑G>0
UG(eiGx + e−iGx) = 2∑G>0
UG cosGx . (2.1)
Using the potential from Eq. (2.1) when solving Hψ = εψ, where H is the hamiltionianand ε is the energy eigenvalues, one obtains an expression for the motion of an electronin a potential. The Schrödinger equation for this potential is written out explicitly(
1
2mp2 + U(x)ψ(x)
)=
(1
2mp2 +
∑G
UGeiGx
)ψ(x) = εψ(x) . (2.2)
The boundary conditions of the potential gives a number of allowed wave vectors, denotedwith k. From the properties of Fourier series the resulting wave function can be expressedas (with k real)
ψ(x) =∑k
C(k)eikx. (2.3)
When using the expression Eq. (2.3) for ψ(x) when solving Eq. (2.2) one obtains awell-known result from solid state physics, namely the central equation(
~2k2
2m− ε)C(k) +
∑G
UGC(k −G) = 0 (2.4)
[2] p.172.Different techniques can be employed to determine the C(k − G), and often only a
few coefficients is needed for one application. When the C:s are determined from Eq.(2.4) we have the wave function
ψk(x) =∑G
C(k −G)ei(k−G)x . (2.5)
Defineuk(x) ≡
∑G
C(k −G)e−iGx .
The Bloch theorem states that the solution of the Schrödinger equation for a periodicpotential will be on the form
ψk(x) = eiqxuk(x) (2.6)where uk(x) has the period of the lattice, meaning uk(x) = uk(x+ a), and eika is a planewave. Rearranging Eq. (2.5) we have
ψk(x) =
(∑G
C(k −G)e−iGx
)eikx = eikxuk(x) .
Evaluating uk(x+ T ) to test for periodicity gives
uk(x+ T ) =∑
C(k −G)eiG(x+T ) = e−iGT(∑
C(k −G)e−iGx)
= e−iGTuk(x) .
From solid state physics we have that e−iGT = 1,[2] p.30, which verifies the periodicity.This was already expected, since we had a Fourier series over the reciprocal lattice vectors.This proves Bloch’s theorem. The result will be used when studying the Kronig-Penneymodel.
5
2.4 Transmission and reflection coefficients for deltapotential
In the models described in this thesis we will primarily consider potential barriers withthe spatial behavior of a delta function. It leads to simplified calculations and gives aconvenient model of a crystal.[2] In section 3.3.2 we will study transmission and reflectionof electron waves through such potential barriers and thus we need to compute thetransmission and reflection coefficients. The transmission and reflection coefficients inquantum mechanics are complex numbers and the probability of an outcome is found bysquaring the magnitude of the coefficient. To determine the transmission and reflectioncoefficient for a delta potential barrier we will consider the time independent Schrödingerequation,
− ~2
2m
d2ψ
dx2+ V (x)ψ(x) = Eψ(x),
where V (x) = V0δ(x). Rearrangement of the terms gives
d2ψ
dx2+
2m
~2Eψ(x) =
2m
~2V0δ(x)ψ(x). (2.7)
The potential will be zero everywhere except in x = 0
d2ψ
dx2+ k2ψ(x) = 0, k =
√2mE
~2. (2.8)
Solving this and taking into account that we have an incident and reflected wave on theleft-hand side and only a transmitted wave on the right-hand side yields
ψl(x) = eikx + reikx x > 0ψr(x) = teikx x < 0.
The wave function must be continuous at x = 0 which gives the condition
1 + r = t. (2.9)
Since the potential is a delta function the derivative of the wave function will not becontinuous. [1] Below ε is arbitrarily small and positive.(
dψ
dx
)ε
−(dψ
dx
)−ε
=
∫ ε
−ε
2m
~2[V (x)− E]ψ(x) dx (2.10)
limε→ 0
∫ ε
−ε
2m
~2[V (x)− E]ψ(x) dx = lim
ε→ 0
∫ ε
−ε
2m
~2V0δ(x)ψ(x) dx− lim
ε→ 0
∫ ε
−ε
2m
~2Eψ(x) dx
=2mV0~2
ψ(0) =2mV0~2
t
(2.11)
limε→ 0
(dψ
dx
)ε
=
(dψr(x)
dx
)x=0
= ikteik·0 = ikt (2.12)
6
limε→ 0
(dψ
dx
)−ε
=
(dψl(x)
dx
)x=0
= ikeik·0 − ikre−ik·0 = ik(1− r) (2.13)
Inserting Eq. (2.11), (2.12) and (2.13) to the original expression, Eq. (2.10) yields
ikt− ik(1− r) =2mV0t
~2. (2.14)
The continuity of the wave function gives us condition (2.9) while the discontinuity of thederivative of the wave function gives us condition (2.14). Using these two conditions onecan find the expressions for the transmission and reflection coefficients. The transmissioncoefficient for a delta potential barrier is found to be
t =ik~2
ik~2 −mV0. (2.15)
If V0 is set to be
V0 =~2
2m
λ
a
the transmission coefficient obtained is
t =2ik
2ik − λa
. (2.16)
When knowing the transmission coefficient the reflection coefficient may be computedusing Eq. (2.9) and this result will be used in the investigation of the Olsen and Vignalemodel.
