Chapter 4. Electronic Structure

Transcription

1 Chapter 4 Electronic Structure The electronic properties of matter determines its macroscopic behavior. The magnetic phenomenon or the superconducting behavior of a metal has its roots in the electronic structure. With the expression electronic structure we mean the energetic and spatial distribution of electrons and how their energy is related to their motion in a solid. The electronic structure of each element is different. Yet, there are close resemblance between elements of the same group, i.e., that lie in the same column of the periodic table. The reason is given by the fact that the electronic shells are filled by one additional electron as we go from left to right in the periodic table, and elements with the same shell occupancy show similar behavior. Instead of dealing with different elements 1 one considers elements in some characteristic groups. Here, we first analyze the behavior of free electrons. In this free-electron model the weekly bound electrons of the atoms move about freely in the metal without the influence of the attractive potential of ion cores. Thus the potential energy is neglected, the total energy is in the form of kinetic energy. The free-electron model gives us considerable information about several electronic properties of the so-called simple metals. The alkali metals or nobel metals can be regarded as simple metals. The free-electron model cannot explain why some elements are metals and others insulators. Besides, in the free-electron model electrons can travel long distances without scattering. As a next step, we place the electrons in a periodic potential of the ion cores and see the formation of energy bands, band gaps, effective masses, etc. At the end, we mention the other extreme, the localized states. We need this approach to account for the nature of the 4f electrons of the rare-earth elements. These are some fundamental subjects of a Solid State Physics course. Subsequently, we apply these ideas to surfaces. Since the symmetry is broken in one direction (it is customary to define the direction perpendicular to the surface as the z-direction) we expect a major response in the electronic behavior because the electronic motion is restricted and new states may be formed that do not exist in the bulk. 1 V.L. Moruzzi, J.F. Janak, and A.R. Williams, Calculated electronic properties of metals, Pergamon, New York,

2 CHAPTER 4. ELECTRONIC STRUCTURE 103 In the following chapter, we will also learn some experimental techniques that are currently used to investigate the electronic properties of the bulk and the surface. 4.1 Free Electrons in Metals Free Electrons in 3DIM An unconfined electron in free space is described by the Schrödinger equation ( h2 2m 2 ϕ = h2 2 ) ϕ 2m x + 2 ϕ 2 y + 2 ϕ = Eϕ, (4.1) 2 z 2 where m is the free-electron mass. The solutions of this equation, are plane waves labelled by the wave vector and correspond to the energy ϕ k (r) = 1 (2π) 3 eik r (4.2) k = (k x, k y, k z ) (4.3) E = h2 k 2 2m = h2 2m (k2 x + k 2 y + k 2 z). (4.4) The vector components of k are the quantum numbers for the free motion of the electron, one for each of the classical degrees of freedom. The number of states in a volume dk = dk x dk y dk z of k-space is N(k) dk = 2 dk (4.5) (2π) 3 with the factor of 2 accounting for the spin-degeneracy of the electrons. To express this density of states in terms of energy states, we use the fact that the energy (Eq. 4.4) depends only on the magnitude of k. Thus, by using spherical polar coordinates in k-space, dk = k 2 sin θ dk dθ dφ, (4.6) where the variables have their usual ranges (0 k <, 0 φ < 2π, and 0 θ π) and integrating over the polar and azimuthal angles, we are left with an expression that depends only on the magnitude k: N(k) dk = 2 (2π) 3 dk = 1 π 2 k2 dk (4.7) By invoking Eq. 4.4, we can perform a change of variables to bring the right-hand side of this equation into a form involving the differential of the energy: 1 π 2 k2 dk = 1 π 2 ( 2mE h 2 ) dk de de = 1 ( ) 2m 3/2 E de (4.8) 2π 2 h 2

