back to homepage

Klick on the images to download an eps version.

Chemical Bonding in solids

(a) Typical interatomic potential for bonding in solids. (b) Resulting force, i.e. $-\mathrm{grad} \phi(\mathbf{r})$.

The energy changes $\Delta E_{\uparrow \uparrow}$(triplet) and $\Delta E_{\uparrow \downarrow}$(singlet) for the formation of the hydrogen molecule.The energy zero is the total energy for two hydrogen atoms in the ground state. The dashed lines represent the approximation for long distances. The two insets show grey scale images of the corresponding electron probability density.

Crystal Structures

Example for a two-dimensional Bravais lattice.

Illustration of unit cells (primitive and non-primitive) and of the Wigner-Seitz cell for a rectangular two-dimensional lattice.

A two-dimensional Bravais lattice with different choices for the basis.

(a) Simple cubic structure; (b) Body-centred cubic structure and (c) Face-centred cubic structure.

Structures of CsCl and NaCl.

Close-packing of spheres leading to the hcp and fcc structures.

The structures for (a) diamond and (b) graphite. $sp^3$ and $sp^2$ bonds are displayed as solid lines.

Construction for the derivation of the Bragg condition. The horizontal line represent the lattice planes which are separated by a distance $d$.

Three different lattice planes in the simple cubic structure characterised by their Miller indices.

Illustration of x-ray scattering from a sample. The source and detector for the x-rays are placed at $\mathbf{R}$ and $\mathbf{R'}$, respectively. Both are very far from the sample.

Upper part: lattice for an infinite chain with a spacing of $a$ as well as the reciprocal lattice, an infinite chain with a spacing of $2\pi / a$. Middle and bottom: Two lattice periodic function $\rho(x)$ in real space as well as their Fourier transformation. The Fourier transformation is given such that the magnitude of the Fourier coefficients $\vert\rho_n\vert$ is plotted on the reciprocal lattice vectors they belong to.

Ewald construction for finding the directions in which constructive interference can be observed.

Mechanical Properties

Illustrations for the definition of (a) stress $\sigma=F/A$ and strain $\epsilon=\Delta l /l$ and (b) shear stress $\tau =F/A$ and shear angle $\alpha$.

Typical stress of a solid as a function of applied strain.

Sketch illustrating the definition of Poisson's ratio.

Young's modulus for different materials. The values are merely a guide as strong variations are possible.

Estimate of the yield stress for shearing a solid (a) atoms in equilibrium position (b) distortion for a small shear stress, (c) meta-stable equilibrium (d) new stable equilibrium for the sheared solid.

An edge dislocation. The plane denoted by S is called the slip plane.

Shearing of a solid in the presence of an edge dislocation. The dislocation moves through the solid by breaking only one row of bonds at a time.

A point defect can pin a dislocation such that it cannot move.

Thermal Properties of the Lattice

(a) One-dimensional chain with one atom per unit cell. (b) Allowed vibrational angular frequencies $\omega$ as a function of wave vector $k$.

Instantaneous position of atoms in a chain for two different wavelengths with $\lambda=10a$ and $\lambda=(10/11)a$. Note that the wave is transversal for illustrative purposes. Otherwise, we have only considered longitudinal waves in one-dimensional chains.

(a) One-dimensional chain with two atoms per unit cell. (b) Allowed vibrational angular frequencies $\omega$ as a function of wave vector $k$.

Vibrational spectrum for a finite chain of atoms with a length of ten unit cells and a unit cell length of $a$. The light grey line represent the vibrational spectrum for an infinite chain of lattice constant $a$. The black markers represent the actual vibrational frequencies which are allowed.

(a) Energy level diagram for one harmonic oscillator. (b) Energy level diagram for a chain of atoms with one atom per unit cell and a length of $N$ unit cells.

Phonon dispersion in Al and diamond along several directions in reciprocal space. The inset shows the Brilloin zone, which is the same for both materials. After B. Grabowski et al. , Phys. Rev. B 76, 024309 (2007) and N. Mounet and N. Marzari, Phys. Rev. B 71, 205214 (2005).

Illustration of the derivation of the interatomic force constant from Young's modulus.

Temperature dependent heat capacity of diamond. Data points from J. .E. Desnoyers and J. A. Morrison, Phil. Mag. 3, 42 (1958) and A. C. Victor, J. Chem. Phys. 36, 1903 (1961).

Temperature dependent heat capacity in the Einstein model compared to a typical experimental result for an insulator (a) linear scale (b) log-log scale.

Two-dimensional cut, showing (a) the points of integers $(n_x,n_y,n_z)$ which represent the allowed vibrational states and (b) the values of wave numbers $(k_x,k_y,k_z)$ corresponding to $(n_x,n_y,n_z)$. The circle represents a cut through a sphere for a certain highest value of $n$ or $k$, such that a total of $N$ states is enclosed.