7
Chapter 3
Investigation
3.1 ProblemThe analytical models considered are the Kronig-Penney model and the method describedby Olsen and Vignale in [3]. We study electron propagation in periodic potentials withthe two mentioned methods. We solve the Schrödinger equation numerically for periodicpotentials.
3.2 Description of Models
3.2.1 The Kronig-Penney model
The Kronig-Penney model makes use of Bloch’s Theorem, see section 2.3, in order toobtain domains of allowed and forbidden values of the wavenumber k.
The electrons in this model are semi-bound, they are sufficiently bound to the ionsso that the wave functions do not overlap more than one or two of the nearest neighbors.We should expect a resemblance of the free electron model.
3.2.2 The Olsen and Vignale model
Another approach that would lead to finding that conduction in a crystal only can occurfor certain energies, namely in the conduction bands, is the model described by Olsen andVignale in [3]. The crystal is modeled as a one dimensional periodic potential composed ofN identical barriers separated by the distance a, the lattice constant. Electrons incidenton this potential have a probability of being either transmitted or reflected. Transmissionof electrons is equivalent to conduction while total reflections is equivalent to insulation.It will be shown that transmission and total reflection can only occur for certain energies,meaning there are certain energy bands were conduction can occur.
8
3.3 Analytical Calculations
3.3.1 The Kronig-Penney model
Classical model
In this classical version of the Kronig-Penney model the electron associated with each ionis represented by a harmonic oscillator, all oscillators have the same angular frequencyω. There are two cases to consider:
1. There are no couplings between neighboring oscillators.2. There are couplings between neighboring oscillators.
The equation of motion in case 1 is
d2xndt2
= −ω2xn. (3.1)
The equation of motion in case 2 is
d2xndt2
= −ω2xn −K[(xn − xn−1) + (xn − xn+1)]. (3.2)
We will solve this for case 2, which is more general. Use a trial solution xn = An cos(Ωt)and insert it into (3.2). This gives
(ω2 − Ω2)An = −K[2An − An−1 − An+1]. (3.3)
Eq. (3.3) is a difference equation. To solve this, assume An = Ln. If A1 = AN+1 we musthave Ln = 1. This equation has N solutions of the form
L = e2πir/N , r = 0, 1, 2, . . .+ (N − 1). (3.4)
Inserting the result of 3.4 into 3.3 gives us
Ω2 = ω2 + 4K sin2(πrN
). (3.5)
The term with K in Eq. (3.2) represents the coupling between nearest neighbors. Hencethe solution is Ω2 = ω2 for an uncoupled system and in this case the frequencies areN -fold degenerate, i.e. all oscillators are moving together with the frequency Ω = ω. Inthe case of couplings we have a band of allowed frequencies, Ω is allowed to have valuesin the interval [ω,
√ω2 + 4K].
This classical analogy is rather illustrative. Assume the electrons undergo harmonicoscillations about their central locations, then the spread of frequencies is analogous toa spread of energy values. [1]
Quantum model
The first quantum model considered in this report is the Kronig-Penney model. Theperiodic potential is modeled by a series of repulsive delta-function potentials. We couldas well use a potential square barrier, but the calculations are much easier when using
9
a delta-function rather than a broad barrier. At sites where x 6= na, the wave functiondescribing an electron is that of a free particle. We have periodic boundary conditions.The goal of the below derivation is to see what restrictions are found for a particle ina periodic potential. In the end, a relation between q in Bloch’s theorem and the wavefunction of the electron is obtained.
V (x) =~2
2m
λ
a
∞∑n=−∞
δ(x− na) (3.6)
We start with solving the time independent Schrödinger equation with Eq.(3.6)
d2ψ(x)
dx2= −2m
~Eψ(x), k2 = E
2m
~=⇒
ψ(x) = A sin(kx) +B cos(kx). (3.7)
In the region Rn defined by (n− 1)a ≤ x ≤ na we have
ψ(x) = An sin k(x− na) +Bn cos k(x− na). (3.8)
Similarly, for the region Rn+1 defined by na ≤ x ≤ (n+ 1)a we have
ψ(x) = An+1 sin k(x− (n+ 1)a) +Bn+1 cos k(x− (n+ 1)a). (3.9)
Continuity at x = na gives the equality
− An+1 sin(ka) +Bn+1 cos(ka) = Bn. (3.10)
For the derivative we have a discontinuity, see Eq. (2.10) - (2.12),
dψ(x)
dx Rn+1
− dψ(x)
dx Rn=λ
aψ(na) =⇒ (3.11)
An+1 cos(ka) +Bn+1 sin(ka)− An = gBn, g =λ
ka. (3.12)
From Eq. (3.10) and Eq. (3.12) solve for An+1 and Bn+1An+1 = Bn(g cos(ka)− sin(ka)) + An cos(ka)
Bn+1 = An sin(ka) +Bn(g sin(ka) + cos(ka)).(3.13)
Rewrite Eq. (3.8) using na = (n+ 1)a− a
ψ(x) = An sin k((x+ a)− (n+ 1)a) +Bn cos k((x+ a)− (n+ 1)a). (3.14)
Make use of Bloch’s theorem ψ(x) = eiqxu(x)
ψ(x+ a) = eiq(x+a)u(x+ a) = eiq(x+a)u(x) = eiqaeiqxu(x) = eiqaψ(x).
