Quantum Mechanical Conduction of Electrons in 1D Fibonacci Quasicrystals

Transcription

1 Theoretical Physics Quantum Mechanical Conduction of Electrons in 1D Fibonacci Quasicrystals Carl-Johan Backman ( ) Johannes Wennberg ( ) SA104X Degree Project in Engineering Physics, First Level Department of Theoretical Physics Royal Institute of Technology (KTH) Supervisor: Jack Lidmar May 20, 2013

2 Abstract In this report we model and study the propagation of electrons in one-dimensional (1D) Fibonacci quasicrystals. The quasicrystals are modeled with delta function potentials and created by either arranging their strengths or their spacings as a 2 letter Fibonacci word. The transmission and reflection amplitudes are calculated using an iterative method. A comparison is made against an ordinary crystal and a semi-random setup. The transmission of electrons are investigated and related to the electrical conduction in the different types of materials. We also investigate how sensitive the quasicrystals are to perturbations, compared to crystals. Finally, a quasicrystal modeled with a 3 letter Fibonacci word is discussed. We find that the Fibonacci structure increases the resistance by a relatively large factor, which makes the quasicrystals act like semiconductors. We also find tendencies that the quasicrystals are more sensitive to perturbations compared to ordinary crystals. The results are in good agreement with similar, earlier studies, both theoretical and experimental. Keywords: Fibonacci quasicrystals, Electron transmission, Dense Fibonacci word, Onedimensional, Band theory of solids.

3 Contents 1 Introduction Fibonacci Words Quasicrystals Experimental Realizations of 1D Fibonacci Quasicrystals Background Material 5 3 Investigation Problem Model Binary Quasicrystals Tertiary Quasicrystals Crystals and Random Distributions of Potentials Method Reflection and Transmission Amplitudes Analytical Calculations Phase Shift due to Tunneling Through a Delta Function Numerical Analysis Results Binary materials Tertiary materials Perturbations Discussion Model and method Next step Summary and Conclusions 27 Bibliography 27 1

4 Chapter 1 Introduction The purpose of this report is to, with illustrative graphs and clear reasoning, qualitatively show the effect that a quasiperiodic structure has on electron transmission and electrical conductivity. The report is structured as follows. First, here in Chapter 1, we give a brief introduction to our subject and explain some vital concepts. In Chapter 2 we describe what background material we have used to carry out our thesis. In Chapter 3 the investigation is carefully explained. The model and method that we use, our calculations and the results that we receive. Also, a discussion of the results, model and method is presented here. Finally, in Chapter 4, the results are summarized with our conclusions. However, before we go any further, let us establish some fundamental concepts. 1.1 Fibonacci Words Number sequences have always fascinated people all around the world. One of the most well known number pattern is the Fibonacci sequence: A sequence where every 1 number is determined by sum of the two previous numbers. That is, for the n:th number we have F n = F n 1 + F n 2. (1.1) Now think of F n as a word instead, a string, and that the addition is carried out as a string concatenation. Then you no longer have a number sequence but instead a letter sequence, and in this particular case, a Fibonacci word (also known as a Penrose chain or a Fibonacci chain). For example, let F 1 = A and F 2 = AB, then the first 6 Fibonacci words are F 1 = A, F 2 = AB, F 3 = ABA, F 4 = ABAAB, F 5 = ABAABABA, F 6 = ABAABABAABAAB. 1 Except the first two integers that are defined to be 0 and 1 in the Fibonacci case. 2

5 The Fibonacci words have several interesting features. They are ordered but not periodic, and the proportion of A s and B s is approximately the same (the golden ratio 2 ) no matter which part of the word you look at. These words found an unexpected use when Dan Shechtman discovered quasicrystals in the 1980 s [1], since they share some of the characteristics with the Fibonacci words, as we will see below. 1.2 Quasicrystals Quasicrystals are a group of solids which are neither crystals nor amorphous. They are ordered, just as a crystal, but lack discrete translational symmetry. As mentioned above they were discovered recently and much is yet to be discovered about them. However, much of our general knowledge about crystals comes from their translational symmetry which makes it possible to use Fourier analysis to examine them. The lack of translational symmetry in quasicrystals makes them far more difficult to analyze, especially when trying to understand the placement of the different atoms in the material [2]. Because of the fact that quasicrystals are ordered, translationally asymmetric and homogeneous, we choose to model them with Fibonacci words (as in Ref. [3]), since they share these characteristics. In addition, quasicrystals structured as a Fibonacci word can be physically realized (Sect. 1.3) [4]. It has also been discovered that there are real quasicrystals that have an internal structure related to the golden ratio [2]. There is a lot of research being done on quasicrystals and especially one area has attracted interest - band theory of solids [4]. When dealing with propagation in solids, typically, allowed energy bands where the transmission is high, and energy gaps where transmission is very low or even forbidden, emerge. As will be shown in this report the structure of these band-gaps in a material has great impact on the electrical conduction. An ordinarily good conductor, such as an aluminum alloy or tin, can increase its resistivity remarkably only by restructuring it as quasicrystal [5, 6]. This can be useful when metal is needed as a material but electrical conduction is not. 1.3 Experimental Realizations of 1D Fibonacci Quasicrystals As stated earlier we treat electron propagation in Fibonacci quasicrystals in this report. Nevertheless, we will briefly mention photon propagation here as well. The reason being that recent experiments of that have been conducted on Fibonacci quasicrystals created in a laboratory environment [4]. Also, since our model and method describes conduction in terms of tunneling and interference (more on this in Sect ), they can be generalized and thus describe photon propagation too. The quasicrystals are grown by stacking dielectric materials with different refractive indices on top of each other, ordered as a Fibonacci word. The transmission is then investigated by shooting photons at the quasicrystal and measuring what comes out on the other side. The results from these measurements show the same band-gap structure as we find in our results (Sect. 3.6) [4]. 2 The golden ratio is defined as φ = =

