Theoretical Studies of Transport Properties in Epitaxial Graphene

Transcription

1 Theoretical Studies of Transport Properties in Epitaxial Graphene Diploma Thesis by Nicolas Ray Supervised by Prof. Dr. Oleg Pankratov Lehrstuhl für Theoretische Festköperphysik Institut für Theoretische Physik Universität Erlangen-Nürnberg June 9, 011

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3 Contents 1 Motivation and Outline 1 Graphene 3.1 Atomic Structure Electronic Band Structure Tight-binding model Dirac-Weyl Hamiltonian Electron Scattering Suppression of backscattering Klein tunneling Graphene on Silicon Carbide Atomic Structure Electronic Band Structure Model for Band Structure with Interface States Electron Scattering due to Interface States Theoretical Background Lattice Vibrations Theory of elasticity Vibrations of a discrete lattice Theory of Conductivity Boltzmann Equation Relaxation time approximation Calculation of the conductivity Calculation of the mobility Calculation of the relaxation time Electron Scattering due to Interface Phonons Formulae for Conductivity and Mobility Transition probability Calculation of the relaxation time Calculation of conductivity and mobility Deformation Potential Scattering Qualitative discussion Numerical results

4 5.3 Resonant Scattering due to a Single Interface State Resonant Scattering due to Multiple Interface States Conclusion 61 Bibliography 63 Acknowledgements 65

5 1 Motivation and Outline The two-dimensional honeycomb lattice of carbon atoms today referred to as graphene was studied until 004 only as a convenient theoretical concept, a building block of graphite as well as of carbon nano tubes and fullerenes. Free standing graphene layers as two dimensional crystals in general were considered to be thermodynamically unstable according to the arguments of Peierls and Landau [1,, 3]. The experimental discovery of graphene came as a surprise in 004, when A. Geim and K. Novoselov produced few-layer and single-layer graphene flakes from graphite by mechanical exfoliation [4]. In 010 they were awarded the Nobel Prize in Physics for groundbreaking experiments regarding the two-dimensional material graphene 1. Graphene is a material of great interest for fundamental research. Due to a special symmetry of the two-dimensional honeycomb lattice remarkable electronic properties arise. Near the Fermi level the electrons are described by an effective Dirac-Weyl Hamiltonian which means that they behave as massless relativistic particles (cf. Chapter ). Hence a variety of quantum mechanic effects which can be observed in graphene are totally new. A novel relativistic Quantum Hall effect, for example, can be measured in graphene at room temperature [5]. The Klein tunelling, which describes the complete transparency of potential barriers for charge carriers with certain angles of incidence was predicted for the electrons in graphene [6] and actually measured [7]. Graphene is not only a subject of fundamental research but also of enormous technological interest [8, 9]. The unique scattering behaviour of the massless carriers gives rise to a high electron mobility, which together with graphene s favourable structural properties makes it a good candidate for nanoelectronic applications. For technological purposes, however, graphene production by mechanical exfoliation is not suitable, as it is a noncontrollable and expensive procedure. More promising is the epitaxial growth on silicon carbide (cf. Chapter 3). The ordered carbon layers are, in this case, generated by sublimation of Si-atoms from the SiC crystal. However, at least the first of these layers is coupled to the substrate and the subsequent layers interact albeit weakly with the SiC surface. Consequently some of the unique properties of free standing graphene do not persist in epitaxially grown graphene layers. A prominent feature is the mobility of epitaxial graphene, which is dramatically suppressed for reasons that are not yet understood. In order to benefit from the promising epitaxial growth technologically, more knowledge about the interaction of graphene with the SiC substrate is needed. In this thesis the influence of the thermal lattice vibrations 1 press release; The Royal Swedish Academy of Science, October 5,

6 1 Motivation and Outline at the SiC-graphene interface will be studied theoretically. Ab-initio calculations show that the electronic band structure changes due to the interface vibrations. This gives rise to additional electron scattering in the graphene layer, which explains the mobility decrease. The outline of the thesis is as follows: In Chapter the fundamental properties of free standing graphene are reviewed. The electronic band structure near the Fermi level is derived from a tight-binding model and the consequences for electron scattering are discussed. Chapter 3 is dedicated to epitaxial graphene on SiC. The difference between free standing graphene and epitaxial layers as well as a model for the interaction of interface phonons with the electrons in graphene are presented. After reviewing the theoretical background for lattice vibrations and conductivity theory in Chapter 4, we calculate the electron mobility and the conductivity of the epitaxial graphene layer accounting the scattering due to interface phonons in Chapter 5.

7 Graphene In this section, the most important properties of graphene mentioned in the previous chapter will be examined. First we review the atomic honeycomb lattice and the corresponding reciprocal lattice. After that we derive the Dirac-Weyl Hamiltonian using a tight-binding approach. Finally some consequences for the electron scattering are discussed..1 Atomic Structure The carbon atoms of graphene are arranged in a hexagonal (honeycomb) lattice, which can be represented by a Bravais lattice with two atoms (A and B) in a unit cell (cf. Fig..1, left). Every atom on sublattice A has three nearest neighbours on sublattice B, and vice versa. The lattice vectors a 1 and a as well as the nearest neighbour vectors δ 1, δ and δ 3 are displayed in Fig..1. In cartesian coordinates they are given by a 1 = a ( ) 3, 3 a = a ( ) 3, 1 δ 1 = a ( 3, 1) = δ 3 + a 1 δ = a ( 3, 1) = δ 3 + a δ 3 = a(0, 1) (.1) where a = 1.4 Å is the distance between neighbouring carbon atoms. The right panel of Fig..1 shows the first Brillouin zone of the corresponding reciprocal lattice. It also has the hexagonal symmetry. The reciprocal lattice vectors are b 1 = π ( ) 3, 1 3a b = π ( ) 3, 1 3a satisfying the relation a i b j = πδ ij. The most important points in the reciprocal space are the inequivalent K and K points, where the electrons are approximately described by the Dirac-Weyl Hamiltonian (cf. Chapter.). These points are located at K = 1 3 (b 1 + b ) K = 1 3 (b 1 + b ) (.) 3