Eq. (3.14) is equal to Eq. (3.9) if An+1 = eiqaAn
Bn+1 = eiqaBn.(3.15)
10
Insert Eq. (3.15) into Eq. (3.13)eiqaAn = Bn(g cos(ka)− sin(ka)) + An cos(ka)
eiqaBn = An sin(ka) +Bn(g sin(ka) + cos(ka))=⇒ (3.16)
An(eiqa − cos(ka)) = Bn(g cos(ka)− sin(ka))
Bn(eiqa − g sin(ka)− cos(ka)) = An sin(ka).(3.17)
Multiply left-hand sides and right-hand sides, respectively
AnBn(eiqa−cos(ka))(eiqa−g sin(ka)−cos(ka)) = AnBn sin(ka)(g cos(ka)−sin(ka)) =⇒
ei2qa − 2eiqa(g
2sin(ka) + cos(ka)
)+ 1 = 0. (3.18)
When solving this quadric equation, the Euler formula eix = cosx+i sinx is used. Takingimaginary parts of Eq. (3.18) gives
2 sin(qa) cos(qa)− 2 sin(qa)(g
2sin(ka) + cos(ka)
)= 0 =⇒
cos(qa) =λ
2
sin(ka)
ka+ cos(ka). (3.19)
The left-hand side is bounded by one and the right-hand side is bounded by 1 + λ2sin(ka)ka
.Thus there are regions where this equality does not hold. These forbidden regions givethe onset of forbidden bands. More comments on this in section 3.6.1.
3.3.2 The Olsen and Vignale model
Classical model
Initially a classical model will be considered. When studying an electron incident ona potential barrier in classical mechanics the electron has a probability of being eithertransmitted or reflected. The probability of being transmitted through one barrier is T,which is also called the transmission coefficient. The probability of being reflected byone barrier is R, the so called reflection coefficient. The uncertainty of the outcome issimply due to our lack of knowledge about the system, but it is known that
T +R = 1.
To begin with, lets consider an electron incident on two potential barriers. The proba-bility of the electron being transmitted through the composition of barriers, called T2, isnot simply T · T but actually
T2 = TT + TRRT + TRRRRT + TRRRRRRT + · · · =
= T 2(1 +R2 +R4 +R6 + · · · ) = T 2
1−R2 .
The first term after the equality sign corresponds to the electron being transmittedthrough both of the barriers at its first try. The following terms correspond to theelectron being transmitted through the first barrier after which it is reflected between
11
Figure 3.1: An electron incident on two identical potential barriers in the classical case. Tis the probability of transmission while R is the probability of reflection. The probability oftransmission through both barriers is obtained by adding all different paths to the outcome.Figure from [3].
the two barriers an even number of times before it is finally transmitted by the secondbarrier. Factorization and the fact that an geometrical sum is obtained leads to thefinal step in the equation above. To obtain the probability of an outcome, in this casetransmission, the probabilities of all different roads to the outcome have to be added.
Now consider a periodic system of N potential barriers. The transmission coefficientfor the system is calculated in the same manner as for the case with two barriers andis denoted TN . TN then includes the probability for all the different ways in which theelectron can be transmitted through the N barriers. If one more identical barrier isadded, the transmission coefficient of the system will be
TN+1 = TNT + TNRRNT + TNR2R2
NT · · · =TNT
1−RNR. (3.20)
To be able to analyze and understand Eq. (3.20) it will be rewritten. If using the relationsR = T − 1, RN = TN − 1 and then dividing by TNT the following is obtained
TN+1 = 11T+ 1TN−1 .
Taking the inverse of the equation above gives1
TN+1= 1
T+ 1
TN− 1
If one is subtracted on each side the relation 1/T − 1 = (1 − T )/T = R/T can be usedon all terms including transmission coefficients. That results in
RN+1
TN+1= RN
TN+ R
T.
By using induction one obtains the result seen below
RN+1
TN+1
=
(RN−1
TN−1+R
T
)+R
T=
(RN−2
TN−2+R
T
)+ 2
R
T= · · · = (N + 1)
R
T. (3.21)
12
The quotient RN/TN would tend to infinity if all electrons were reflected and it wouldtend to zero if all electrons were transmitted. These properties may seem familiar andwe realize that the quotient can be compared with resistance. Eq. (3.21) gives usthat resistance increases linearly with N , the number of potential barriers, which seemsintuitive.
The classically calculated value of the resistance is much lower than the measuredvalue. Therefore, it is not suitable to consider conduction as a classical phenomenon. [3]
Quantum model
In the quantum case the waves of the incident electron will be considered. Part of thewave can be transmitted and part can be reflected and multiple pieces of transmittedand reflected waves will interfere.
The parts of the electron wave being transmitted and reflected will have differentprobability amplitudes given by
t = |t|eiδ r = ±i|r|eiδ.
The probability amplitude is a complex number. The probability of a specific outcome isgiven by squaring the magnitude of the probability amplitude and getting the coefficientsas in the classical case. The relation between the probability amplitudes is given by
|t|2 + |r|2 = T +R = 1 r is shifted ±π2relative to t.
Now consider the model of N identical potential barriers, whose centers are separatedby the distance a. When a wave travels the distance a between two barriers it will gainin phase. The total transmission and reflection coefficient for passing one barrier andtraveling the distance a is t and r. Where
t = |t|eiδeika = |t|eiθr = ±i|r|eiδeika = ±i|r|eiθ.