3 CHAPTER 4. ELECTRONIC STRUCTURE 104 From this equation, we deduce the well-known density of states N(E) of a free electron gas in three dimensions: N(E) = 1 ( ) 2m 3/2 E (4.9) 2π 2 h 2 Notice the characteristic square-root dependence on the energy (Fig. 4.1). This results from the fact that in 3DIM, the surfaces of constant energy in k-space are spheres of radius 2mE/ h. Figure 4.1: The densities of states N(E) of an ideal 3DIM electron gas. The density of states weighted by the amplitude square of the wave function is denoted as the local density of states. N(E, r) = i ϕ i (r) 2 δ(e E i ), (4.10) where density of states is obtained by integration: N(E) = N(E, r)dr DIM Electron Gas If we restrict electron motion in one dimension, for instance in the z-direction, and let the electrons move freely in the other two directions, we encounter a quantum well in the z-direction. The Schrödinger equation is the same as that in Eq. 4.1 for free electrons in 3DIM h2 2m 2 ϕ = Eϕ, (4.11)

4 CHAPTER 4. ELECTRONIC STRUCTURE 105 but the boundary conditions on ϕ are different. In the direction of confinement along the z-direction we locate the confining planes with infinite potentials at z = 0 and z = L. The boundary conditions for ϕ are ϕ(x, y, 0) = ϕ(x, y, L) = 0 (4.12) Thus, a natural way to solve the Schrödinger equation is by the method of separation of variables. By writing the wave function ϕ as ϕ(x, y, z) = φ(x, y)ψ(z) (4.13) and substituting into Eq. 4.2 we obtain the Schrödinger equations for φ ( h2 2 ) φ 2m x + 2 φ = E 2 y 2 k φ (4.14) and for ψ, with the boundary condition h2 d 2 ψ 2m dz = E nψ (4.15) 2 ψ(0) = ψ(l) = 0 (4.16) Figure 4.2: First three energy levels and wave functions of electron confined to z-direction within a sheet of thickness L. The wavelengths are indicated on the wave functions. From CK.

5 CHAPTER 4. ELECTRONIC STRUCTURE 106 Notice that the equation and boundary condition for ψ are precisely those for a particle in a 1DIM box with infinite barriers at the edges. The solution for the not-normalized eigenfunction is ψ n (z) = sin(k n z), (4.17) where k n = nπ/l and n a positive integer. The energy associated with these functions are E n = h2 kn 2 (4.18) 2m By solving the Schrödinger equations in Eq and Eq. 4.15, we find that the energy eigenvalues for the quantum well in all three directions are given by E n,k = h2 2m (k2 n + k 2 x + k 2 y), n = 1, 2,... (4.19) This shows that this energy dispersion has features of both the 2DIM electron gas (associated with the motion in the x-y plane) and the 1DIM electron gas (associated with the confinement along the z direction). In the ground state of a system of N free electrons with an electron concentration of N/V, where V is the volume, the occupied states may be represented as points inside a sphere in k space. The energy at the surface of the sphere is the Fermi energy E F with the wave vector k F such that E F = h2 2m (k2 F). (4.20) Considering that there is one allowed wave vector for the volume element in k space and two allowed values of m s, the spin quantum number, for each allowed value of k, we find that E F depends on the mass and electron concentration: Fermi-Dirac Distribution Law E F = h2 2m (3π2 N/V ) 2/3. (4.21) The ground state is the state of the system at absolute zero. The situation that happens as the temperature is increased is given by the Fermi-Dirac distribution function. The kinetic energy of the electron gas increases as the temperature is increased, and some energy levels are occupied which were vacant at absolute zero, and some levels are vacant which were occupied at absolute zero. The situation is illustrated in Fig. 4.3, where the plotted curve is of the function f(e) = (e (E µ)/k BT + 1) 1. (4.22) The Fermi-Dirac distribution function gives the probability that a state at energy E will be occupied in thermal equilibrium. The quantity µ is called the chemical potential, which is equal to E F at absolute zero; it is a function of temperature. At all temperatures f(e) is equal to 1 2 when E = µ. As usual, E F is defined as the topmost filled energy state at absolute zero.