8 Graphene ' ky y x k x Figure.1: [10] Left: The real space honeycomb lattice generated by the two sublattices A (blue) and B (yellow) with lattice vectors a i and nearest neighbour vectors δ i. Right: The first Brillouin zone in reciprocal space with reciprocal lattice vectors b i. The high symmetry points are marked as Γ, K and K.. Electronic Band Structure Every carbon atom in the graphene lattice brings four valence electrons. Three of them are sp hybridized and connect the neighbouring atoms with stiff covalent σ-bonds. The remaining electron occupies a p z orbital, which is directed perpendicular to the graphene sheet. In graphite, where the carbon layers are stacked, these π electrons provide the weak bonding between the neighbouring layers. In a single graphene layer however, the π electrons form a half-occupied delocalized energy band. While the σ-electrons are trapped in the strong C-C bonds far below the Fermi energy, the π-electrons have a low energy excitation spectrum and are responsible for the charge transport in graphene. Hence in the following, only the π-bands will be examined. For low energy excitations, the dispersion of the π-electrons can be approximated by a Dirac-Weyl Hamiltonian, which can be derived using a tight-binding approach as follows...1 Tight-binding model In the tight-binding model the electron wave function is approximated by a linear combination of atomic orbitals. The Hamiltonian allowing only hopping between nearest 4

9 . Electronic Band Structure neighbours 1 is Ĥ = ɛ 0 (â R+δ1 â R+δ1 + ˆb Rˆb ) R + t R B R B 3 i=1 (ˆb RâR+δ i + â R+δ iˆbr ) (.3) An operator ˆb R (ˆb R ) adds (removes) an electron at site R of sublattice B. The operators â and â operate in the same way, but on sublattice A. The Hamiltonian consists of two parts. The first sum accounts for the on-site energies by counting the electrons of sublattices A and B. As the sites are equivalent, every electron contributes the same energy ɛ 0. The second sum describes the nearest-neighbour hopping. Note, that the hopping always involves a change of sublattice. The hopping integral is denoted by t. The eigenstates of the Hamiltonian are the linear combinations ( ) α ψ k = α ψk A + β ψk B =: β of the Bloch waves ψk A and ψb k defined on the two sublattices. ψ A k = 1 N R B ψ B k = 1 N R B e ik(r+δ 1)â R+δ 1 0 The tight-binding Hamiltonian can be expressed in this basis as Ĥ = ( ψ A k Ĥ ψa k ψa k Ĥ ψb k ) ψ B k k Ĥ ψa k ψb k Ĥ ψb k Combining (.4) with (.3) yields for the diagonal elements e ikr â R 0 (.4) ψ A k Ĥ ψa k = ψ B k Ĥ ψb k = ɛ 0 The shift ɛ 0 of the eigenvalues can be chosen to be zero without loss of generality. For the off-diagonal elements with (.4) and (.3) we obtain ψ A k Ĥ ψb k = ψ B k Ĥ ψa k = t ( e ikδ 1 + e ikδ + e ikδ 3 ) = t e ikδ 3 ( 1 + e ika 1 + e ika ) where the relations (.1) are used. =: t f(k) (.5) 1 Hopping between next nearest neighbours and beyond can be neglected for most purposes; the next nearest neighbour hopping integral is approximately 100 times smaller [10] than the nearest neighbour hopping integral and leads to perturbation of the band structure only on a small energy scale. 5

10 Graphene E(k) [ev] Γ K M k vector Figure.: Electron energy dispersion of the π electrons in graphene along high symmetry axis ΓKM (cf. Fig..1). The dispersion was calculated with the tight binding model (.7) considering only nearest neighbour hopping with a hopping energy t = 3 ev. The resulting secular equation ( 0 t f(k) t f (k) 0 ) ψ k = E k ψ k (.6) yields the eigenenergies E ± k = ±t f(k) We insert (.5) and the lattice vectors from (.1) to obtain an explicit expression for the energy dispersion: E ± k = ±t f(k) = ±t 1 + e ika 1 + e ika ( ) ( ) ( ) = ±t 3a 3a 3 + cos 3akx + 4 cos k x cos k y (.7) This result is plotted along the high symmetry axes ΓK and KM (cf Fig..1, right) in Fig... At zero temperature the lower band is completely filled and the upper band is completely empty. There is no band gap however, since the positive and the negative 6

11 . Electronic Band Structure energy band touch each other at the K point. This can be shown easily by inserting (.) into the energy dispersion (.7): f(k) = 1 + e i 1 3 (b 1+b )a 1 + e i 1 3 (b 1+b )a = 1 + e i π 3 + e i 4π 3 = 0 (.8) We will refer to the states in the upper (lower) band as electrons (holes)... Dirac-Weyl Hamiltonian We now focus on electron energies in the low-energy range E k < 1 ev. According to Fig..7 the states are restricted to the area near the K and the K points and the dispersion is approximately linear. It is reasonable to Taylor-expand the function f(k) around the K and the K point up to first order with respect to k. As the zeroth order term vanishes (.8) we obtain f(k + k ) [ k f(k)] k=k k = [ ia 1 e ika 1 + ia e ] ika k=k k = i a ( ) 3k x + 3k y e i π a ( ) 3 + i 3k x + 3k y e i 4π 3 = i a ( ( 3k x i sin π ) ( )) π + 3k y cos 3 3 = 3a ( k x ik y) With the Fermi velocity v F = 3at/( ) = 10 6 m/s (experimental value) the Hamiltonian (.6) near the K point can be written as Ĥ K+k = v F ( 0 k x ik y k x + ik y 0 ) (.9) Similarly, near the K point one obtains Ĥ K +k = v F ( 0 k x ik y k x + ik y 0 ) (.10) With the Pauli spin matrices σ = (σ x, σ y ) = (( ) ( 0 i, i 0 and the momentum p = k the Hamiltonians (.9) and (.10) can be written as )) Ĥ K+k = v F σ p Ĥ K +k = v F σ p 7