Furthermore tN and rN are the coefficients for all the N barriers together. If one moreidentical potential barrier is added to the series of N barriers the new coefficients will becalculated as in the classical case
tN+1 = tN t+ tN rrN t+ tN r2r2N t+ . . . =
tN t
1− rN r. (3.22)
In Eq. (3.22) the first term after the equality sign corresponds to the case when the waveis transmitted through all the first N barriers and the immediately is transmitted throughthe last barrier. Every term except the first term in Eq. (3.22) represent the wave beingtransmitted through all the first N barriers and then being reflected a multiple timesbefore it finally is transmitted through the last barrier.
rN+1 = rN + t2N r + t2N r2rN + . . . = rN +
t2N r
1− rN r. (3.23)
In the reflection coefficient, Eq. (3.23), the first term after the equality sign correspondsto the wave being reflected from the N first barriers. Every term except the first term in
13
Eq. (3.22) corresponds to the wave being transmitted by the N first barriers after whichit is reflected multiple times between the last barriers before it is transmitted back.
Now let N → ∞. It is of interest to examine the case where t → 0 and r → 1,meaning a total reflection of the incident electron wave and therefore no conduction. Eq.(3.23) and the fact that t → 0 gives r∞+1 = r∞. Thus r tends to a constant, constantmagnitude and phase, and so does t. Lets consider Eq. (3.22) again. If using t = |t|eiθ,r = ±i|r|eiθ and limN→∞ rN = ±ieiθ∞ the following is obtained
limN→∞
tN+1
tN= lim
N→∞
t
1− rN r= lim
N→∞
|t|eiθ
1± i|r|eiθrN. (3.24)
Finally
limN→∞
tN+1
tN=
|t|e−iθ ± |r|eiθ∞
. (3.25)
Knowing that the reflection and transmission coefficients tends to constants gives thattheir phases tends to constant. Thus the imaginary parts in the left hand side of Eq.(3.25) cancel and the Eq. (3.25) must be real. Only the numerator in Eq. (3.25) hasimaginary parts and the constraint gives
=e−iθ ± |r|eiθ∞ = 0. (3.26)
Expanding with the Euler formula gives
=cos θ − i sin θ ± |r|(cos θ∞ + i sin θ∞) = 0 (3.27)
− sin θ ± |r| sin θ∞ = 0 =⇒ sin θ
|r|= ± sin θ∞. (3.28)
Since the magnitude of a sine cannot be larger than one we get the following condition
| sin θ| ≤ |r|.
The condition for the transmission coefficient to be nonzero is then
| sin θ| > |r|. (3.29)
This condition can be rewritten by squaring, subtracting one on both sides and usingthe relation |t|2 + |r|2 = 1 and the Pythagorean trigonometric identity. The conditionobtained is
| cos θ||t|
< 1. (3.30)
The basic idea of this model is thus to explain the behavior of electrons in a crystal bystudying interference of transmitted and reflected electron waves in the crystal. Fromthe above equation it is concluded that there are certain energies for which the proba-bility of transmission is non-zero while there are gaps in energy where the probability oftransmission tends to zero and thus there will be no propagation of electron waves. Theintervals of energy for which the probability of transmission is non-zero are equivalent tothe allowed energy bands found by using the Kronig-Penney model.
14
3.4 Numerical AnalysisIn our investigation of electron propagation in periodic potentials, we wanted to simulatethe phenomena with a computer program to further expand our analysis. We decidedto use the program package Matlab for the simulation. The reason for this is becausewe have some proficiency in the Matlab programming language from earlier courses, butalso for its flexibility.
3.4.1 The Olsen and Vignale model
To calculate the transmission probability for different numbers of potential barriers, wewrite a code which sums the transmission and reflection coefficients for N barriers ac-cording to Eq. (3.22) and Eq. (3.23). By constructing a function that uses the latesttransmission and reflection coefficients to calculate the next coefficients we can iterativelyobtain the total transmission coefficient for any number of barriers. The transmissionprobability through N barriers is then obtained by |tN |2.
The same method may be used to see how the probability of transmission, |tN |2,depends on the wave vector, k. For a specific number of barriers we calculate the trans-mission coefficient for a range of values of k.