6 This is remarkable since the electron and the photon are quite different particles, e.g. their masses, their charges and the fact that one is a fermion (electron) and the other a boson (photon). This fact clearly shows that interference, and therefore the phase of the particle, plays an important part in all conduction. 4

7 Chapter 2 Background Material For the interested reader who wishes to deepen his or her knowledge in the subjects presented in this report we here suggest further reading for some central topics. However, the field of quasicrystals is young and evolving and this list will therefore soon be outdated and serve only as an introduction. We would therefore like to encourage the reader to try and find out how the field has developed since this work was conducted. Ref. [1] is the original report by D. Shechtman where he confirms the existence of quasicrystals. Ref. [2] report on an investigation where an exact solution of the atomic structure of a quasicrystal has been calculated. It also talks about the challenges of describing the structure of a quasicrystal through its x-ray diffraction and similar crystallographic methods. Ref. [3] explains the idea to use Fibonacci words to model quasicrystals in a 1D Kronig-Penney model. Here the matrix method (which we will mention several times in this report) for calculating the transmisson through a quasicrystals is derived. It also presents boundary conditions for tunneling through a delta potential. Ref. [4] describes a physical realization of an optical Fibonacci quasicrystal and the experiments conducted on it. Ref. [7] is a summary and it describes the present situation in the field of quasicrystals as well as a collection of other literature giving introductions to different parts in the study of quasicrystals. Ref. [8] presents a general iterative method for calculating the transmission and reflection of electrons in a general setup of potentials. It begins with the classical case and extends through quantum mechanics to asymmetric setups and quasicrystals. It also compares the classical results with different quantum mechanical results. For a clarification of the model as well as a comment on the article and a reply, see Ref. [9]. 5

8 Chapter 3 Investigation 3.1 Problem The problem discussed in this report is to model 1D Fibonacci quasicrystals with delta function potentials using MATLAB, and study how electrons transmit through and reflect off the quasicrystals. The model should be able to distribute the strengths of the potentials and the spacings (as explained in Sect ) around them as a Fibonacci word. This will be done for a binary material, i.e. a material with 2 types of atoms. In our model this is represented by either using 2 different strengths or 2 different spacings. The transmission and reflections amplitudes are calculated using an iterative method that takes the electron s interference with itself into account. The behavior of the total transmission is then to be analyzed when the difference between the strengths of the potentials and the spacings is varied. The results are also to be compared with other materials such as pure crystals, materials with a semi-random setup, and tertiary materials consisting of 3 types of atoms. 3.2 Model Binary Quasicrystals The model we use is a modified Kronig-Penney model. The ordinary Kronig-Penney model is represented by 1D periodic potentials with finite lengths between each potential. We modify it to hold for aperiodic systems as well, similar to the model described by D. Kiang and T. Ochiai [3]. Before we describe the model in more detail we need to define something that we, in the rest of the report, will call the spacing of each delta function potential. This is merely a geometric property. Think of it as adding empty space on each side of the delta function, where no other potential can be placed and no other spacing can overlap. For example, place a delta function δ 0 in x = 0, and forbid any other delta function or spacing to exist any closer than a distance a on each side of the delta function δ 0. Then that specific delta function (δ 0 ) has spacing 2a. With this in mind, we can now describe the model used in this investigation. The model consists of i = 1, 2,..., N potentials, represented as delta functions. The i:th potential has strength g i and spacing d i (Fig. 3.1, 3.2). The potential is always placed in the middle of its spacing d i. In this model the quasicrystal is then built by adding one potential at the time, left of the previous one. That is, the first potential is 6