6 CHAPTER 4. ELECTRONIC STRUCTURE 107 Figure 4.3: Plot of Fermi-Dirac distribution function f(e) versus E/µ for zero temperature and for a temperature k B T = 1µ. The value of 5 f(e) gives the fraction of levels at a given energy which are occupied when the system is in thermal equilibrium. From CK. 4.2 Band Theory of Metals Every solid contains electrons which are arranged in energy bands separated by regions for which no electron energy states are allowed. Such forbidden regions are called energy gaps. If the number of electrons in a crystal is such that the allowed energy bands are either filled or empty, then no electrons can move in an electric field and the crystal will behave as an insulator. If the bands are partially filled, the crystal will act as a metal. To understand the difference between insulators and conductors the free-electron model must be extended to take account of the periodic lattice of the solid Periodic Lattice A crystal consists of a periodic repetition of a set of atoms in space. As seen in Section 2.2, periodic structures have long-range translational order and can be described by a lattice and a basis (unit cell). A unit cell is a collection of atoms at each point; this collection is identical at each lattice point. In simple solids, the basis consists of one single atom, complicated organic structures may have thousands of atoms. We need three basis vectors a. b, and c, which define the unit cell. With the basis and the lattice, all structures are uniquely defined. 3DIM structures can be described by symmetry operations which map the structure in itself. Translation is a parallel shift of the structure by T = pa + qb + rc,

7 CHAPTER 4. ELECTRONIC STRUCTURE 108 which is called translational invariance. p, q, and r are integers, analogous to the Miller indices. Any point in the crystal can be reached from the origin using T. The basis vectors a, b, c span the unit cell. The sum of all translations for any p. q, and r is called the translational group of the structure. The translational group defines the 3DIM periodicity of the structure. Rotation and mirror reflection are called point operations. They leave 1 point or 1 line unchanged. Point group consists of operations which leave 1 point unchanged and the structure invariant. In periodic structures there are two possible operations: mirror reflection around a line (mirror plane) and rotation through 2π/n (n = 1, 2, 3, 4, 6) around a point. The numbers give the n-fold rotation around a point. Only these are compatible with translational properties. In general, there are only some selected translation symmetries that are compatible with a given point group. There are fourteen different lattice types, Bravais lattices, which fulfill this requirement (see Fig. 2.7). More precisely, we can state that the choice of the unit cell in a crystal is not unique, the basis has to be defined accordingly. If the smallest possible unit cell contains just one atom, such a lattice is called a Bravais lattice. It is functional and appropriate, that any periodic function f(r) is represented by its Fourier transform. Then, we deal with Fourier components of the function rather than with the function itself in real space. Any function defined for a crystal, such as the electron density, is periodic with the same translation vector T as the basis vectors. We can write f(r + T) = f(r). (4.23) The crystal itself transforms into the reciprocal lattice with the vectors r, such that G = ha + kb + lc. (4.24) The function f (r ) in the reciprocal space has the property f (r + G) = f (r ) (4.25) We define a unit cell in the reciprocal space beyond which f(r) repeats itself. The smallest such unit cell is called a Brillouin zone. It is defined by the area surrounded by the planes that are perpendicular bisectors of the vectors from the origin to the reciprocal lattice points. It is remarkable that the value of k enters into the conservation laws for collision processes of electrons in crystals. For this reason k is called the crystal momentum of the electron. In a process involving a momentum transfer K, we can write with G being the reciprocal lattice vector. k + K = k + G (4.26)

