The idea of this lecture is to remind you about the basic ideas of solid state physics and to establish a link to our actual subject, the physics and chemistry of solid surfaces. I assume that you have already followed a basic course on solid state physics. Hence, any detailed mathematical treatment is omitted and I just focus on some highlights.
When trying to learn something about solids the biggest problem one encounters is that a macroscopic solid contains very many (1023) atoms. It is therefore impossible to solve any equations of motion, classical or quantum, in a direct way. On surfaces the situation is better (1015 atoms or so) but still absolutely hopeless. From the beginning on, physicists have therefore treated the electrons in a solid in a statistical way, first by applying kinetic gas theory (Drude) and later by using the correct Fermi-Dirac statistics (Sommerfeld).
The key-concept to a quantitative description of the electronic and vibrational properties of solids is the fact that most solids are crystals. One can use periodic boundary conditions and solve the problem in reciprocal space. The first section reminds you about the description of crystals and about reciprocal space.
In the following two sections we divide the properties of a solid into electronic contributions and lattice vibrations. This division is not without problems: In principle one would have to solve the Schrödinger equation for the whole system, with the co-ordinates of all the electrons and all the atoms. The reason why it works anyway is that the atoms are so much heavier and slower than the electrons. When the atoms move out of their equilibrium-position the electrons follow quickly but they stay in their ground state. They just move to another ground-state with higher energy. When the atoms are moved back the electrons follow to their initial ground state. This is called adiabatic or Born-Oppenheimer approximation.
The mass difference is also reflected in the different energy scales in electron and ion motion: typical kinetic energies of electrons are in the region of several eV while the phonon energies are several meV. The strategy to follow is therefore to solve the electronic structure assuming a rigid crystal. Then the vibrational properties can be calculated from the known electronic properties. Finally, the influence of the vibrational states on the electronic system can be considered: it is usually just a very small (but potentially important!) change.
Many solids exist in a crystalline form. Not only the ones which appear as crystals in nature (like diamond, many minerals and salts) but also metals grow as crystals. In solid state physics one can idealize the crystal structure and use the perfect periodicity to facilitate many of the problems.
The most fundamental definition is that of a Bravais lattice. It is defined as a lattice of points with position vectors
| (3.1) |
Examples for a Bravais lattice are the body centred cubic lattice and the
face centred cubic lattice, like shown in Fig. 3.1.
Figure 3.1: The body-centred cubic (bcc) and the face-centred cubic (fcc) Bravais lattice.
Given the Bravais lattice the primitive unit cell can be defined: it is any volume of space which when translated through all the vectors of the Bravais lattice fills all of space without either overlapping itself or leaving voids. There are of course many possible choices for this primitive unit cell.
One very common choice is the Wigner-Seitz cell. This cell has the full symmetry of the lattice and is defined as the region of points closer to any given lattice point than to any of the other lattice points. The Wigner-Seitz cells of the bcc and fcc lattice are given in Fig. 3.2.
Sometimes it is also
convenient to think in terms of non-primitive unit cells, e.g. in
case of the two above cubic structures. When the unit cell is though
of as a cube then the bcc cell contains two atoms and the fcc cell
contains four.
Figure 3.2: The Wigner-Seitz cells for the body-centred cubic (bcc) and the face-centred cubic (fcc) Bravais lattice.
Finally, a real crystal can be described by a Bravais lattice and a so-called basis. The basis is a fixed arrangement of atoms or molecules which is sitting on each point of the bravais lattice. It can just be one atom or it can be a whole protein in biological crystals.
A central concept in solid state physics is the reciprocal lattice. It is defined as the set of all wave vectors G that yield plane waves with the periodicity of a given Bravais lattice. This means that if G belongs to the reciprocal lattice of a Bravais lattice with points R then the relation
| (3.2) |
or
| (3.3) |
must hold. In other words: let a1,a2,a3 be the vectors defining the primitive direct lattice. Then the reciprocal lattice vectors g1,g2,g3 are defined by
| (3.4) |
where
| (3.5) |
Note that the reciprocal lattice is also a Bravais lattice. The fcc lattice, for example, has a bcc lattice as its reciprocal lattice and vice versa.
The concept of the reciprocal lattice allows to re-write many solid state problems in Fourier space and this is the way to solve them because of their periodic nature. Take for example a periodic one dimensional function
| (3.6) |
Then ρ can be written in a Fourier series
| (3.7) |
In the three dimensional case the sum is taken over the reciprocal lattice vectors G
| (3.8) |
ρ could be anything with the periodicity of the lattice. An important example is the charge density in the solid. In reciprocal space ρ can be described by a few Fourier coefficients whereas it has to be defined at by its value for every point in real space.