9 the rightmost one and the last is the leftmost one. This distinction, that the quasicrystal is built from right to left, will prove to be of great importance later in this section. In order to model a quasicrystal with its characteristics we can either Fibonacci distribute the strengths of the potentials or the spacings of them. For example, let us say we have N = 5 potentials with strengths g a and g b and that all potentials have the same spacing. Then, if the strengths are distributed as a Fibonacci word, the strengths would be distributed like g a g b g a g a g b. A picture of this setup is shown in Fig If the spacings are to be Fibonacci distributed then it is performed in analogy with how we distribute the strengths, with the modification that the strengths are kept at the same value and the spacings are Fibonacci distributed instead. A picture of this setup can be seen in Fig Note that while Fibonacci words have several similarities with quasicrystals they are not the only suitable words. In fact, the infinite Fibonacci word is a member of the so called Sturmian words, of which all more or less share some characteristics with quasicrystals [10]. Figure 3.1: 5 equidistant potentials where the strengths are Fibonacci distributed. 7

10 Figure 3.2: 5 potentials with equal strengths where the spacings are Fibonacci distributed Tertiary Quasicrystals Most of the known stable quasicrystals are tertiary, i.e. a solid that consists of 3 different types of atoms, even though there are exceptions [2]. To generalize our model and be able to represent tertiary materials as well we use dense Fibonacci words, as described by A. Monnerot-Dumaine [11], to get 3 letter Fibonacci words. There are several other variants of 3 letter Fibonacci words and for one alternative see A. Ghosh [12]. The dense Fibonacci words are created by a simple mapping from 2 letter Fibonacci words. The mapping is made by grouping letters into pairs and then using the rules presented in Table 3.1. Table 3.1: Mapping from a Fibonacci word (2 letters) to a dense Fibonacci word (3 letters). Combination AA AB BA Maps to α β γ For example, F 6 = ABAABABAABAAB maps to F6 d = αβγγαβ, where the superscripted d stands for dense. This mapping is possible even for the infinite Fibonacci word since the combination BB never occurs in the 2 letter Fibonacci word. The result is an ordered but not periodic 3 letter word. The non-periodicity can be proved when considering that a periodic 3 letter word would imply a periodic 2 letter Fibonacci word Crystals and Random Distributions of Potentials Since we want to compare the behavior of electrons inside a quasicrystal with ordinary crystals and materials with a random distribution of potentials, we have to model these 8

11 last two cases in addition to the quasicrystals. This is done in the following manner: Say that we have 2 different potentials, a and b, then the crystals are modeled (in the 2 letter case) as g a g b g a g b g a g b... and d a d b d a d b d a d b..., i.e. they are periodic. The random setup is created in such a way that strength or the spacing (depending on which case we examine) of each new added potential has a 50% probability to be any of the potentials a and b. The 3 letter case is structured in analogy with its 2 letter counterpart. For a crystal with potentials α, β and γ we have g α g β g γ g α g β g γ... and d α d β d γ d α d β d γ..., while the random distribution is created by letting each new added potential have equal probability to be any of the 3 potentials. 3.3 Method Reflection and Transmission Amplitudes Ultimately we want to calculate the reflection and transmission coefficients, i.e. the probability of reflection and transmission. To do this, we first need to calculate the probability amplitudes, since the coefficients and the amplitudes are related by { R = r 2, (3.1) T = t 2, where R is the reflection coefficient, r is the reflection amplitude, T is the transmission coefficient and t is the transmission amplitude. In order to calculate the amplitudes we need to take into account that an incident electron does not either transmits through the entire array of potentials or reflects off it. There are more possibilities for how it travels through the quasicrystal. Actually, there are an infinite number of ways that the electron can travel through the material. To get a formula for all these possibilities we use a method described by R.J. Olsen and G. Vignale [8]. The most important parts of the derivation of the method follow below. To understand this method think of a very small crystal consisting of only 2 potentials. Then, for example, the electron can transmit through the first potential, then reflect two times before it transmits through the second potential. The electron can also transmit through the first potential, reflect four times and then transmit through the second potential and so forth. Because of this we can write all possible ways of transmission through a crystal with 2 potentials as an infinite series. Let t denote the total probability amplitude of transmission, i.e. the phase shift due to the spatial displacement as well as the phase shift due to tunneling through the potential. Let r denote the corresponding values due to reflection off the potential. We then find the total transmission amplitude for both potentials to be t 2 = t t + t r r t + t r r r r t + t r r r r r r t +... = t 2 (1 + r 2 + r 4 + r ) = t 2 1 r 2, (3.2) where we in the last step use that Eq. (3.2) is a geometrical series. The subscript indicates that this holds for 2 potentials. 9

12 Once we have the transmission amplitude for the 2 potential crystal we can easily add a third potential and calculate the new amplitude using the same method. Eq. (3.2) tells us that as long as we know the transmission amplitude of an existing array we can add a new potential to that array and calculate a new amplitude for it. This is the iterative nature of this method. Eq. (3.2) can then be generalized, and the transmission amplitude for an array consisting of N + 1 potentials can be determined using t N+1 = t t N 1 r r Nl, (3.3) where t N is the transmission amplitude for N potentials, r Nl is the reflection amplitude for N potentials given an electron incident from the left. The subscript Nl is necessary since the setup is irregular and asymmetrical which means that we cannot assume that r Nl = r Nr [8]. The reflection amplitude is calculated using the same method, and it results in the following formula r N+1l = r + t 2 r Nl 1 r r Nl. (3.4) Once we have the amplitudes we can calculate the reflection and transmission coefficients with Eq. (3.1). This setup is general and can be applied to distributions of different lengths and strengths corresponding to quasicrystals as well as crystals and materials with random compositions. As mentioned earlier in this section we use a Fibonacci word to distribute the strengths or the spacings in an ordered and aperiodic way. 10