8 CHAPTER 4. ELECTRONIC STRUCTURE 109 Figure 4.4: Plot of energy E versus wave vector k (a) for a free electron and (b) for an electron in a monatomic linear chain with a lattice constant a. The energy gap E g is associated with the first Bragg reflection at k = ±π/a. From CK Motion of Electrons in a Periodic Potential In many situations the band structure of a crystal can be accounted for by the nearly-free electron model for which the band electrons are treated as perturbed only weakly by the periodic potential of the ion cores. Often the gross overall aspects of the band structure can be explained on this model. We consider, for the sake of simplicity, a linear chain of atoms with a lattice constant a. In Fig. 4.4 the band structure is shown (a) for free electrons and (b) nearly-free electrons with an energy gap at k = ±π/a. The Bragg condition (k + G) 2 = k 2 for diffraction of a wave with the wave vector k becomes in 1DIM k = ± 1 G = ±nπ/a, (4.27) 2 where G = ±2nπ/a and n is an integer. The first reflection and the first energy gap occur at k = ±π/a. The reflections at these particular values of k arise because the wave reflected from one atom in the linear lattices interferes constructively with the wave reflected from a nearest-neighbor atom. The difference in phase between the two reflected waves is just ±2π for these two values of k. The region in k space between π/a and π/a is called the first Brillouin zone of this lattice Band Gaps At k = ±π/a, when the Bragg condition is satisfied, a wave traveling in one direction is Bragg-reflected and then travels in the opposite direction. Thus, at k = ±π/a, the wave functions are made up equally of waves traveling to the right and to the left and form two different standing waves. If the traveling waves are in the form e iπx/a and e iπx/a, the standing waves are: ψ(+) (e iπx/a + e iπx/a ) = 2 cos(πx/a); ψ( ) (e iπx/a e iπx/a ) = 2 cos(πx/a). (4.28)

9 CHAPTER 4. ELECTRONIC STRUCTURE 110 Figure 4.5: (a) Variation of potential energy of a conduction electron in the field of the ion cores in a linear lattice. (b) Distribution of probability density in the linear lattice for ψ(+) 2 cos 2 (πx/a), ψ( ) 2 sin 2 (πx/a), and for a traveling wave. The wave function ψ(+) piles up electronic charge on the cores of positive ions, thereby lowering the potential energy, while ψ(+) piles up charge in the region between the ions removing it from the ion cores and thereby raising the potential energy. From CK. The standing waves ψ(+) and ψ( ) are even and odd, respectively, when x is substituted for x. Figure 4.5(a) indicates schematically the variation of potential energy of a conduction electron in the field of the positive ion cores of a monatomic linear chain. The potential energy of an electron in the field of a positive ion is negative. In Fig. 4.5(b) the distribution of electron density corresponding to the standing waves ψ(+), ψ( ), and to a traveling wave is sketched. The traveling wave ψ e ikx distributes electrons uniformly with ψ 2 = 1, while standing waves distribute electrons preferentially either midway between ion cores [ψ( )] or on the ion cores [ψ(+)]. The potential energy of the three charge distributions is different in such a way that it is higher for ψ( ) than for a traveling wave and lower for ψ(+) compared to a traveling wave. If the potential energies of ψ( ) and ψ(+) differ by E g, there is an energy gap of width E g between the two solutions at k = π/a in Fig. 4.4 or between the two solutions at k = π/a. The wave functions at points A in Fig. 4.4 will be ψ(+), and the wave functions above the energy gap at points B will be ψ( ).

10 CHAPTER 4. ELECTRONIC STRUCTURE Bloch Functions Bloch 2 has proved that the solutions of the Schrödinger equation with a periodic potential are of the form ψ = u k (r)e ik r, (4.29) where u is a function, depending in general on k, which is periodic in x, y, z with the periodicity of the potential (that is, with the period a of the lattice). This amounts to that the plane wave e ik r is modulated with the period of the lattice. In order to justify the Bloch theorem, we can follow some simple arguments. Consider N lattice points on a ring of length Na, and suppose that the potential is periodic in a, so that V (x) = V (x + ga), (4.30) where g is an integer. Because of the symmetry of the ring we look for eigenfunctions ψ such that ψ(x + a) = Cψ(x), (4.31) where C is a constant. Then and, if eigenfunction is to be single-valued. so that C is one of the N roots of unity, or We have then ψ(x + ga) = C g ψ(x); (4.32) ψ(x + Na) = ψ(x) = C N ψ(x), (4.33) C = e i2πg/n ; g = 0, 1, 2,..., N 1. (4.34) ψ(x) = e i2πg/na u g (x) (4.35) as a satisfactory solution, where u g (x) has periodicity a. Letting we have k = 2πg/Na, (4.36) ψ = e ikx u k (x), (4.37) which is the Bloch result. A function in the form of the form Eq is known as a Bloch function. All one-electron wave functions in an ideal crystal are of the Bloch form. 2 F. Bloch, Z. Physik 52, 555 (1928).