3.4.2 Numerical solution of the Schrödinger equation
Discretization of the Schrödinger equation
When solving a partial differential equation, PDE, there are several numerical methodsavailable. For our purposes we need to solve a hyperbolic PDE. Since we need a timeevaluated solution of a well defined initial condition we use the Method of Lines. Theadvantage of this method is that we then can use one of the many built in solvers ofsystems of ordinary differential equations, ODEs, that exist in Matlab. We start with adiscretization of the time dependent Schrödinger equation
− ~2
2m
∂2u(x, t)
∂x2+ V (x)u(x, t) = i~
∂u(x, t)
∂t(3.31)
in space for N inner points with the central difference approximation for a second orderderivative
∂2u
∂x2≈ ui−1 − 2ui + ui+1
h2x, u(xi, t) ≈ ui(t). (3.32)
Insert the difference approximation into Eq. (3.31) and solve for ∂u∂t
− ~2
2m
ui−1 − 2ui + ui+1
h2x+ V (xi)ui = i~
∂ui∂t
=⇒
∂ui∂t
=i~
2mh2x︸ ︷︷ ︸const
ui−1 − i[
~mh2x
+V (x)
~
]︸ ︷︷ ︸
const+V (x)~
ui +i~
2mh2x︸ ︷︷ ︸const
ui+1. (3.33)
By doing this we obtain a system of N first order ordinary differential equations of theform
du
dt= Au. (3.34)
15
Eq. (3.34) can be solved by one of the built in solvers for systems of ODEs in Matlab.When this discretization is obtained we want to be able to implement bounded boundaryconditions, e.g. particle in a box, as well as periodic boundary conditions. This is repre-sented by a small change of elements in the coefficient matrix A. For bounded boundaryconditions we have that u0 = 0, uN+1 = 0 whereas we have u0 = uN+1 for periodicboundary conditions. Note that in the case of bounded boundary conditions there are Nunknowns and and for periodic boundary conditions there are N + 1 unknowns. [5]
Particle in a Box
The particle in a box is a standard problem in quantum physics. In short, the problemis to fit in a plane wave within a domain which has boundaries of infinite potential. Theplane wave can be described by
ψ(x) = A sin(kx) +B cos(kx) (3.35)
where ψ(x) is the solution of the time independent Schrödinger equation and k2 = 2mE~ .
The potential has the form
V (x) =
∞ if a < x
0 if 0 < x < a
∞ if x < 0
=⇒ ψ(0) = ψ(a) = 0. (3.36)
From ψ(0) = 0 we see that B ≡ 0 and from ψ(a) = 0 we see that ka = nπ =⇒ En = ~2k22m
.Our function now reads
ψ(x) = A sin(nπx
a
)which is normalized to
ψ(x) =
√2
asin(nπx
a
). (3.37)
As previously mentioned this is the solution of the time independent Schrödinger equa-tion. What we need now is to consider time development of these eigenfunctions. Asproposed by [1] we have that
Ψ(x, t) =∑
Anun(x)e−iEnt/t (3.38)
where un(x) ∝ sin(nπxa
)are the eigenfunctions for the particle in a box. By implementing
this time dependence on a selected set of eigenfunctions and using Matlab, we can followhow ψ(x) develop in time by plotting the function value at selected points in time. Whatwe want conclude with this is that we can get the same solution by using a numericalmethod as with analytical calculations. By using ψ(x) = 2√
asin(πxa
)as initial state for
the built in Matlab method ode23 and let it solve for a certain time interval, we can seeif the numerical solution yields the same result as the analytical solution in this case.
Derivation of wave packet
For the numerical analysis a wave packet is needed as an initial state to model theelectron. A wave packet is a superposition of plane waves with a variety in amplitudesand k-values and the general form is [1]
16
ψ(x, t) =
∫ ∞−∞
A(k)ei(kx−ωt) dk.
As an initial value, at t = 0, we consider a gaussian wave packet of the form
ψ(x, 0) = Ae−(x−x0)
2
2α eik0x (3.39)
where the function is centered about x0 and α is related to the spacial spreading or thewidth of the function as ∆x = 2
√α. The function must be normalized for the total
probability to integrate to one. We have the condition∫ ∞−∞|ψ(x, 0)|2 dx = 1. (3.40)
In order to determine the normalization constant, A, the normalization condition Eq.(3.40) is used, which yields∫ ∞
−∞|A|2e−
(x−x0)2
α dx = 1 ⇒ A =1
(πα)1/4. (3.41)
Thus, as an initial state the following gaussian wave packet is used
ψ(x, 0) =1
(πα)1/4e−
(x−x0)2
α eik0x. (3.42)
−8 −6 −4 −2 0 2 4 6 8
x 10−9
0
0.5
1
1.5
2
2.5
3x 10
9
space [m]
|ψ|2
−8 −6 −4 −2 0 2 4 6 8
x 10−9
−4
−3
−2
−1
0
1
2
3
4
5
6x 10
4
Re
(ψ)
space [m]
Figure 3.2: (a) Plot of |ψ(x, 0)|2 versus distance [m], meaning the probability density of thewave. ψ(x, 0) is the gaussian wave packet used as initial value. (b) Plot of <(ψ(x, 0)) versusdistance [m].
In time evolution the wave packet will move with the group velocity vg and it will alsospread as time increases.
Numerical simulations
We make a program to simulate electron propagation over periodic potentials. To do this,we use our numerical solver of the time-dependent Schrödinger equation with an easilymanipulated potential. We have the ambition to be able to get result within 30 minutes
17
of computational time on a personal computer. Beacuse we are unable to implement thedirac potentials correct numerically, we approximate the dirac potentials with smoother
potentials constructed with gaussian functions f(x) = ae−(x−b)2
2c2 , a, b and c are constants.A gaussian function is easily adjusted to have the properties wished for. Our programallows for easy adjustment of the number of potentials.
To simulate the propagating electron, we use the gaussian wave packet obtained inEq. (3.42) as initial state for the Schrödinger equation
Ψ(x, 0) = ψ(x) =1
(πα)14
e−(x−x0)
2
α eik0x .