12 Graphene This result should be compared with the right- and left-handed Dirac-Weyl Hamiltonian, which one obtains (e.g. for neutrinos) from the Dirac Hamiltonian with a particle mass of zero [11]. Ĥ R = c σ p Ĥ L = c σ p The two components of the true Dirac-Weyl Hamiltonian account for spin. In the lowenergy graphene Hamiltonian the two components correspond to the wave function projection onto the two sublattices A and B. Due to the similar mathematical structure this degree of freedom is often referred to as pseudo-spin. Note that the speed of light c occuring in the Dirac-Weyl Hamiltonian is replaced by the Fermi velocity v F c/300 in the graphene case and that the K point Hamiltonian does not exactly match the lefthanded Dirac-Weyl Hamiltonian due to the complex conjugation of the Pauli matrices σ. From the structure of the Hamiltonian one can immediately conclude that for electrons and holes in graphene the chirality is a good quantum number. The chirality is in this case the projection of the pseudo spin σ on the direction of the momentum p/p. At the K point electrons (holes) have positive (negative) chirality (cf. Fig..3)..3 Electron Scattering The fact that the low-energy electrons in graphene are governed by the Dirac-Weyl equation (instead of the Schroedinger equation) profoundly affects their scattering behaviour. Two interesting phenomena, namely the suppression of backscattering and the Klein tunneling, will be discussed below. First the eigenvectors of (.9) and (.10) are calculated. With the angle θ k between the vector k and the x-axis (.9) becomes ( ) 0 e iθ k Ĥ K+k = v F k e iθ k 0 with the eigenvectors k = 1 ( e iθ k / ±e iθ k/ Similarly near the K point one obtains k = 1 ( e iθ k / e iθ k/ ) = e iθ k/ ψ A k ± eiθ k/ ψ B k for E k = ± v F k (.11) ) = eiθ k/ ψ A k e iθ k/ ψ B k for E k = ± v F k The eigenstates for positive (negative) energies are electron (hole) states. In continuum approximation the atomistic structure of the sublattices A and B is neglected and the 8

13 .3 Electron Scattering space representation of the basis vectors can be written as r ψ A k = 1 V e ikr A r ψ B k = 1 V e ikr B where only the orthogonality of the sublattices A B = 0 persists and V is the area of the graphene sheet..3.1 Suppression of backscattering. With the eigenstates (.11) the electron scattering near the K point due to an impurity potential U(r) can be calculated. k U(r) k = 1 d r e ( ( ) ( ) i(k k )r e iθ k, e iθ ) U(r) 0 k e iθ k V 0 U(r) e iθ k = 1 ( ) d r e i(k k )r θk θ k U(r) cos V The cosine function resulting from the pseudospin structure of the electron wave function suppresses the backscattering. For θ k θ k = π the matrix element yields zero. That means that impurity scattering, which often dominates the electronic properties of a material, is not effective in graphene. Consequently the electron mobilty in graphene is very high. The most important scattering mechanism in case of suspended graphene is due to flexural (i.e. out of plane) phonons and limits the mobility to about 10 4 cm /(Vs) at room temperature. The flexural phonon scattering can be suppressed by lowering the temperature or by applying strain to the sample. Then mobilities of 10 5 cm /(Vs) can be achieved [1]..3. Klein tunneling. The Klein tunneling, also known as Klein paradox, describes the transmission of a relativistic particle with the energy E through a potential barrier of height V 0 > E, as sketched in Fig..3 (b). While for a non-relativistic electron governed by the Schroedinger equation the transmission probability T decays exponentially with increasing barrier height, for a relativistic electron of graphene, the transmission probability can take values of T = 1 for certain angles of incidence (e.g. normal incidence) and very high potentials V 0. These results are obtained by setting up general plane waves of the type (.11) for the ranges in front of, inside and behind the potential barrier and fitting the coefficients such that the functions match at the boundary of the barrier [6]. Note that this argument applies only to impurity potentials U(r) that do not break the sublattice symmetry, i.e. to sufficiently long range potentials. 9

14 Graphene Figure.3: [6] a) Low energy spectrum of graphene. The blue filled areas indicate the occupied states. The Dirac cone is shifted according to the potential barrier displayed in (b). Electrons (holes) have positive (negative) chirality. b) Electron with energy E approaching a potential barrier of height V 0 and width D. The perfect transparency of the barrier for electrons with normal incidence can be explained with the pseudospinor structure of the wave functions: In Fig..3 (a) the band structure corresponding to Fig..3 (b) is displayed. Within the potential barrier, due to the shift of the band structure by V 0, the Fermi energy is not located in the upper electron branch but it is shifted to the lower branch of hole states. The incoming wave function of the right-moving electron state matches a wave function of a left-moving hole state inside the barrier, which behind the barrier connects to a wave function of a right-moving electron state. The crucial point is the conservation of pseudo spin. For all three sections the direction of the pseudospin is the same. In contrast, backscattering of the electron wave function from k to k at the barrier would change the pseudospin. However, the potential considered here acts in the same way on the two sublattices A and B and therefore does not affect the pseudospin [6]. Hence, the conservation of pseudo spin results in a perfect transmission of the electron wave through the potential barrier. 10