13 3.4 Analytical Calculations Phase Shift due to Tunneling Through a Delta Function The phase shift due to tunneling through a delta function is derived as follows. Figure 3.3: Dirac delta potential. A general, time-independent wave function that corresponds to an electron incident from the left and the right of the delta function potential shown in Fig. 3.3, placed at x = 0, can be written as { Ae ikx + re ikx x < 0, Ψ(x) = (3.5) te ikx + Be ikx x 0, where A, B C are normalizing constants, r, t C are the reflection and transmission amplitudes and k is the wave vector. The boundary conditions for a delta potential (retrieved from D. Kiang and T. Ochiai [3]) are Ψ(0 ) = Ψ(0 + ), Ψ (0 + ) Ψ (0 ) = 2mg Ψ(0), (3.6) 2 where g is the area/strength of the potential. Note that even though we kept the factor m/ in the derivation, we put this ratio to 2 = 1 in order to simplify our calculations. This only affects the absolute values of m the potential strengths and spacings. That is not a problem since the model s purpose is to qualitatively show the behavior of electrons in quasicrystals, not to provide exact numbers. Also, the initial phase of the wave function is completely arbitrary and is set to be zero in the numerical calculations. 11

14 Electron Incident from the Left For an electron incident from the left of the delta potential placed at x = 0, with known phase and wave vector k, we can put B = 0. This can be interpreted as if there are no electrons incident from the right of the delta potential. We can then determine the value of A by normalizing the wave function. Since the value of A only scales up or down the wave function and does not change the behavior of it we here put A = 1 for simplicity. The boundary conditions Eq. (3.6) give for this case 1 + r = t r = t 1, (3.7) and Substituting Eq. (3.7) into Eq. (3.8) yields Some further algebra results in the equations and ikt (ik ikr) = 2mgt 2. (3.8) 2ik(t + 1) = 2mgt 2. (3.9) t = r = t 1 = ik 2 ik 2 mg, (3.10) mg ik 2 mg, (3.11) where t, r C, g R is the area/strength of the delta potential and m is the mass of the electron. In the formulae that we later use in the numerical calculations (Eq. 3.3 and 3.4) the transmission and reflection amplitudes due to tunneling through the potential is combined with the phase shift due to the travelled distance between the potentials as and t = t e ikd, (3.12) r = r e ikd, (3.13) where d is the spacing (as explained in Sect ) of the potential. Hence, Eq. (3.13) and Eq. (3.12) give us the total reflection and transmission amplitudes [8]. 12

15 3.5 Numerical Analysis All numerical calculations are made using MATLAB. Due to the iterative setup of the method that we use, the transmission is calculated for each new added potential. Depending on the investigation different results are extracted and analyzed. In many cases a comparison between a quasicrystal, a crystal and a random distribution of potentials is made. How the materials are structured is described in detail in Sect When we calculate the resistance in the materials we use the result found by Landauer [13] R 1 T T, (3.14) where R is the effective resistance (the resistance one electron experiences when it travels through the entire material) and T is the transmission coefficient. The designations may be a bit confusing here but note that resistance R R (reflection coefficient) and temperature T T (transmission coefficient). It is important to point out that Eq. (3.14) does not account for the total resistance, but rather one component of it. We will not derive Eq. (3.14) explicitly here, but the derivation is made possible by assuming that the wave vector k (or the energy if you like since E k 2 ) is constant in magnitude during the electron s passing through the material. Physically this means that the electron does not exchange any energy with the atomic nuclei by collisions, something that is possible only when the temperature T = 0. Consequently Eq. (3.14) does not account for the temperature s addition to the resistance. This limitation in Eq. (3.14) affects our results in such a way that the resistance sometimes increases faster than an experiment would show. This can be seen for the case of the random distribution in Fig. 3.6C, 3.7C. Here the resistance increases exponentially with the length of the material, which contradicts Ohm s law. The explanation for this is that in reality where T > 0, electrons can, and do, exchange energy with phonons. They do this increasingly with higher temperature, and this energy exchange destroys some of the quantum mechanics involved. To be precise, it destroys the electron s interference with itself, making the resistance increase slower. Actually, when taking the resistance s temperature dependence into account and investigating lengths much longer than the mean free path 1, the resistance increases linearly with the length of the material and thus is consistent with Ohm s law [8]. 3.6 Results All plots in this section are presented with the relevant quantity on the axes but without any units. The reason for this is our simplification 2 = 1 (Sect ) which implies m that we no longer work in SI units, but rather some kind of natural units. Besides, since our purpose is to qualitatively show tendencies in order to compare them with known behavior, the exact numbers are not as relevant. 1 When dealing with distances smaller than the mean free path Ohm s law does not always hold. 13