For a physical interpretation on electrical conduction, correct choices of parameters areimportant. When discussing electrons propagating over one or many potentials, it iscommon to study electron energies of the same order of magnitude as the energy of thepotential barrier. In the program, we implemented this using
E =~2k2
2me
, k = k0 =⇒ k0 =
√2meE
~2. (3.43)
An allowed wave vector is taken from Eq. (3.19), valid for a delta potential. We approx-imate this wave vector for our gaussian potentials. This gives us E. The user enters Eas an parameter. From the uncertainty relation ∆k∆x > 1
2we have
∆k =1
2∆x(3.44)
which is used to determine the wave vector spread, when ∆x is entered as a parameter[1].
Our intention is to compare our numerical simulations to the result obtained by usingthe Olsen and Vignale method. When studying the Olsen and Vignale method we foundthat the probability of transmission is non-zero for some ranges of energy and wave vectorswhile there are ranges of wave vectors where the probability of transmission tends to zero.By letting the wave packet model an electron and be incident on a number of periodicpotential barriers we can, in our simulation, see whether the wave packet is mostlyreflected or transmitted. If choosing wave vectors that are mainly in the allowed regionsthe wave packet would be mostly transmitted to agree with the result from the Olsenand Vignale method, but if choosing wave vectors mainly from the forbidden regions wewould hope that most of the wave packet would be reflected to get an agreement betweenthe methods.
3.5 Results
3.5.1 The Kronig-Penney model
From the Kronig-Penney model we have the condition
cos(qa) =λ
2
sin(ka)
ka+ cos(ka) (3.45)
which yields allowed and forbidden values of k, the wave number of a propagating electron.This is valid for delta potentials. In figure 3.3 below cos(ka) + λ
2sin(ka)ka
vs. ka is plotted.
18
Regions where cos(ka) + λ2sin(ka)ka≤ 1 are marked with "*". The horizontal lines of ±1
show the bounds of cos(qa).
0 2 4 6 8 10 12 14−2
−1
0
1
2
3
4
5
6
ka [ 0 , 5pi ]
y
Figure 3.3: Plot of cos(ka) + λ2sin(ka)ka
vs. ka showing the allowed and forbidden bands.
3.5.2 The Olsen and Vignale model
By studying interference of transmitted and reflected electron waves the behavior ofelectrons in a crystal can be explained. From Eq. (3.30) it is concluded that there arecertain energies for which the probability of transmission is non-zero while there are gapsin energy where the probability of transmission tends to zero. In other words, we foundthat there are allowed and forbidden energy bands for electrons in a crystal.
By using Eq. (3.22) and Eq. (3.23) the probability of transmission may be calculatediteratively for different numbers of potential barriers. The barriers studied are deltafunction potentials. The result is shown in figure 3.4 for the quantum cases where theenergy is either inside or outside an allowed energy band. For the case where the energyis outside the allowed energy band, in the band gap, the probability decreases rapidly asthe number of barriers increases. For the case with the allowed energy the probabilityis high and repeats periodically. The same figure shows the probability of transmissionversus the number of barriers for the classical case where the probability decreases as1/N [3].
19
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
N
Pro
babi
lity
Figure 3.4: Probability of transmission versus the number of delta potential barriers, N , cal-culated using Eq. (3.22). The dashed line shows the classical case where the probability tendsto zero as N increases. The dotted line shows the quantum case outside of an allowed energyband where the probability tends to zero exponentially. The unbroken line shows the quantumcase inside an allowed energy band where the probability of transmission is high and repeatsperiodically.
Eq. (3.22) and Eq.(3.23) allow us to study the probability of transmission dependenceon the electron energy for a specific number of potential barriers. The result is shown infigure 3.5. The figure shows the transmission probability versus the electron wave vector,k, which is related to the electron energy. Inside the allowed energy bands the probabilityis high and in the band gaps the probability tends to zero. For certain wave vectorsthe probability is one which means the resistance is zero for electrons with that wavevector. For larger number of potential barriers, meaning larger crystals, the transmissionprobability gets closer to zero and the resistance for electrons tends to infinity. Fromfigure 3.5 it is seen that the width of the allowed energy bands increases as the electronwave vector and the energy increases. The width increases with the magnitude of theprobability amplitude, |t|, which increases with electron energy. [3]
20
0 0.5 1 1.5 2 2.5
x 1010
0
0.2
0.4
0.6
0.8
1
Wave vector k
Pro
babi
lity
Figure 3.5: Probability of transmission versus wave vector, k, for a periodic potential consistingof N delta potential barrier. The unbroken line shows the case where N = 3, the dashed lineshows the case where N = 6 and the dotted line shows the case N = 1000. Inside the allowedenergy bands the transmission probability is high for all cases and outside the allowed energybands the probability tends to zero. As k increases the width of the allowed energy bandsincreases while the forbidden energy bands gets narrower.
3.5.3 Numerical solution and simulations
Comparing the analytical and numerical solution of particle in a box, we find the solutionsdisplayed in figure 3.6. The result of correspondence adds credibility to our numericalsolution, and constitutes a validation of the numerical methods we use.