11 CHAPTER 4. ELECTRONIC STRUCTURE Reduced-Zone Scheme It is often convenient to select the wave vector k of the Bloch function so that it always lies within the first Brillouin zone. This procedure is known as the reducedzone scheme. If we encounter a Bloch function written as ψ k (r) = e ik r u k (r), (4.38) with k outside the first zone, we may always find a suitable reciprocal lattice vector G such that k = k G (4.39) lies within the first Brillouin zone. Then Eq may be written as ψ k (r) = e ik r u k (r) = e ik r( e ig r u k (r) ) e ik r u k (r) = ψ k (r), (4.40) where we have defined u k (r) e ig r u k (r). (4.41) Figure 4.6: (a) The result of the perturbation associated with the nuclear potentials on the free-electron levels. Gaps are opened up at ±nπ/a. (b) The bands using the extended-zone scheme may be folded into the first zone of (a). From J.K. Burdett, Chemical Bonding in Solids, Oxford University Press, Oxford, Both e ig r and u k (r) are periodic in the crystal lattice so u k (r) is also, hence ψ k (r) is of the Bloch form. Even with free electrons it may be useful to work in the reduced-zone scheme, as is seen in Fig It follows also that any energy E k for k outside the first zone is equal to an E k in the first zone, where k is related to k by Eq Thus we need solve for the energy only in the first Brillouin zone, for each band. The opening of the band gap is illustrated in Fig. 4.7 for the reduced-zone scheme.

12 CHAPTER 4. ELECTRONIC STRUCTURE 113 Near the top or bottom of a band the energy is generally a quadratic function of the wave number, so that by analogy with the expression E = (h 2 /2m)k 2 for free electrons we may define an effective mass m such that 2 E/ k 2 = h 2 /m. The motion of the electron is characterized by this effective mass, m. Figure 4.7: (a) E k relationship drawn in the reduced-zone scheme for free electrons. The branch AC is reflected in the vertical line at k = π/a gives the usual free-electron parabola for E k vs k for positive k. (b) A crystal potential U introduces band gaps at the edges of the zone, but the overall features of the band structure remain. The dashed portions are due to free-electron parabola. Two energy bands α and β are shown separated by E g at k = ±πa. From CK. 4.3 Tight-Binding Approximation A wide question relates to the relative importance of band structure and atomic correlation effects in solids. The conduction electrons of free-electron like metals, such as alkali metals or Al, are shared between atoms for conduction, and may be treated by the above methods. The potential in which they move is rather smooth, and they can be well represented by plane waves. In band calculations, available electrons are successively filled into the calculated bands with different l- components and the Fermi level is obtained by the state filled by the last electron.