16 3.6.1 Binary materials Transmission The following plots show how the effective transmission T, in a binary material depends on the wave vector k with a given number of potentials N. The transmission is plotted for three different structures: A) a quasicrystal, B) a crystal and C) a random distribution of potentials. This is done for two cases: First, in Fig. 3.4, the strengths are Fibonacci distributed and then, in Fig 3.5, the spacings are altered instead. By studying Fig. 3.4A, 3.4B and Fig. 3.5A, 3.5B we can clearly see the conduction bands and the band-gaps in the quasicrystal and the crystal. The width of the transmission bands is approximately the same in both cases, but the transmission in the quasicrystal fluctuates much more. We can also see that the width of the bands increases with increasing k. Both these results are consistent with earlier studies made using a different method, a fact that supports our method [3]. The transmission of the randomized distribution (Fig. 3.4C, 3.5C) also behaves as one would expect, i.e. it has no clear transmission bands but instead some arbitrary placed peaks. Comparing Fig. 3.4A and 3.5A we can see tendencies that the conduction bands are split up into narrower bands in Fig. 3.4 and therefore decreases the transmission. This could mean that the total transmission is affected more by Fibonacci distributing the strengths of the potentials than the spacings of them. Figure 3.4: Transmission of electrons as a function of wave vector k when the strengths of the potentials are Fibonacci distributed. Here g a = 1, g b = 2 and d a = d b = 5. Shown for 3 cases: A) a quasicrystal, B) a crystal and C) a random distribution. In all three cases a material of N = 377 potentials were used. 14

17 Figure 3.5: Transmission of electrons as a function of wave vector k when the spacing of the potentials are Fibonacci distributed. Here d a = 5, d b = 4 and g a = g b = 1. Shown for 3 cases: A) a quasicrystal, B) a crystal and C) a random distribution. In all three cases a material of N = 377 potentials were used. Resistance Here we present the results of our calculations of the effective resistance R (Eq. 3.14) in a binary material as a function of N potentials. The wave vector k is fixed and chosen in such a way that transmission is high. Once again this is performed for three structures: A) a quasicrystal, B) a crystal and C) a random distribution of potentials. In Fig. 3.6 the strengths are Fibonacci distributed while Fig. 3.7 shows the situation where the spacings are Fibonacci distributed. Note that Fig. 3.6C and 3.7C are logarithmic plots. When we look at Fig. 3.6, 3.7, we can see some very interesting results. With a material consisting of only a few potentials the resistance in the quasicrystal (Fig. 3.6A) is approximately 10 times higher than the resistance in the crystal (Fig. 3.6B) of the same length. As of today, this feature is one of the most interesting facts about quasicrystals in general, and a lot of research is being conducted in this area. Experiments show that alloys that consist of up to 70% aluminum can increase their resistance with a factor of 20 simply by arranging the alloys as quasicrystals [5]. Consistent with our results for the transmission, Fig. 3.6A and 3.7A display the tendencies that the varying of the strengths has bigger impact on the conduction than the varying of potential spacings, i.e. the resistance is higher in the case of Fibonacci distributed strengths. As was discussed in Sect. 3.5 the random distributions Fig. 3.6C and 3.7C illustrate the exponential increase in resistance. 15

18 Figure 3.6: Effective resistance, 1 T T, as a function of potentials N when the strengths of the potentials are Fibonacci distributed. Shown for 3 cases: A) a quasicrystal, B) a crystal and C) a random distribution. Here k = 2.3, g a = 1, g b = 2 and d a = d b = 5. Figure 3.7: Effective resistance, 1 T T, as a function of potentials N when the spacings of the potentials are Fibonacci distributed. Shown for 3 cases: A) a quasicrystal, B) a crystal and C) a random distribution. Here k = 2.46, d a = 5, d b = 4 and g a = g b = 1. 16

19 Transmission for Varied Potentials Given a quasicrystal of N potentials with two different potential barriers a and b, placed at equal distances apart, the relative strength (g a /g b ) of the potentials are varied within an interval. The final transmission T through all potentials is then calculated for each combination of the strengths as a function of the wave vector k. The final results are presented in a point plot with the transmission of a specific strength relationship on the y-axis and k on the x-axis. All points in the plot represent cases where the transmission T > This is what Fig. 3.8 shows. In Fig. 3.9 the strengths of the potentials are equal and instead the ratio of the spacings (d a /d b ) are varied. Note that ga = 1 and d a db = 1 correspond to a crystal. Being a numerical method limits the number of potentials possible to use in this investigation, however 377 potentials should be sufficient since the behavior when the number of potentials goes to infinity can be seen already with as few as 10 potentials [14]. As expected, in Fig. 3.8 and 3.9 we can see the gradual change in the conduction bands and band-gaps as the quasiperiodic nature of the setup gets more accentuated, i.e. the ratio deviate more from 1 (which corresponds to a crystal). In the case with the varied potentials strengths (Fig. 3.8) the bands break up into more and narrower bands the further from 1 the ratio gets. However, the edge of each conduction band corresponding to a larger k is preserved. This has been observed at every band examined with this method, and is consistent with the results of D. Kiang and T. Ochiai [3] where a matrix method was used instead of the iterative one used in this report. No such band edge preservation can be seen in Fig. 3.9 where the spacings around the potentials are varied. While it still shows a repeating pattern for each band no obvious conclusions can be made in this case. The way the spacings are dealt with in our model (Sect ) is different from how D. Kiang and T. Ochiai [3] model them for example, and even though the results show similarities they are not easily compared. g b 17