15 3 Graphene on Silicon Carbide A technologically promising way of producing graphene is the epitaxial growth on SiC [13, 14, 15, 16, 17, 18]. The advantage of this method over the mechanical exfoliation is that it allows to produce flakes of greater area in a more controlled way. By heating the SiC crystal to temperatures above 1000 K one can achieve that the Si atoms sublimate from the surface whereas the remaining C atoms reorganize to form graphene layers. However, some of the unique properties of graphene are modified in presence of the SiC substrate, one of them being the extremely high mobility. While the electron mobility reaches values up to 10 5 cm /(Vs) in exfoliated suspended graphene flakes [8, 19, 0, 1], for epitaxial graphene layers on SiC(0001) only mobilities of around 10 3 cm /(Vs) are measured [1]. This drastic mobility suppression is clearly caused by the influence of the substrate, however the exact mechanism responsible for this effect is not known yet. A decrease of mobility is also observed for graphene sheets on other substrates. For example on SiO substrate the mobility is about 10 4 cm /(Vs) [0]. Here mobility suppression is due the electric field induced by the optical surface phonons in the substrate []. This field causes additional electron scattering in the graphene layer. This effect, however, cannot explain the decrease of mobility in epitaxial graphene on SiC for two reasons: (i) The optical surface phonons in silicon carbide have much higher energies than in silicon dioxide and therefore should not play an important role at room temperature and (ii) the charge is more evenly distributed on a SiC surface, than on a SiO crystal and the phonon-induced electric field is much weaker. Hence for epitaxial graphene on SiC there should be other scattering mechanisms which lead to the dramatic drop of the mobility. 3.1 Atomic Structure First we take a closer look at the atomic structure of the graphene layers on the SiC surface. Of course, the processing time and temperature influence the morphology of the interface. But a decisive role is played by the type of the silicon carbide surface. One distinguishes two cases: the C-terminated surface (0001) and the Si-terminated surface (0001), which remarkably lead to completely different results after heating is applied [17, 18, 3]: On the C-face, the graphene growth is not self-limited. This typically results in multiple graphene layers, which are rotated with respect to each other. The rotation is very 11

16 3 Graphene on Silicon Carbide Figure 3.1: Left: [18] Top view of the atomic structure for the ( 3 3) reconstruction. The C-C bonds of the buffer layer are artificially stretched by about 8 %. Right: [4] Side view of one unit cell for the (5 5) reconstruction. yellow: Si-atoms od SiC, black: C atoms of SiC; red: C atoms of buffer layer, blue: C atoms of graphene layer. important, since it electronically decouples the layers from each other such that the electronic structure of a single graphene layer is preserved in the whole stack, which may be several hundred carbon layers thick. On the Si-face the growth is slower and self-limited. Hence, it is easier to generate single layers. If multiple graphene layers are produced, they arrange in a stable AB stacking, as in graphite. An important feature of the epitaxial graphene grown on the Si-face is the existence of a buffer layer. This is the first carbon layer on the Si-face which is strongly coupled to the substrate. This covalent coupling destroys the unique graphene band structure of the buffer layer and only the subsequent C-layer shows the genuine electronic properties of graphene. Nevertheless the buffer layer has the same honeycomb atomic structure as graphene, with practically the same lattice constant [18, 3]. This is no surprise, as the σ-bonds connecting the carbon atoms in the buffer layer are very strong. Consequently the buffer sheet cannot be stretched or compressed considerably to fit the periodicity of the SiC(0001) surface. The commensurate structure comprising the periodicities of both lattices is (as found in experiment [18]) a ( )R30 reconstruction of the SiC(0001) surface. This unit cell nearly perfectly matches a (13 13) cell of the buffer layer. However, such a large supercell contains several hundred atoms per layer and is therefore not suitable for accurate ab-initio computations. Models with smaller unit cells have been used in the past, that on the downside always require stretching or compression of the buffer sheet. To fit a ( ) buffer layer cell to a ( 3 3)R30 surface unit cell (see Fig. 3.1, left), stretching of the C-C σ-bonds by about 8 % is necessary [18]. Another model (see Fig. 3.1, right) which requires practically no stretching, but contains many 1

17 3. Electronic Band Structure more atoms per cell uses a (5 5) surface superstructure which is commensurate with ( ) graphene units [4]. Note, that in the ( 3 3) model there is one Si atom per unit cell with a dangling bond pointing to the buffer layer, while the other two Si atoms have a carbon atom of the buffer layer sitting directly on top of them. For the (5 5) model, the situation is more complicated. In the larger unit cell, a higher number of dangling bonds is present, although the dangling bond density per unit area is smaller than in the ( 3 3) model. 3. Electronic Band Structure The objective of this section is to introduce a band structure model which captures the most important features of the electron spectrum of the realistic ( ) interface. For the ( 3 3) and (5 5) models, the ab-initio band structure was explicitly calculated in [5] and [4] respectively. The most important properties of these results compared to the calculated dispersion of the free standing graphene layers will be pointed out. After that it will be discussed which results from the models can be transfered to the ( )R30 structure. The left panel of Fig. 3. shows the results for the ( 3 3)R30 model. The topmost plot shows the band structure for a single carbon layer on top of the SiC(0001) surface. No π-bands with linear dispersion at the K point are present close to the Fermi energy. This shows that the π electrons of the first layer are needed to saturate the dangling bonds of the SiC substrate. Hence the first carbon sheet is a buffer layer which does not posess the Dirac spectrum of free standing graphene. The next carbon layer (see the middle plot), however, yields electron states similar to those of graphene. It can therefore be regarded as weakly coupled to the substrate. Similarly, if a third layer is added (lowermost plot), the two top layers behave like a free standing graphene bilayer, while the lowest layer still serves as a buffer. Although the graphene-like bands are easily spotted, some important differences have to be noted: (i) The Dirac cone is intersected by an additional flat state. This localized state results from the dangling Si bonds at the interface. (ii) The Fermi level is pinned by this interface state, which is located about 0.5 ev above the tip of the Dirac cone. This results in n-doping of the graphene layers due to charge transfer from the substrate. Similar results are obtained in the (5 5) model (Fig. 3., right). The K and K point are folded back to the Γ point in this case. Instead of a single flat interface state, several such states appear. This reflects the larger unit cell and the increased number of dangling bonds. The Fermi energy is pinned at about 0.4 ev above the tip of the Dirac cone. Two other notable features of the band structure are recovered by zooming into the region near the Dirac point (see Fig. 3.3). 13