13 CHAPTER 4. ELECTRONIC STRUCTURE 114 In contrast, if electrons, or pairs of electrons, are localized in covalent bonds, they are in a state associated with a specific atom. In the extreme case of the innershell electrons that are localized at atomic sites, the dominant process of conduction is the motion of electron from one atomic site to other. This motion is governed by correlation effects came about by the interplay between repulsive electron-electron interaction and wave-function hybridization. Usually, this interplay is in favor of charge interaction, and in the extreme case of heavy Fermions the system shows Fermi-liquid behavior with a large effective mass m. The 4f electrons of most rare-earth metals and narrow-band d electrons of transition metals behave as if they were unfilled core levels and atomic interactions dominate their behavior. In this case, the binding energy of electrons is a strong function of the band occupancy. Such bands show predominantly occupations by an integer number of electrons. For these localized-electron systems, interatomic type charge fluctuations may be responsible for electronic conduction in the form dad n B n da n 1 db n+1, where A and B are atomic sites and n the electron occupancies. Roughly, for the conduction to occur, the energy required for this process should be less than the d-band width w for transition metals. If the d-d Coulomb and exchange interaction U is larger than the dispersional band width w than it is impossible for the above charge fluctuation to occur and the material becomes an insulator in spite of its unfilled d shell. These are the so-called Mott-Hubbard insulators. If U < w, the system will be a metal as predicted in the elementary band theory. The localized-electron behavior is well accounted for by the approximation which starts out from the wave functions of the free atoms which is known as the tightbinding approximation. It is quite good for the inner electrons of atoms and used to describe the d bands of some of the transition metals. Consider an atomic orbital φ(r l) with a well-defined character, e.g., 1s, 2s, 2p orbitals centered on an atom at position l in the crystal. In this approximation, the wave functions are constructed using such orbitals and obey Bloch theorem φ k (r) = l e ik l φ(r l). (4.42) In the orthogonalized plane waves method the localized and extended characters of the wave function are combined. An atomic region around each atom is defined where the wave function is described in terms of atomic orbitals. Outside this region, the relatively smooth parts of wave functions are expanded in terms of plane waves. Also in this method, one constructs a set of Bloch functions using occupied core states of the ions in each atom. 4.4 Surface Electronic Structure Fundamental aspects of Surface Science have their roots in electronic properties. These include the charge density in the neighborhood of the vacuum interface, the

14 CHAPTER 4. ELECTRONIC STRUCTURE 115 difference of the electron states near the surface compared to those in the bulk, chemical bonding states in the first few atomic planes, the electrostatic potential felt by surface atoms. Surface states that are present at the surface and not in the bulk give us mostly the clue about the behavior of surfaces. The macroscopic behavior of surfaces and interfaces, like oxidation, heterogeneous catalysis, or crystal growth, strongly depend on its electronic properties. Figure 4.8: Electron density profile with a positively charged uniform background n for z 0. From N.D. Lang and W. Kohn, Phys. Rev. B1, 4555 (1970). In order to study the electronic structure of surfaces, one starts with a semiinfinite crystal which has total number of electrons N at positions R. We have to consider the kinetic energy of all the electrons, the electron-ion interaction for all N and R, and the electron-electron repulsion for all electrons. The final term is not straightforward to handle, it is namely the exchange-correlation term, and since 1940 s several approaches have been used to satisfactorily solve the problem. In the so-called jellium model the discrete ion potential of a semiinfinite lattice is approximated by an averaged uniform positive charge density, n. Inverse charge density of the background is often related to a spherical volume: (4π/3)r 3 s = 1/ n. Figure 4.8 displays the electron density profile n(z) for two choices of the background density r s. As a result of uncertainty principle electron density may not abruptly change from zero to its finite value n as we enter the solid from the vacuum side, this means that there is no sharp edge to the electron distribution. As a consequence, there is an exponentially decaying probability to find electrons outside the solid. In other words, electrons spill out into the vacuum region for z > 0 and thereby create an electrostatic dipole layer at the surface. We also notice that n(z) oscillates