In the reciprocal lattice it is of course also possible to define a primitive unit cell. Of special importance in the theory of electronic and vibrational states is the Wigner-Seitz cell in the reciprocal lattice. It is called the first Brillouin zone. The first Brilloin zones for the bcc and fcc lattice look like the Wigner-Seitz cells for the fcc and bcc lattice in Fig. 3.2, respectively.
Another point worth mentioning in connection with our actual subject,
the surface physics, is the definition of the Miller indices
. These
indices are used to define a lattice plane or a surface orientation.
A plane can be
conveniently defined by a vector perpendicular to the plane and the
Miller indices use the reciprocal lattice vectors:
the lattice plane with the Miller indices h,k,l is the plane
perpendicular to the reciprocal lattice vector
hg1+kg2+lg3. In a simple cubic
lattice the reciprocal lattice is also simple cubic and the
orientation of the reciprocal lattice vector is obvious (see Fig.
3.3). For the
definition of lattice planes the bcc and fcc lattice are treated as
simple cubic.
Figure 3.3: Three lattice planes and their Miller indices in the simple cubic lattice.
A surface can be formed by cleaving a bulk crystal (two surfaces, actually). The first thing one could assume is that all the atoms stay at the same positions as before. This turns out not to be the case. Imagine the forces on an atom at the new surface. In the most simple picture it looses some of its nearest neighbours and an entirely new energetic situation arises. The first layer atoms could move further away from the remaining neighbours or closer towards them. Such a change of the first interlayer spacing is a called a relaxation. On many surfaces, especially on semiconductors, the atoms try to find new “partners” for the broken bonds sticking into the vacuum. This can lead to a reconstruction of the surface where the periodicity parallel to the surface is not the same as in the bulk. For details see section 7.2.
If we now just consider the two-dimensional periodicity parallel to the surface then we can define a two dimensional Bravais lattice, a Wigner-Seitz cell and a basis for the lattice in exactly the same way as in three dimensions. The definition of the basis is somewhat open in that we can include more than one layer of atoms. We can also define a two-dimensional reciprocal lattice and a two-dimensional Brillouin zone, the surface Brillouin zone (SBZ). The high symmetry points in the SBZ have similar names as the points in the bulk Brillouin zone but they carry a bar over the letter. The centre of the zone, for example, is called Γ in the bulk and Γ on the surface.
An example illustrating the usefulness of reciprocal space again is
given in Fig. 3.4. The Fig. shows a
Scanning Tunnelling Microscopy (STM) image of a Pt(111) surface together
with a Fourier transform of this image. Pt is an fcc metal and the
(111) surface is a closed-packed surface with a hexagonal symmetry.
In a very simple picture, the STM image corresponds to the
charge density on this surface (at the Fermi level). A Fourier
transform of the charge density should basically be an image of equ.
3.8 with the intensity at the reciprocal lattice
spots being equal to the Fourier coefficients of the charge density.
This is indeed the case. We can see that the six spots around the origin are
by far the most intense features. They alone give already a decent
description of the image. Because of symmetry, it is basically one
Fourier coefficient describing the whole STM image. This makes the
usefulness of reciprocal space obvious. When looking closer, more
weaker features at other reciprocal lattice points can be seen.
Figure 3.4: Scanning tunnelling microscopy image of Pt(111) (left) and the Fourier transform of this image (right) [10].
The first theory for describing the properties of metals was given by Drude. Drude transferred the successful kinetic gas theory to the electrons in the metal. In this theory, the electrons are moving freely through the solid, not interacting with each other or with the rigidly fixed ion cores. The only interaction which takes place is an instantaneous scattering from the ion cores (but no scattering from other electrons). This theory was giving the right order of magnitude for the Hall coefficient of most metals and it seemed to explain the Wiedemann-Franz law that the ratio κ/σ of the thermal and electrical conductivity is proportional to the temperature. It gave the right proportionality constant (but for the wrong reasons). The Drude model totally failed to describe the properties like the heat capacity of a metal. In the kinetic gas theory the electronic contribution to the heat capacity would be very high, in the experiment it was not.