We have made a few simulations of an initial gaussian wave packet incident on anumber of potential barriers. The following function is used to generate all potentials
V (x) = 5 · 1.6 · 10−19 · exp
((x− x0)2
2 · 10−22
). (3.46)
Our first observation is that our program is very demanding. A timespan from 0 to 10−14
seconds takes more than 12 hours to complete on the modern personal computers we areusing, and is limiting our simulations. By lowering the tolerance level and shorteningthe time span, we can obtain a solution within an acceptable computational time. Infigure 3.7 we see the result of the wave packet incident on ten barriers. It can be seenin the figure that there is an interference between reflected and incoming parts of thewave. The wave packet also integrates to one within the precision 0.9999994 . . . ≈ 1 atthe sample time t = 10−15 s.
21
0 1 2 3 4 5
x 10−10
0
1
2
3
4
5
6
7x 10
4
space [m]
Re(
Ψ)
0 1 2 3 4 5
x 10−10
−200
−150
−100
−50
0
50
100
space [m]
Im(Ψ
)
Figure 3.6: Plot of time dependent wave function with initial state Ψ(x, 0) =√
2a sin
(πxa
). The
left plot shows the real part of Ψ(x, t) and the right shows the imaginary part. The red linein the figure is generated analytically. The blue dots in the figure is a snapshot in time fromour numerical solution of the Schrödinger equation. In the plot of the imaginary part of thenumerical solution we have numerical instability but we still have correspondence between theanalytical and numerical solutions.
−8 −6 −4 −2 0 2 4 6 8
x 10−9
0
2
4
6
8
10
12x 10
8
Distance [m]
Pro
babi
lity
dens
ity
−8 −6 −4 −2 0 2 4 6 8
x 10−9
0
2
4
6
8x 10
−19
Distance [m]
Ene
rgy
[J]
Figure 3.7: Numerical solution of gaussian wave packet incident on ten potential barriers.The upper plot of the figure shows |Ψ(x, t)|2 after 10−15 s. The wave packet integrates to0.9999994 . . . ≈ 1. The lower plot shows the position of the potential barriers.
3.6 Discussion
3.6.1 The Kronig-Penney model
As seen in figure 3.3, there are allowed and forbidden values of k, and thus, in the energyE = ~2k2/2m. Clearly the equality never holds for regions where λ
2sin(ka)ka
+ cos(ka) > 1
22
and this is where the band gaps are found.There are two situations to consider in Bloch’s theorem:
ψ(x+ a) = eikaψ(x) for a free particle, i.e. q = k.
ψ(x+ a) = eiqaψ(x) in general.(3.47)
In the general case, q corresponds to the shift in phase due to the passage of the potentialand traveling a distance a. q can be seen as a parameter determined by the materialspotential and structure. For a free particle V (x) = 0 and the wave function correspondingto a given energy E is
ψ(x) = eikx.
A displacement of a distance a gives
ψ(x+ a) = eik(a+x) = eikxψ(x).
Thus a displacement is expressed as a phase factor multiplied with the original solution.This implies that |ψ(x+ a)|2 = |ψ(x)|2 and that observables will be the same at x andat x+ a.[1]
3.6.2 The Olsen and Vignale model
Studying the model of an idealized crystal with periodic boundary conditions is slightlydifferent from studying a real crystal with boundaries to the surroundings. It seems thatby modeling the crystal as a periodic potential, you do not take the boundaries to thesurrounding into account. But the property of the transmission probability being largeand sometimes complete inside the allowed bands can only occur when studying morethan one potential barrier, except the trivial case with one barrier with a transmissioncoefficient that equals one. This means that the first potential barrier in the series mightrepresent the boundaries with the surroundings of the crystal. [3]
The method described by Olsen and Vignale in [3] has the advantage that it can easilybe extended to studying disordered systems. The transmission and reflection coefficientscan be changed and impurities can be added. With similar reasoning as before equationscorresponding to the Eq. (3.22) and Eq. (3.23) may be derived.
3.6.3 Numerical solution and future work
Our program is capable of performing the simulations we wanted, but with limitations.The limitations are of computational character, more specifically that we cannot use thetime span we would like to. This is mainly because the program is computationallydemanding and less usable for everyday simulation. We could possible wait 24 hours forthe simulations to finish, but that would not be flexible enough for our original purposewith the program. Considering our results of the numerical simulations, we concludethat we have a working numerical method.
To compare results from the simulation with e.g. the ones from figure 3.4 we need tobe able to simulate 100 barriers, with such a discretization that the boundary conditionsare not immediately affecting the solution. This is since we want to observe how thesolution evolves depending on which and how many barriers we use, not depending oneffects from the boundaries, e.g. wave superposition.
23
In [3] the probability of transmission and reflection through a series of delta func-tion barriers is calculated. It is of scientific interest to compare results from differentsources intended to describe the same phenomenon. Comparing results from the Olsenand Vignale model with our numerical simulations would strengthen the validity of oursimulations. This is not possible at this stage, mainly because we are not able to modeldelta function potentials, apart from the computational difficulties. If we approximatethat our gaussian potentials resembles the one used in the Kronig-Penney model, wecould compare the results. Our idea of how this might be done is to let the numericallysimulated wave function pass through a number of potentials. Because of the probabilityinterpretation, we can integrate |Ψ(x, t)|2 over the interval after the potentials. We thenhave a probability that the electron is localized after the barriers, meaning that the elec-tron has transmitted through the complete series of barriers. In [3] an infinite numberof reflections are accounted for in the coefficients. To obtain this numerically, we wouldneed to wait an infinite time. However, one can argue that the results could be comparedafter a given time, approximating that only a small number of reflections will occur afterthat given time, and that those reflections would not affect the results significantly.