18 3 Graphene on Silicon Carbide Figure 3.: Ab initio calculation of the electrondispersion for one (a,b), two (c,d) and three (e) carbon layers on SiC(0001); Left:[5] ( 3 3)R30 reconstruction; right: [4] (5 5) reconstruction. 14

19 3. Electronic Band Structure Figure 3.3: [6] Ab-initio calculation of the electronic band structure for the (5 5) reconstruction. The distance between the graphene and the buffer layer was increased artificially by 0 Å, 0.5 Å, 0.5 Å, 0.75 Å and 1.0 Å (from left to right). (i) While for free standing graphene the π-bands meet exactly at the K and at the K point, here a small gap of 30 mev opens. A model describing this effect via symmetry considerations has been presented in [4]. (ii) The interface states interact with the Dirac cone states, causing a hybridization of the bands. However, the interaction does not affect the cones which originate from K and K points equally. In Fig. 3.3 is seen that only one of the two cones interacts with the lowest interface state, while the other cone interacts with the next interface state (not shown in the figure). Near the interface states the degeneracy of K- and K -derived states is therefore lifted. The selection rules, which are responsible for these effect are related to the interface symmetry, but as yet have not been derived from the microscopic model. For the commonly observed ( ) reconstruction no band structure data are available. However, the results for ( 3 3) and (5 5) models can be qualitatively transfered to this case. In particular, a set of flat interface states interacting with the Dirac state and the Fermi energy pinning at 0.4 ev above the tip of the Dirac cone should persist. The exact number of the interface states and their energies, however, will depend on the model. It is reasonable to assume, that the strength of the interaction between the interface states and the Dirac states is similar for the (5 5) model and for the real ( ) structure. 15

20 3 Graphene on Silicon Carbide 3.3 Model for Band Structure with Interface States Next, the influence of the interface states on the electronic properties shall be studied in more detail. We start by setting up a simple model for the electronic band structure. Although the calculated interface states have some small dispersion, they will be treated as completely flat in this model. As derived in Chapter, the energy levels for the free standing graphene layer are E ± k = ± v F k near the K point. In the basis Ψ ± k = 1 ( e iθ k / ψ A k ± e iθ k/ ψ B k ) (3.1) the Hamiltonian is diagonal: Ĥ K ( E + k 0 0 E k ) We extend the band structure model by introducing a constant energy level E 1 > 0. The eigenstate ϕ (1) k associated with the energy level E 1 is an additional basis vector, assuming that ϕ (1) k and the graphene states Ψ± k are orthogonal. The interaction of the energy level E 1 with the positive branch of the Dirac cone E + k via a potential V 1 is expressed by the matrix E + k V 1 0 Ĥ K V 1 E 1 0 (3.) 0 0 E k Diagonalization of this matrix generates three energy levels ɛ l,k with the eigenvectors 1 ψ l,k = C + l,k Ψ+ k + C l,k Ψ k + C(1) l,k ϕ(1) k 1 The eigenenergies and eigenstates of the Hamiltonian (3.) are ɛ 1,k = E+ k + E 1 ɛ,k = E+ k + E 1 + (E + (E + k E 1 k E 1 ) + V 1, ψ 1,k = ) + V 1, ψ,k = γ + k γ + k Ψ + k + 1 ϕ (1) γ k γ k ɛ 3,k = E k, ψ 3,k = Ψ k γ + k k Ψ + k + 1 ϕ (1) γ k k with the abbreviation γ ± k = 1 V 1 E+ k E 1 (E + k ± E ) 1 + V1 16

21 3.4 Electron Scattering due to Interface States The generalization to a band structure with several interface states is straightforward. Consider N + interface states E j > 0 interacting only with the positive branch E + k of the Dirac cone and N interface states E j < 0 interacting only with the negative branch E k of the Dirac cone. Instead of the matrix (3.) a N + N + Hamiltonian has to be diagonalized, where N = N + + N. One obtains N + energy branches ɛ l,k with the eigenvectors N ψ l,k = C + l,k Ψ+ k + C l,k Ψ k + C j l,k ϕj k The positive and negative energy levels E + k and E k in the model do not mix, however. Therefore there are N + eigenvectors with C l,k = 0 and N eigenvectors with C + l,k = 0 Switching to the basis ψk A, ψb k and ϕj k of the two graphene sublattices and the interface states in the substrate, by applying (3.1) the eigenvectors become ψ l,k = e iθ k/ C + l,k ψa k + eiθ k/ C + l,k ψb k + ψ l,k = e iθ k/ C l,k ψa k eiθ k/ C l,k ψb k + j=1 N C j l,k ϕj k, (l = 1,..., N +) j=1 N C j l,k ϕj k, (l = N + + 1,..., N) For a more compact notation the index σ {+, } is introduced: j=1 ψ l,k = e iθ k/ C σ l,k ψ A k + σ eiθ k/ C σ l,k ψ B k + N C j l,k ϕj k (3.3) The pinning of the Fermi level at an energy different from the tip of the Dirac cone is taken into account in this model by adding a shift to all energy levels. j=1 3.4 Electron Scattering due to Interface States The above calculations hold only for fixed time-independent interaction potentials V j and a fixed general shift. These parameters, however, depend on the distance d between the graphene layer and the buffer layer. One expects the interaction strengths V j as well as the shift to decrease, when the distance d increases (cf. Fig. 3.3). Thermal vibrations will cause fluctuations of the distance d around its equilibrium value. The fluctuation δu of the energy levels resulting from these oscillations will give rise to electron scattering. In the following the displacement vector u(x, y) will be used to describe the displacement of the substrate (SiC + buffer) surface from the equilibrium position. The distance d 17