A quantum mechanical treatment of the electronic structure in a solid starts with considering free electrons in a box. First we calculate the possible states for one electron and then we fill them with electrons according to the right statistics. The electron-electron interaction is neglected. The Schrödinger equation is simply
| (3.9) |
with the boundary conditions that the wave function must vanish at the border of the box. The solutions are
| (3.10) |
with energies
| (3.11) |
and with restrictions for the k vector
| (3.12) |
where L is the length of the box. If we now go to periodic boundary conditions, i.e. to
| (3.13) |
the solutions of (3.9) are
| (3.14) |
and the k-points lie less dense in space:
| (3.15) |
If we consider the number of states inside a sphere in k-space it is
| (3.16) |
and with
| (3.17) |
it is trivial to see that the density of states is
| (3.18) |
In the free and independent electron model the states in equation 3.10 are filled up with two electrons each (to account for spin) up to the Fermi level according to the Fermi-Dirac distribution
| (3.19) |
where µ is the (temperature-dependent) chemical potential which is equal to the (temperature-independent) Fermi energy EF at T=0. Note that the important point with this distribution is that most of the electrons “sit” deep in the Fermi-sea and can not participate in any small excitations at all. This explains immediately why the Drude model gave a heat capacity which was much too large. For the heat capacity but also for transport by electrons only the electrons close (in the order of kT) to the Fermi energy are relevant.
Sommerfeld modified the Drude model such that the electrons have to obey the right statistics. This model then gives the correct prediction for the specific heat of the conduction electrons
| (3.20) |
where D(EF) is the density of states at the Fermi level. In a classical gas the specific heat would be constant and 3nkB/2. The linear contribution of the electrons is very small and can only be measured at very low temperatures since the lattice contribution goes faster to zero ∝ T3.
In a real crystal the potential is not zero or constant. It has the same periodicity as the lattice. The Schrödinger equation is
| (3.21) |
where U(r)=U(r+R)=∑G UG ei G r is the potential. The solutions of this equation are Bloch waves with the form
| (3.22) |
where uk(r)=uk(r+R) is a lattice periodic function. A general property of the Bloch waves is that
| (3.23) |
where G is a reciprocal lattice vector. This means that a Bloch wave does not change when it is shifted by a reciprocal lattice vector. Setting this into the Schrödinger equation gives that also
| (3.24) |
Since both the wave-functions and the energies are periodic in reciprocal space it is sufficient to look at both in the first Brilloin zone.
For a very simple approach to the electronic structure of metals one can assume that the potential is very weak, i.e. that the electrons are nearly free. Formally this is done by taking only the first two coefficients in the Fourier series for the potential
| (3.25) |
The solution for this potential are similar to the free electron case. The only difference is that now gaps appear in the band structure because the bands split at the Brillouin zone boundaries. The size of the splitting is about twice the magnitude of the second Fourier coefficient in the potential.
The other limiting case is that the electrons are not free at all but rather bound almost as in the corresponding atoms with only a small overlap between the wave functions of neighbouring atoms. Let the Schrödinger equation for the atom
| (3.26) |
be solved. Then the Hamilton of the solid can be written as
| (3.27) |
where Δ U(r) describes the deviation of the potential from the superposition of the atomic potentials, i.e. Δ U(r) very small close to the atomic cores. This is known as the tight-binding approximation. The wave functions must be Bloch functions, of course, and can be written as
| (3.28) |
where the Φ and the ψat wave functions are very similar. In the most simple case of an s state it leads to solutions of the form
| (3.29) |
i.e. to a band of width γ which is shifted from the atomic level position by the amount β. The band-width γ is given by the overlap of the atomic orbitals. Close to the bottom of the band this has again a nearly quadratic dispersion.
The local density approximation can be used to study some fundamental properties of surfaces. The most simple model for a surface is the so-called jellium model . Here, the ions are replaced by a uniform positive background charge. If each atom donates q electrons to the valence band then the charge of the ion is e(Z−q). Smearing it out leads to a average charge of n = e(Z−q)/V = e/(4/3 π rs3) where V is the volume of the Wigner-Seitz cell and the inverse-sphere radius rs is just another way of expressing the electron density.
A surface in the jellium model is an abrupt change of the positive charge density at the surface
| n+(z)= |
| | z≤ 0 , n+(z)=0 | z> 0 (3.30) |
The charge density of the electrons does not follow n+ exactly,
of course. The result of a calculation is given in Fig. 3.5.
Figure 3.5: Electron density at the surface of jellium as a function of distance from the surface for two values of rs (after Ref. [9]).
One observes two things: the electron density spills out into the vacuum and, when approaching the density 1 inside the crystal, it shows small oscillations as a function of distance.
The spill out of the electrons into the vacuum happens because the electrons try to lower their kinetic energy. This energy gain is balanced by the loss of potential energy. The depletion of electrons just below the surface and the increase of density just above the surface sets up a dipole layer which leads to a change of electrostatic energy Δ Φ when moving an electron across the surface.