Different simulations of similar kind are available in the program package developed byprof. em. G. Lindblad at Theoretical Physics, KTH. [6] Performing various simulationsfor periodic potentials using [6] shows resemblance to our simulations. This indicatesvalidity of our simulations.
It is worth to consider to use another program package for similar problems. This ismainly since Matlabs symbolic mathematical functions has certain limitations when theyare combined with explicit numerical calculations, which in some cases could be quitetroublesome to overcome. In our simulation we have had problems when integratingthe built in delta function of Matlab explicitly, as mentioned above. The built in deltafunction is not fully compatible with the most used built in ODE solvers in Matlab. Otherprogram packages, such as Mathematica or Maple, might have handled this issue in abetter way. Considering that in the simple case of the particle in box yielded the sameresult in analytical and numerical calculations we are likely to have a fully functionalalgorithm for solving the time dependent Schrödinger equation. Inspecting the resultswe have obtained from our simulations, it seems that another computational approachis needed. One of the goals with the program is to be able to perform simulations on amodern personal computer, for both convenience and accessibility.
Information search in publications return among other things the paper [4], by Viss-cher. In the paper, Visscher presents an algorithm for solving the time dependentSchrödinger equation. He shows that one can successfully solve the Schrödinger equa-tion numerically by separating the real and the imaginary part. However, one mustmake a slight displacement of the steps in time and space to avoid instability. The stepdisplacement is the main point in [4]. To continue the work on our program, it seemsreasonable to proceed by implementing the method by Visscher. The method has beenproven efficient and has the advantages of both implicit and explicit methods, meaningstable solutions and a shorter computational time.
24
Chapter 4
Summary and Conclusions
In this thesis we have studied the propagation of electrons in crystals. The crystallinestructure was modeled as a one dimensional periodic potential consisting of identicalpotential barriers symbolizing the space between ion cores. Initially, two methods wherestudied, namely the Kronig-Penney model and a model described by Olsen and Vignalein [3] and they lead us to the same result.
In the analysis with the Kronig-Penney model we studied a series of repulsive deltapotential barriers. By using the continuity of the wave function and the discontinuityof the derivative when passing a barrier two equations where obtained. Finally Bloch’stheorem was used to eliminate all unknowns and we arrived at the result
cos qa = cos ka+λ
2
sin ka
ka. (4.1)
The left hand side of of Eq. (4.1) is bounded by one which gave us the result that thereare restrictions on the value of the electron wave vector, k. This means that there areallowed and forbidden ranges of the electron energy. With this result we could constructthe plot in figure 3.3 showing the allowed and forbidden energy bands.
To begin the analysis with the Olsen and Vignale model we looked at the transmissionand reflection amplitudes for the potential barriers that were the building blocks of theone dimensional periodic potential. When looking at a series consisting of N + 1 barriersthe equations for the total transmission and reflection coefficients where found to be
tN+1 =tN t
1− rN r, rN+1 = rN +
t2N r
1− rN r. (4.2)
We examined under what circumstances the transmission coefficient tended to one andfound the condition
| cos θ||t|
< 1. (4.3)
It could be concluded that there are certain energies for which the probability of trans-mission is non-zero while there are gaps in energy where the probability of transmissiontends to zero and thus there will be no propagation of electron waves. The intervalsof energy for which the transmission probability was non-zero where equivalent to theallowed energy bands found when using the Kronig-Penney model. With the result wewhere able to produce a plot showing the transmission probability versus the number of
25
barriers, figure 3.4, and a plot of the transmission probability versus the wave vector fora specific number of barriers, figure 3.5.
When we solved the simple case of the particle in a box we could with certainty tellthat our numerical algorithm was working as we intended. Due to limitations we couldnot use our methods in order to reach as far as obtaining results that strengthen theconclusions from the two studied models. Our limitations narrowed down to that ourmethod was too computationally demanding. We had to run our simulations for a veryshort time interval and with a low tolerance in order for the computational time to bewithin acceptable limits.
To complete our comparison of results we would need more time for our numeri-cal simulations with the present algorithm. However, another approach to solve the timedependent Schrödinger equation would be to implement the algorithm described by Viss-cher in the article [4]. This algorithm has been used and proven efficient, yielding stablesolutions in shortened time.
26
Bibliography
[1] S. Gasiorowicz, Quantum Physics , Third edition, John Wiley & Sons, Inc., (2003).
[2] C. Kittel, Introduction to Solid States Physics , Eight edition, John Wiley & Sons,Inc., (2005).
[3] R.J. Olsen och G. Vignale, The quantum mechanics of electric conduction in crys-tals , Am. J. Phys. 78 (9), 954 (2010).
[4] P.B. Visscher, A fast explicit algorithm for the time dependent Schrödinger equation,Computers in Physics 5 (6), 596-598 (1991).
[5] L. Edsberg Introduction to Computation and Modeling for Differential Equations,First edition, John Wiley & Sons, Inc., (2008).
[6] G. Lindblad, Schrodinger program package for Matlab, available athttp://www.theophys.kth.se/mathphys/schrodinger.html, (2004)
27