22 3 Graphene on Silicon Carbide between the substrate and the graphene layer depends on the component u z of the displacement vector perpendicular to the surface. The parameters V j and therefore are functions of u z. The scattering potential δu can be Taylor-expanded in terms of u z. One obtains where the first coefficient is given by δu l,k = α l,k u z + α () l,k u z +... α l,k = ɛ l,k u z = u z + N j=1 ɛ l,k V j V j u z and the indices l and k denote energy band number and wave vector respectively. Using the first order term as the scattering potential can only be a good approximation, if α () k is small compared to α k. It should also be noted, that the model considers the displacement u z at one specific location (x, y) to calculate the scattering potential at that location. Variation of the displacement vector over the xy-plane is neglected. This local approximation holds for surface excitations slowly varying in space, i.e. excitations with long wavelength. To perform quantitative calculations, one has to guess by how much the energy of each band changes with the change of u z. For the simple model described here this information is contained in the few parameters / u z and V j / u z. The numerical values of these parameters can be extracted from the ab-initio band structure calculations for the (5 5) model shown in Fig By measuring the energy changes in the band structure for different displacements of the graphene layer from the buffer layer, one obtains an estimate of 0.1 ev/å for / u z and 0. ev/å for V j/ u z. The interaction strengths V j are about 50 mev. Remember however, that (i) the exact electronic band structure is different for the ( ) structure and (ii) the electronic band structure behaves much more complicated upon changing u z in reality than can be expressed by the few parameters used in this simple model. Still, these numbers provide a guideline for the following calculations. Finally note, how the two parameters / u z and V j / u z enter the scattering potential δu l,k = α l,k u z : The derivative / u z is independent of k. It therefore describes a mechanism of global scattering affecting electrons at all energy levels in the same way. In contrast the derivatives V j / u z are multiplied by a factor ɛ l,k / V j which is k-dependent and considerably different form zero only near the interface state. This resonant scattering mechanism therefore only affects electron states near the energy of the interface state. 18

23 4 Theoretical Background In this chapter the principles of the theory which are necessary for treating the model described in the previous section, will be reviewed. 4.1 Lattice Vibrations As the scattering potential in the model described above depends on the distance of the graphene layer to the substrate, it is important to understand, how the surface of the substrate is vibrating when it is thermally excited. The equilibrium positions of the atoms forming the crystal are defined by the configuration which has the minimal potential energy. The quantity that will be studied in this chapter is the displacement vector u l which tells us how far the l-th atom shifts from its equilibrium position Theory of elasticity 1 Before actually examining the vibrations of a crystal lattice by considering the interactions of single atoms located at the lattice sites, a much simpler model is studied in this chapter. The crystal is replaced by a homogeneous elastic medium. Obviously the properties, which directly result from the atomistic structure of the lattice cannot be reproduced by this approximation. However, for acoustic vibrations where the displacement vector u does not change considerably over the length of several interatomic spacings, the microscopic structure of the lattice is unimportant and the theory of elasticity produces the same results as the more accurate theory of lattice vibrations sketched in the next section. An advantage of the elasticity theory is that all microscopic information needed to describe the interaction between single lattice sites is contained in few macroscopic parameters, the elastic constants. The equations describing the motion of the elastic medium are quite simple compared to the ones obtained from a treatment of the discrete lattice. 1 The derivations in this chapter follow the presentation in the textbook Theory of Elasticity by L. D. Landau and E. M. Lifschitz [3]. 19

24 4 Theoretical Background σ zz σ xz σ yz σ zy σ zx x z y σ xx σ yx σ xy σ yy Figure 4.1: Components of the strain tensor σ ik Strain and stress. As the homogeneous elastic solid has no discrete lattice sites, all the occuring quantities are defined on a continuous space. The vector u(x, y, z) gives the displacement of the point (x, y, z) from its equilibrium position. For small displacements the symmetric strain tensor is defined as u ik = 1 ( ui + u ) k x k x i (4.1) If the elastic medium is deformed, internal forces arise which drive the medium back to equilibrium. The information about these forces is contained in the stress tensor σ ik. The component σ ik of the stress tensor is the i-th component of the force acting on a unit area perpendicular to the x k -axis. (See fig. 4.1) The i-th component of the force acting on a particular volume element inside the body is given by the sum f i = σ ik / x k (the Einstein summation convention is implied). Now an equation connecting the strain tensor and the stress tensor has to be derived. If the deformation is small, the internal forces are proportional to the deformation amplitude, according to Hooke s law, i.e. the relation between the two tensors is linear. This relation can be derived via the free energy F [3]. On one hand σ ik = F/ u ik at constant temperature, on the other hand the free energy is expressed in terms of the strain tensor and some elastic constants explicitly: F = F (u ik ). By combining these two relations one obtains σ ik = E ( u ik σ σ ) 1 σ u llδ ik with the two elastic constants Young s modulus E and Poisson s ratio σ. (4.) 0