The small oscillations of the charge density are Friedel oscillations . They have a periodicity of twice the Fermi wavelength (π / kF) and arise because the electron gas is not able to screen a perturbation with Fourier components larger than 2kF. The step is sharp and has high-k Fourier contributions and therefore it creates Friedel oscillations.
We can also calculate the surface energy
of jellium as a function of rs (or electron density).
Fig. 3.6 shows
the result of such a calculation. At low densities the results are
reasonable but at high density the result is completely
inappropriate: the surface energy becomes negative.
It might be reassuring to know that re-introducing the lattice
solves this problem in the local-density approximation.
While the surface energy is of
fundamental importance in a total-energy calculation, it
is very hard to measure. One can, for example, extrapolate the
surface tension
of liquid metals to 0 K but one can not measure the
work done on a macroscopic crystal when cleaving it.
Figure 3.6: Experimental surface energies compared to the result of a jellium calculation (after Ref. [9]).
If we now go back to a crystal with a lattice we can discuss the fundamental properties of the electronic wave functions. The wave functions at the surface do, of course, have to obey the symmetry parallel to the surface but the translational invariance perpendicular to the surface is broken. This symmetry breaking can actually lead to new solutions of the Schrödinger equation, electronic states which reside only at the surface.
A simple qualitative picture is given in the following. Consider the Bloch wave functions in the solid:
| (3.31) |
Nothing in the derivation of the Bloch waves requires the k-vector to be real. A k with an imaginary part does, however, lead to problem in the bulk: it gives solutions which are growing without bound in the crystal. If however, only the component of k perpendicular to the surface is non-real one can try to match the solution which is exponentially growing inside the crystal with an exponential decay outside the crystal. In this way a surface-state wave-function is created. Inside the crystal it can be written as
| (3.32) |
outside it is exponentially decaying.
We write κ instead of k⊥ to make clear that we do not
deal with a conventional k⊥ vector but with a complex one.
Fig. 3.7 shows such a solution.
We want to calculate the motion of the atoms in a solid in the so-called harmonic approximation. The interatomic potential is given by the ion-ion interaction and by the screening done by the electrons. We assume that we know this potential. Its precise form is of no importance for the following considerations. The interatomic potential is expanded around the equilibrium position. The constant term is irrelevant and the linear term is vanishing. We keep only the quadratic term. Then the equations of motion look like those for coupled harmonic oscillators (hence the name harmonic approximation):
| (3.33) |
This is confusing because of all the indices. n=(n1,n2,n3) and m=(m1,m2,m3) denote the unit cell. α and β are the numbers of the atoms in the unit cell. s is the displacement and i and j are the direction of the displacement. The term Φn α im β j sm β j is the force on the α atom in the n unit cell in direction i when the β atom in the m unit cell is moved in direction j. Basically these are the second derivatives of the potential with respect to the displacements.
Now this complicated system of many coupled equations can be solved by a plane-wave ansatz:
| (3.34) |
This has the form of a plane wave which is only defined on the lattice points r. Putting this back into the equations of motion leads to a simple system of equations:
| (3.35) |
The D is called dynamical matrix and contains mainly the force constants. The initial equation system with more equations than atoms in the crystal has been transformed into a system with 3r equations where r is the number of atoms in the unit cell. It has 3r solutions for each (q) when the determinant vanishes.
The 3r solutions are called "branches". For one atom per unit cell there is only one branch, called the acoustic branch, which has a linear dispersion in the vicinity of q=0 and goes to 0 at q=0. This branch corresponds to the long-wavelength acoustic waves in the crystal. For each additional atom in the unit cell one gets more so-called optical branches which have a non-zero energy at q=0.
The excitations can be viewed as quasiparticles with a momentum q and an energy ℏ ω. They are called phonons. Note that these phonons only exist in an harmonic solid.
For vibrations at surfaces we can make exactly the same argument as for electronic states at surfaces. Breaking the translational symmetry makes room for new solutions because we can have qs which are imaginary perpendicular to the surface. It is easy to imagine that the forces between the atoms in the first layers are different from the forces in the bulk and hence the vibrational frequencies are also different. A genuine surface vibration exists if the frequency at the surface does not show up in the frequency spectrum of the bulk (at the same q∥, strictly spoken).
Surface-located vibrations (and electronic states) can of course also be created by adsorbing something on the surface. If we put, for example, CO molecules on the surface then the stretch frequency between the oxygen and the carbon is, by definition, a surface vibration .
Most of the contents of this lecture is explained in much more detail in standard solid state physics books. I can particularly recommend those by Ashcroft and Mermin [11] and by Ibach and Lüth [12]. More detailed information about the electronic structure at surfaces can be found in [5] and is also worth considering the original article by Lang and Kohn [9].