20 Figure 3.8: A setup with 377 potentials placed 5 l.u. apart (d a = d b = 5) with strengths g a and g b distributed as a Fibonacci word. Their relative strengths are g b = 1 and g a = y g b. Every point represents that an electron with a wave vector of size k = x has the probability of being transmitted larger than g a gb = 1 corresponds to a crystal. Figure 3.9: A setup with 377 potentials placed with g a = g b = 1 and with spacings d a and d b distributed as a Fibonacci word. Their relative spacings are d b = 1 and d a = y d b. Every point represents that an electron with a wave vector of size k = x has the probability of being transmitted larger than d a db = 1 corresponds to a crystal. 18

21 3.6.2 Tertiary materials The following plots all concern a tertiary material. To model the tertiary material we use a dense Fibonacci word, as described in Sect In analogy with the investigation for binary materials Fig and 3.11 plot transmission as a function of wave vector k. Further on, in Fig and 3.13, resistance as a function of N potentials is shown. Both transmission and resistance of the quasicrystal (A) is compared with an ordinary crystal (B) and randomly distributed potentials (C). Finally, the transmission s dependence on the ratio between the potentials α, β and γ s strengths and spacings, is presented in Fig and Transmission Comparing the transmission of the 3 letter Fibonacci quasicrystal, Fig. 3.10A and 3.11A, with the 2 letter counterpart, Fig. 3.4A and 3.5A, we see that the structure of the conduction bands and band-gaps are similar. The fact that the plots resemble those of the 2 letter quasicrystal and crystal speaks in favor of our choice of algorithm for creating the 3 letter Fibonacci word. To determine whether the effects due to tunneling has greater importance for the transmission than the spatial displacement is more difficult in the tertiary case because of the increased fluctuations of the transmission within the conduction bands. Compared to the transmission in the binary material. This makes it hard to find a value of k that gives the same amount of transmission in all cases, thus making the results more problematic to compare. Figure 3.10: Transmission of electrons in a tertiary material as a function of wave vector k when the strengths of the potentials are Fibonacci distributed. Here N = 304, g α = 1, g β = 2, g γ = 1.5 and d α = d β = d γ = 5. Shown for 3 cases: A) a quasicrystal, B) a crystal and C) a random distribution of potentials. 19

22 Figure 3.11: Transmission of electrons in a tertiary as a function of wave vector k when the distances between the potentials are Fibonacci distributed. Here N = 304, d α = 5, d β = 4, d γ = 4.5 and g α = g β = g γ = 1. Shown for 3 cases: A) a quasicrystal, B) a crystal and C) a random distribution of potentials. Resistance One very interesting feature with the 3 letter Fibonacci word is the fact that the high resistance in the binary quasicrystal is inherited also to the tertiary. This is seen when studying Fig. 3.12A and 3.13A. As in the binary case, we get the expected exponential behavior of the resistance for the random distribution of potentials (Fig. 3.12C and 3.13C). Note that Fig. 3.12C and 3.13C are logarithmic. 20

23 1 T Figure 3.12: Effective resistance, T, in a tertiary material as a function of potentials when the strengths of the potentials are Fibonacci distributed. Here k = 2.27, g α = 1, g β = 2, g γ = 1.5 and d α = d β = d γ = 5. Shown for 3 cases: A) a quasicrystal, B) a crystal and C) a random distribution of potentials. 1 T Figure 3.13: Effective resistance, T, in a tertiary material as a function of potentials when the distances between the potentials are Fibonacci distributed. Here k = 4.3, d α = 5, d β = 4, d γ = 4.5 and g α = g β = g γ = 1 Shown for 3 cases: A) a quasicrystal, B) a crystal and C) a random distribution of potentials. 21