15 CHAPTER 4. ELECTRONIC STRUCTURE 116 as it approaches an asymptotic value that exactly compensates the uniform bulk background charge. The wavelength of these Friedel oscillations is π/k Fr, where k Fr = (3π 2 n) 1/3. The oscillations arise because the electrons try to screen out the positive background charge distribution which includes a step at z = 0. The formation of a surface dipole layer is a result that the electrostatic potential in vacuum is larger than the mean value in the crystal. This potential step keeps the electrons within the crystal. The exchange and correlation is a bulk effect which makes neighboring electrons stay away from each other and thus lowers the potential energy of each electron. The work function, Φ, which is the minimum energy required to remove an electron from the bulk to a point away from crystal, is given by the dipole layer. Whenever the atomic density at the surface is large, the spilling out is similarly large, and the work function has a smaller value. Because of this surface contribution, Φ depends sensitively on the exposed crystalline plane as well as on the impurity effects at the surface. Surface geometric effects, like reconstruction or relaxation, also modify surface dipole and consequently the work function, as expected. In calculations a reasonably accurate polycrystalline work function of some metal surface can be obtained using a uniform positive background jellium model. The jellium model of a metal surface neglects the electron-ion interaction and emphasizes the smooth surface potential barrier. Tamm 3 has investigated a linear chain of 1DIM atoms possessing delta-function like positive potentials. For the bulk, he obtained solutions for the Schrödinger equation in the form of ψ(z) = u k (z)e ikz, where u k (z + na) = u k (z) reflects the periodicity of the linear chain. For the real values of k(z) the solutions for ψ(z) are the common Bloch waves, extended over the entire chain. There are also complex values of k(z) which are associated with surface states, present at the surface and decay exponentially inside the surface for the z < 0. Shockley 4 has similarly considered atomic potentials arranged in a linear chain with the interatomic distance a. For large a the energy values resemble that of free atoms, and there are no surface states. As a is deceased, energy values come together and broaden to form bands. For adequately small nearest-neighbor distances, two discrete states move away from the bands to form surface states. In contrast to the jellium model, these models emphasize the lattice aspects of a linear chain of atoms and simplify the surface barrier. In fact, in any proper model, it is the surface barrier that makes the electrons reflect at z = 0 and leads to formation of surface states. Lets now consider a linear 1DIM chain of atoms periodically spaced in the z-direction starting from the surface. We make the assumption that the formation of the surface has no effect on the interatomic distance. In the light of formation of dimmers at the surface as a result of reduced coordination and surface reconstruction, we know that this assumption is unrealistic. The second 3 J.E. Tamm, Z. Physik 76, 849 (1932). 4 W. Shockley, Phys. Rev. 56, 317 (1939).

16 CHAPTER 4. ELECTRONIC STRUCTURE 117 simplification is the modelling of the potential. We consider the step function at z = 0, but for the chain of ion cores, we consider a weak and smoothened periodic potential: V (z) = V o + 2V g cos gz, where g = 2π/a. Hence g is the reciprocal lattice vector of the chain. The solution of the Schrödinger equation ( h2 2m d 2 ) dz + V (z) ϕ = Eϕ, (4.43) 2 using the screened ion-core potentials V (z) and neglecting the electron-electron interactions, leads to ϕ vac,1 = e +kz and ϕ vac,2 = e kz on the vacuum side with k = 2m(V o E)/ h. The solutions must be finite, and we remain with ϕ vac,2 = e kz, because otherwise ϕ for large values of z. Figure 4.9: The periodic potential V (z) for 1DIM semi infinite lattice with a step at the surface, z = 0. The dashed curve is more realistic. In the bulk the solutions have the Bloch form ϕ k (z) = u k e ikz because the potential is periodic: V (z + a) = V (z). We may assume for metals with nearly-free electrons that the potential is weak, and we expand V (z) around its average value V o : V (z) V o = V g e igz (4.44) g Away from the zone boundaries we have in units of h/2m: and ϕ k = e ikz + E k = k 2 + V ge i(k g)z k 2 (k g) 2 (4.45) V g 2 k 2 (k g) 2, (4.46) where E k is measured relative to V o. Near the zone boundary, k k g, and above equations are not valid any more. One has to use degenerate perturbation theory with ϕ k = αe ikz + βe i(k g)z. (4.47)

17 CHAPTER 4. ELECTRONIC STRUCTURE 118 The coefficients can be found as (k 2 E)α + V g β = 0 and V g α + ( (k g) 2 E ) β = 0. (4.48) Eq leads to E k = 1 ( ) (k2 ) k 2 + (k g) 2 2 ± (k g) Vg 2. (4.49) with the wave functions near the band gap: At the zone boundary, k = 1 g, thus, 2 ϕ k = e ikx + E k2 V g e i(k g)z. (4.50) E ± = ( 1 2 g)2 ± V and ϕ ± = e i g 2 z ± V g V e i 2 z. (4.51) Figure 4.10: Relevant energies in the energy gap. The wave functions are standing waves determined by the sign of V : Energy V > 0 V < 0 E + cos 1gx 2 sin 1gx 2 E sin 1gx 2 cos 1gx 2