25 4.1 Lattice Vibrations Equation of motion. With the tensors introduced above, the equation of motion of the elastic medium can be set up easily. In case no external forces are applied, the only forces accelerating a volume element are due to internal stress. The equation of motion is ρü i = f i = σ ik / x k where ρ is the mass density of the material. The internal stresses are generated by the local current deformation. This is expressed by replacing the stress tensor with expression (4.) and then using the definition of the strain tensor (4.1). Finally the differential equation for the displacement vector u is: ρü i = E u i E u k + (1 + σ) x k x k (1 + σ)(1 σ) x i x k The coefficients are constants which depend only on the material. In the following we introduce α = E/(ρ(1 + σ)) and β α = E/(ρ(1 + σ)(1 σ)). Then the above differential equation (in vector notation) reads ü = α u + (β α) ( u) (4.3) Depending on the shape of the body, the corresponding boundary conditions have to be fulfilled. Such condition can be that there are no forces acting at the surface. To get an idea, what kind of vibrations can be predicted by elasticity theory, a few examples will be examined next. Infinite isotropic body. By far the simplest case treated here first is an infinite isotropic body. The equation of motion has to be fulfilled with a condition that the displacement remains finite at infinity. The solutions correspond to elastic vibrations that can be excited in a medium, far away from the surfaces. In order to solve the equation (4.3) it is helpful to decompose the displacement vector u in a transverse part v and a longitudinal part w: Equation (4.3) becomes Taking the divergence of both sides leads to u = v + w, v = 0, w = 0 (4.4) v + ẅ = α v + α w + (β α) ( w) (4.5) (ẅ β w) = 0 ẅ = α ( w) + (β α) ( w) ẅ = α ( w) + (β α) ( w) 1

26 4 Theoretical Background Not only the divergence of the term in brackets is zero, but because of (4.4) also its curl is zero. Therefore the term in brackets is a constant, which can be chosen to be zero. Taking the curl of both sides of equation (4.5) gives v = α ( v) ( v α v) = 0 Also the divergence of the term in brackets is zero, because of (4.4) and therefore the term in brackets is a constant again. So, by decomposition (4.4) the equation of motion (4.3) is converted into six (by components) decoupled wave equations v = α v ẅ = β w Hence the solutions are superpositions of independent plane waves v q (r, t) = v q0 e iqr e iωq,tt (4.6) w q (r, t) = w q0 e iqr e iω q,lt (4.7) Here v q0 and w q0 are polarization vectors, the direction of which is dictated by the conditions v = 0 and w = 0 to be v q0 q and w q0 q. Thus for a given wave vector q there is one longitudinal wave w q and two independent transverse waves v (1,) q. Dispersion relations for these waves are obtained by inserting the plane waves (4.6) and (4.7) into the equation of motion: ω q,t = q α = qc t, ω q,l = q β = qc l (4.8) Note, that the dispersion relations for all modes are linear. The velocities c t and c l are determined by the elastic constants of the medium. Half-space medium. Now the surface waves of a half space z < 0 filled with an elastic medium will be considered. The boundary condition is that there are no stresses at the free surface, that is σ xz = σ yz = σ zz = 0 (4.9) Apparently it must hold only at the surface, i.e. at z = 0. The expression (4.9) can be translated into conditions for the displacement vector by applying (4.) and (4.1).

27 4.1 Lattice Vibrations The surface waves propagating along the xy-plane, moreover, are restricted by u = 0 at z =. This results in plane waves that are damped towards the interior of the medium. u = u 0 e i(qxx+qyy ωqt) e κqz The dependence of κ q > 0 and ω q on the wave vector q is given by the equation of motion and the boundary condition (4.9). The equation of motion (4.3) can again be decomposed into a transverse and a longitudinal part. But here the two resulting equations will be still coupled by the boundary conditions, i.e. the transverse wave v and the longitudinal wave w can not be combined arbitrarily. The condition (4.9) allows only such combinations which result in vanishing total stress at the surface. In [3] it is shown, that for these surface waves again a linear dispersion relation is obtained. ω q = c t ξq (4.10) with c t being the velocity of the transverse bulk waves. The parameter ξ depends on the ratio of the bulk wave velocities c t /c l in a complicated way and takes values from around to for different materials. What should be kept in mind from this result, is that the surface waves according to theory of elasticity have linear dispersion, just like the bulk waves, and that they even have a velocity similar to the one of transverse bulk waves Vibrations of a discrete lattice In this section the vibrations of the atoms in the crystal lattice will be studied, taking into account the structure of the lattice and not approximating it as a homogeneous medium as in the section before. First, a lattice with only one atom per unit cell is considered. The total potential energy is at its minimum, when every atom is located at its equilibrium position R l (The index l counts the atoms of the lattice). But if the crystal gains energy the atoms can oscillate around their equilibrium position due to thermal excitation. It is assumed that the displacement u l of an atom from its equilibrium position is small, so that it is sufficient to expand the potential energy up to terms quadratic in u l : U = U 0 + l U u l u l α + 1 α l,l U u l α u l β u l αu l β The derivations in this chapter follow the presentation in the textbook Condensed Matter Physics by M. P. Marder [7]. 3