24 Transmission for Varied Potentials Fig and 3.15 were made in a manner similar to Fig. 3.8 and 3.9 except a dense Fibonacci word was used instead of an ordinary Fibonacci word. The figures were made by keeping one potential constant and then varying the ratios between that potential and the other two. Each sequent plot represents a higher value of k. The axes of the plots are linear but the values are not shown in order to avoid redundant numbers. Fig and 3.15 are difficult to analyze due to the large amount of data and many variables involved. However, some general trends can be seen. If Fig and 3.15 are placed orthogonally to Fig. 3.8 and 3.9 we can consider them as 2D projections of a common 3D space. The line y = 1 in the binary Fibonacci word (Fig. 3.8 and 3.9) would then be seen as a point in Fig and The observed pattern seems to indicate that the criteria for transmission is that the combination of x, k and y, k values that permit transmission in the binary Fibonacci word setup also, approximately, permit transmission in the tertiary setup. This could mean that the principles which determine transmission in the Fibonacci quasicrystals carry over to other quasiperiodic distributions as well. The similar pattern is perhaps most clearly seen if we compare Fig and 3.8. In both cases we see with increasing k: First, a band gap, then a band with varying transmission which has a clear cut at the edge of larger k but a more split up behavior at edge of smaller k. Figure 3.14: A setup with a 3 letter dense Fibonacci word with y = g a /g b and x = g c /g b with g a, g b and g c being the strength of the 3 different potentials. Every diagram has a different value of k starting in the top left corner with k=0.35. Each successive diagram has an increased k with a step k = Every point represents transmission T >

25 Figure 3.15: A setup with a 3 letter dense Fibonacci word with y = d a /d b and x = d c /d b with d a, d b and d c being the spacing of the 3 different potentials. Every diagram has a different value of k starting in the top left corner with k=0.35. Each successive diagram has an increased k with a step k = Every point represents transmission T > Perturbations The results of adding a small perturbation in the potentials phase shift are presented in Fig This is demonstrated for three cases: A) a binary crystal B) a binary quasicrystal and C) a tertiary quasicrystal. In Fig the corresponding unperturbed materials are shown. This perturbation in the phase shift can be interpreted as both a perturbation in the potential strength as well as the potential spacing. Note that no tertiary crystal is shown, since they act very similar to the binary crystal and would therefore be a redundant plot. The introduced perturbation is made using a normal distribution with a mean µ = 0 and a variance σ = k. We choose this variance because we want the perturbation to be small, so that when it is combined with a large number of potentials the effects that might arise from random variations are reduced. The results presented are means over 100 perturbed systems/materials. Each point in the figure represent the mean of 100 potentials each in order to suppress fast oscillations. There are some very interesting trends showing in Fig The figure tells us that quasicrystals may have a tendency to be more sensitive to perturbations compared to ordinary crystals, especially the tertiary quasicrystal. Exponentially fitting the curves in Fig gives the decrease constants a C = , a QC,binary = and a QC,tertiary = We see that the binary quasicrystal drops off approximately 1.5 times faster than the crystal and the tertiary decreases approximately 2.7 times faster. 23

26 This is a reasonable result since the quasicrystals have a kind of perturbation already, due to their aperiodic nature. Nevertheless, this is an important result that needs to be taken into account in the process of artificially creating quasicrystals. In order for the material to get the typical quasicrystal behavior, and not that of a randomized distribution, the quasicrystals need to be created with high quality. Figure 3.16: Transmission as a function of potentials for A) a binary crystal, B) a binary quasicrystal and C) a tertiary quasicrystal. The plotted results are means over 100 unperturbed potentials. k = 0.5 in all plots. For cases A) and B) the potentials are t a = 0.9e 0.56i, r a = 0.436e 2.13i, t b = 0.6e 0.97i and r b = 0.8e 2.54i. In C) there is an additional potential t c = 0.75e 0.787i r c = 0.9e 2.36i. Figure 3.17: The mean transmission for 100 perturbed systems as a function of potentials for A) a binary crystal, B) a binary quasicrystal and C) a tertiary quasicrystal. The plotted results are means over 100 potentials. k = 0.5 in all plots. For cases A) and B) the potentials are t a = 0.9e 0.56i, r a = 0.436e 2.13i, t b = 0.6e 0.97i and r b = 0.8e 2.54i. In C) there is an additional potential t c = 0.75e 0.787i r c = 0.66e 2.36i. Every potential has an added normal distributed perturbation with mean µ = 0 and a variance σ = k. 3.7 Discussion Model and method Our model, as any other model, has both strengths and weaknesses. One very interesting feature is that because of its general design it can easily, by changing the phase shift so it corresponds to that of a photon, be used for simulating photon propagation in quasicrystals (Sect. 1.3) instead [4]. Further on, one aspect that needs some attention is how we place the potentials. They are always placed with an equal amount of empty space on each side, where no other potential is allowed to exist. What we in this report refer to as the potential s spacing. This can become a problem if you want to Fibonacci distribute the distances of between potentials, as in the report by D. Kiang and T. Ochiai [3]. The problem is that our model imposes a restriction on the distances between the potentials. This is best explained by some mathematics. First, say you have a material with 2 potentials, with spacing A or B. Now say you want to have either distance D 1 or D 2 between two sequent potentials. 24