18 CHAPTER 4. ELECTRONIC STRUCTURE 119 In Eq we introduce k which is measured relative to the zone boundary: k = 1 2 g + k and we obtain E = E(k 2 ) = ( 1 2 g)2 + k 2 ± V 2 g + g 2 k 2 (4.52) For positive values of k 2 this equation is compatible with the standard solution. The novelty about this equation is that it has real solutions E even for k 2 < 0, that is there are real solutions for Vg 2 /g 2 < k 2 < 0, namely in the energy gap. Figure 4.10 illustrates this situation, where the Brillouin zone boundary k = 0 corresponds to the energy gap. Schrödinger equation has acceptable solutions also for imaginary values of k : 0 < k < Vg. This region of imaginary g k is shown in the figure as dashed line. These solutions are not valid for the bulk, because for z the values diverge 5. Figure 4.11: Energy bands in complex k-space. Ref. [5]. E(k 2 ) is a continuous function of k 2. For negative values there is one real function E, and we introduce a complex wave vector k = g + iµ. Thus we have 2 results in the complex k-plane as shown in Fig In order to determine the wave functions for the solutions in the band gap, we introduce Eq in Eq and use k = iµ. It follows: 5 V. Heine, Proc. Phys. Soc. 81, 300 (1962). ϕ k = e µz cos ( gx 2 + φ) (4.53)

19 CHAPTER 4. ELECTRONIC STRUCTURE 120 with µ max = Vg g. Now, we have to perform wave matching for coupling the wave functions in order to find acceptable solutions. This means that ϕ, ϕ and ϕ /ϕ (logarithmic derivative) must both be continuous. ϕ ϕ = µ g 2 tan( gz 2 + φ) (4.54) At the vacuum side we had ϕ = ϕ vac,2 = e kz, and we find ϕ ϕ = k = 2m(V o E)/ h. (4.55) We put the origin, z o = a/2 on a surface atom and obtain: ϕ ϕ = µ g π 2 tan( 2 a + φ) = V o E (4.56) The Eq must be solved graphically and yields solutions for V g > 0. Figure 4.12 shows the surface state decaying both towards vacuum and bulk. The decay towards bulk is determined by e µz (z < 0). At the edge of the gap µ 0, and the surface state reaches relatively deep into the bulk. At the mid gap µ = µ max and the surface state decays within a few atomic distances. Figure 4.12: The atomic potential of the 1DIM linear chain and the surface state at the surface. The evanescent wave for z < 0 is called the Shockley state. Even the simple model, we have used, has shown the existence of electronic states localized at the surface. In the real case the problem has to be considered in 3DIM. We introduce the notion of surface-projected band structure and surface Brillouin zone. The wave vector can be split into the components k perpendicular to the surface, k = k z, and k in the surface, k = k xy. In Fig we see on the right-hand side a surface that represents E(k ). The dashed area is the projected

20 CHAPTER 4. ELECTRONIC STRUCTURE 121 bulk band structure at the surface. Every energy can be identified with a bulk state in 3DIM k-space. The lines at the not-dashed areas stand for the surface states, showing some dispersion in the k -space, which is termed a band of surface states. In the figure, the lower band of surface states mixes with a bulk state, depicted in the shaded circle, and produces an unusually large intensity observable in experiments. These states are called surface resonances. On the left-hand side of the figure, we have the E(k ) plane. At the lower energy gap far right, there is one point in k where the surface state exists. The encircled shaded area highlights the formation of another gap, namely a hybridization gap. Hybridization of wave functions occurs in order to avoid band crossing. Also in the hybridization gap surface states may exist as indicated in the upper gap of E(k ) plane. Figure 4.13: Projected bulk band structure at the surface of a metal. From AZ. In the following chapter, we will make use of photoelectric emission, a standard tool to investigate the bulk and surface electronic structure of matter. There has been two Nobel prizes for photoelectric emission work, A. Einstein (1921) and K. Siegbahn (1981), which shows the impact of the method on our todays understanding of the electronic structure of matter.