28 4 Theoretical Background The summation over the indices α and β, which account for the three coordinates of the displacement vector, is implied. The ground state energy U 0 is not important for vibrations. The first derivatives of the potential energy are all zero at the equilibrium configuration, because the potential energy has a minimum there. Hence, only the second derivatives of the potential energy have to be examined. The motion of every single atom is described by its own equation of motion. Using the periodicity of the crystal lattice and applying periodic boundary conditions, one finds an equation of motion which describes the movement of any bulk atom: M u l α = U t u l l α u l β u l β (4.11) where M is the mass of one atom. The term on the right hand side is the α-th component of the force that accelerates the l-th atom. This differential equation is solved by plane waves u l = p e iqrl iω qt where the polarization vector p determines the direction of the displacement. The wave vector q points in the direction of the propagation of the wave. The general solution for the displacement u l is a linear combination of plane waves with all possible wave vectors and all possible independent polarization vectors. Dispersion relations. The dispersion relation between the wave vector q and the frequency ω q is calculated by inserting the plane wave into the equation of motion: Mωqp α e iqrl = U p u l β e iqrl l α u l β ωqp α = 1 U e iq(rl R l) p M u l β l α u l β ωqp α = ( ) Φ l q p αβ β (4.1) In the last step the 3 3 dynamical matrix Φ l q is introduced. Because of the periodicity of the lattice and the periodic boundary conditions it is the same for any atom, i.e. any value of l. The equation is solved by diagonalizing the dynamical matrix. For three independent eigenvectors p ν it yields three frequencies ω q,ν. The same technique can also be applied to calculate the phonon dispersion for crystals with more than one atom in a unit cell. Each atom is described by its own 3-component equation of motion (4.11). But as the right hand side sums over all atoms, these equations are coupled. The problem can again be converted into an eigenvalue equation 4

29 4.1 Lattice Vibrations Figure 4.: [8] Phonon dispersion of 3C-SiC along the high symmetry axes. A phonon with frequency ν has an energy of ω = πν. For example, the frequency ν = 10 THz corresponds to the energy 41 mev. similar to (4.1), but for a lattice with n atoms per unit cell the dynamical matrix has dimensions 3n 3n. Therefore a set of 3n independent polarization vectors p ν with eigenfrequencies ων results from it. To actually calculate the dispersion relations one needs the second derivatives of the potential energies. For simple examples they can be calculated by e.g. assuming that the potential energy is dominated by the interaction of nearest neighbours only. This interaction can be parametrized by a corresponding spring constant. In Fig. 4. the phonon dispersion curves for SiC are plotted. For large wave lengths, i.e. small values of q, the microscopic structure of the lattice should not play an important role. So for small values of q the results calculated here should agree with the linear dispersion obtained from the elasticity theory in (4.8). Indeed, the acoustic branches in Fig. 4. can be approximated by ω q = cq for small values of q. The optical branches, however, have an approximately constant energy for small values of q. These branches occur only for lattices with more than one atom in a unit cell and therefore cannot be reproduced by the theory of elasticity. Quantum mechanical description of the lattice vibrations. It was derived above, that the displacement vector of the l-th atom can be written in terms of plane waves as u l = q,ν p ν e iqrl iω qt (4.13) 5

30 4 Theoretical Background The index ν labels the different modes and can take values from 1 to n d, where d is the number of spatial dimensions and n is the number of atoms per unit cell. These lattice vibrations shall now be described quantum mechanically. Starting point is the lattice Hamiltonian 3 Ĥ = l (ˆp l ) M + 1 l,l U u l α u l β û l αû l β (4.14) The potential energy U is Taylor-expanded with respect to the displacement vectors û up to the second order. For the reasons discussed above the zeroth order term is unimportant and the first order term vanishes. If there were no summation over the lattice sites, the Hamiltonian would describe a single harmonic oscillator Ĥ = ˆp M + Mω ˆx which is solved by introducing the annihilation and creation operators Mω 1 â = ˆx + i Mω Mω 1 â = ˆx i Mω Similarly, for the Hamiltonian (4.14) one defines the annihilation and creation operators â q,ν = 1 N ( ) Mωq,ν 1 e iqrl p ν N ûl + i ˆp l Mω q,ν l=1 â q,ν = 1 N ( ) e iqrl p Mωq,ν 1 ν N ûl i ˆp l Mω q,ν l=1 These operators annihilate/create a phonon of the mode ν and with the wave vector q which corresponds to a plane wave of the classical solution (4.13). The Hamiltonian (4.14) can be written as Ĥ = q,ν ω q,ν (â q,νâ q,ν + 1 ) where the operator â q,νâ q,ν is simply counting the phonons in the mode ν with the wave vector q. The displacement vector expressed in terms of the phonon creation and annihilation operators reads û l = 1 ) (p ν e iqrl â q,ν + p νe iqrl â q,ν N Mωq,ν q,ν 3 For the moment a lattice with only one atom per unit cell is considered. 6

31 4. Theory of Conductivity For later calculations it is helpful to define the displacement vector not only on the lattice sites, but continuously over the whole space: û(r) = 1 ( ) pν e iqr â q,ν + p νe iqr â q,ν N Mωq,ν q,ν (4.15) Obviously at the lattice sites R l this function matches the previously derived one û(r l ) = û l. 4. Theory of Conductivity Boltzmann Equation To calculate transport properties e.g. electrical conductivity or thermal conductivity one has to know how the particles of the system accelerate in external fields and how they loose their momentum due to scattering processes. The most important function in this context is the distribution function f k (r, t). It describes the probability that in the neighbourhood of r and at time t a particle in momentum-state k is found. With this density, the transport properties can easily be calculated. The Boltzmann equation is an important tool to determine f k, including problems with external electromagnetic fields and temperature gradients. To derive the Boltzmann equation one has to examine how the quantity f k changes in time. Three mechanisms have to be considered: diffusion, scattering, and the change of the k-state due to external fields: [ ] [ ] [ ] f k t = fk fk fk + + (4.16) t diff. t field t scatt. The diffusion is described by the continuity equation. [ ] fk = t diff. r (v kf k ) = v k f k r (4.17) with v k being the velocity of a particle in state k. A similar equation holds in momentum space. If k is the rate of change of the k-state due to external fields, then [ ] fk = t field k ( kf k ) = k f k (4.18) k 4 The derivations in this chapter follow the presentation in the textbooks Condensed Matter Physics by M. P. Marder [7] and Theory of Solids by J. M. Ziman. [9]. 7