27 In our model this can be expressed as (remember that the combination BB never occurs in any Fibonacci word) A = D 1, (3.15) and A + B = D 2. (3.16) 2 Combining Eq. (3.15) and Eq. (3.16) gives and since B 0 we get the following restriction B = 2D 2 D 1, (3.17) 2D 2 D 1. (3.18) The reason for our placement is the simple fact that it is a placement as good as any other. We have not found anything that is obviously wrong with our method (physically speaking), besides the limitation above. Of course, we could have, for example, placed the potentials in the beginning of each spacing. This would imply that the spacing of each potential becomes synonymous with the distance between the potentials. That way of placing the them has the advantage of letting us explicitly Fibonacci distribute the distances between every potential. However, this does not change the structure of the quasicrystal significantly compared to our model, but it does complicate the calculations. Eq. (3.3) and (3.4) would have to be derived in another manner. Therefore, we have chosen to place the potentials in the middle of their spacing. The method itself ought to be discussed as well. Throughout our investigation we have come to realize that it is quite difficult to see the advantage of using this method before the matrix method, described by D. Kiang and T. Ochiai [3] (which has been mentioned earlier in this report), since the calculations are more efficient in the matrix method. Also, when studying other reports with research similar to ours, it seems that the matrix method is the one most commonly used. The features of the iterative method that do speak in favor of it though are: it is a very simple and intuitive method and it clearly shows the great impact that interference has on particle propagation in solids. Another factor that affects the results is the way we construct the randomized distributions. This is something that can be done in many different ways. In our model, when randomizing with respect to strength for example, we let each new added potential have equal probability of becoming any of the available strengths 2. This makes the randomized distribution look more like a crystal than a quasicrystal. If we would have made the ratio between the probabilities equal to the golden ratio instead, then the randomized distribution would resemble a quasicrystal more than a crystal. We chose the first method because if the results from that method differs a lot from the results of an ordinary crystal it tells us more about how sensitive the system is to perturbations than it would if the randomized distribution was more like a quasicrystal. Since they initially have a kind of perturbation already because of their aperiodic nature. 2 2 for a binary word and 3 for a tertiary. 25

28 3.7.2 Next step The method used here can be refined and further developed in a number of ways to suit a more detailed or specific problem. For example, D. Hennig, G.P. Tsironis, M.I. Molina, and H. Gabriel added a nonlinear term in the Kronig-Penney model [15] and F. Domínguez-Adame and A. Sanchez studied relativistic effects [16]. They did however use the matrix method described in [3], but with some additions it should be possible to use the iterative method for this purpose as well. Another field of interest could be to investigate how the electric conduction acts in tertiary materials generated by some other algorithm than the one used in this report, i.e. the dense Fibonacci words. For example, as mentioned in Sect , there are other binary and tertiary words which could prove useful to investigate. Due to the many variables involved in the many possible setups, our investigation is just an overview and can be conducted in an almost infinite number of variations. The goal of this investigation has only been to examine the general behavior of electron propagation in quasicrystals and for a more thorough investigation a refined model is probably more suitable. 26

29 Chapter 4 Summary and Conclusions Even if the model we use is one of the simplest models used to represent solid materials, our investigation shows a clear difference between quasicrystals, crystals and amorphous materials. Both in a binary material as well as in a tertiary material. We find that quasicrystals have conduction bands and band-gaps very similar to ordinary crystals, even though the probability of transmission fluctuates significantly within the conduction bands in the quasicrystal case. We also compare the impact that the effects due to tunneling and the effects that comes from the spatial displacement of the potentials have on the transmission and the resistance in quasicrystals. We find that the tunneling effects accentuate the quasicrystallic behavior more. When studying the resistance our results show that it is remarkably higher in quasicrystals compared to ordinary crystals, even for a few potentials. A resistance as much as 10 times higher can be seen with approximately 50 potentials. This is probably the most powerful feature of the quasicrystals. The mere act of restructuring a material with good conducting properties as a quasicrystal can increase the resistance notably. We also find results for tertiary quasicrystals analogous to those of the binary quasicrystals. In other words, the quasicrystals behaves similar whether the material is binary or tertiary. Another important observation is that quasicrystals are more sensitive to perturbations than crystals, and especially the tertiary quasicrystal. This means that when artificially constructing quasicrystals, high quality is demanded in order to get the characteristics of these materials. The iterative method used here, while simple, is primarily suitable for education and illustrating the principle behavior of electron transmission in quasicrystals as well as other solid materials. We ran simulations with both the iterative method and the more widely used matrix method and for all cases that we tested the results are the same. Still, for a more detailed investigation our conclusion is that the matrix method is probably a more befitting choice, since the algorithm is more efficient and it is also more accepted by the scientific society. Acknowledgements The authors wish to thank Jack Lidmar for his guidance and advice, and Magnus Andersson and Östen Rapp for the help of finding good references. Also thanks to Filip Allard, David Blomqvist, Oscar Olofsson, Erik Larsson, Olle Jacobson and Niclas Höglund for interesting discussions. 27

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