Temperature-dependent thermal conductivity of Si. After C. J. Glassbrenner and G. E. Slack, Phys. Rev. 134, A1058 (1964)

Classical picture for the thermal expansion of a solid. The interatomic potential $\Phi$ is shown as a function of interatomic distance. The grey line marks the temperature dependent mean interatomic distance.

Gibbs free energy for two competing phases, A and B. At the temperature $T_C$ a phase transition occurs.

(a) Melting temperature as a function of cohesive energy for the elements. (b) Melting temperature as a function of the estimated melting temperature from the Lindemann criterion.

Electronic Properties of Metals: Classical Approach

Measured and calculated electrical conductivity of metals as a function of conduction electron density for two different temperatures. The measured data are marked by the element's names; the calculations are the solid lines.

(a) Illustration of the Hall effect. (b) Equilibrium between Lorentz force and force caused by the Hall field for electrons passing through the sample. (c) Equilibrium between Lorentz force and force caused by the Hall field for positively charged carriers passing through the sample.

Electronic Properties of Metals: Quantum Mechanical Approach

The formation of energy bands in solids. (a) Bonding and antibonding energy levels and their occupation for a molecule constructed from two Na atoms. (b) The molecule's energy levels as a function of interatomic separation. (c) The energy levels for a cluster of many Na atoms as a function of their separation. (d) For very many energy levels there is a quasi-continuum between the lowest and highest energy level. The band is half-filled with electrons as illustrated by the bar.

Band formation in Si. The lower band corresponds to the $sp^3$ states and is completely filled.

Electronic states in the free electron model. The increasing energy separations between the points at higher energies in a artefact caused by holing $n_y=n_z=0$.

(a) Density of states for a free electron gas $g(E)$. (b) Fermi-Dirac distribution function at a finite temperature $f(E,T)$. (c) Density of occupied states $g(E) f(E,T)$. Note that the relative width of Fermi distributions 'soft zone' ( $\approx 4k_BT)$ is exaggerated in the sketch for temperatures around room temperature.

Illustration of the 'Fermi trap' effect. Most of the electrons in a metal cannot change their energy by a small amount because the reachable states are already occupied by other electrons.

Qualitative sketch of the electronic and lattice contributions to the heat capacity at constant volume. At sufficiently low temperatures the electronic contribution dominates.

Potential due to a positive point charge in a metal compared to the Coulomb potential of the same charge in free space.

Electronic states in the nearly free electron model for a one-dimensional chain with unit cell length $a$. (a) solutions for three equations using a nearly vanishing $U=U_1=U_{-1}$. (b) solutions for five equations using a nearly vanishing $U=U_1=U_{-1}$. (c) same as (b) but for a larger value of $U$. (d) same as (b) but for larger values for both $U_1=U_{-1}$ and $U_{2}=U_{-2}$. The grey bars symbolise the ranges where a quasi-continuum of energies is available (bands) and the gaps in between them.

Qualitative explanation for the gap openings at the Brillouin zone boundary. Shown are two possible standing electron waves with $k$ corresponding to the zone boundary $\pi / a$ There probability density is either accumulated or depleted in the vicinity of the ion cores compared to a travelling free electron wave which has a constant probability density.

(a) Electronic energy bands in Al along the $\Gamma-\mathrm{X}$ direction only. The inset shows the first Brillouin zone. (b) Energy bands in different directions given by the dashed path between high symmetry points of the Brillouin zone. The horizontal dashed line represents the fictitious Fermi level for aluminium with the same structure but only one valence electron instead of three. Band structure from H. .J. Levinson et al. , Phys. Rev. B 27, 727 (1983).

Electronic energy bands for Si and GaAs. These materials have the same Brillouin zone as Al (see Fig. 1.6). The bands below the grey zone are completely filled and the bands above the grey zone are completely empty at zero temperature. The grey region represents an absolute band gap in the electronic structure. Band structures from M. Rohlfing et al. , Phys. Rev. B 48, 17791 (1993).

Illustration of the difference between metals and semiconductors / insulators. (a) Density of states in the free electron model. (b) Qualitative density of states in the nearly free electron model with the appearance of gaps (c) Occupied density of states for a metal: The chemical potential (or the Fermi energy) cuts through a finite density of states. (d) Occupied density of states for a semiconductor / insulator: the chemical potential is in a region of vanishing density of states.

Simple picture of conduction in a metal. The circles symbolise filled electron states. (a) Situation for a partially filled band without an applied field. (b) Situation for a partially filled band with an applied field. After some time $\delta t$, all electrons are shifted by an amount $\delta k$ due to the applied electric field. The asymmetric distribution in the electrons group velocity gives rise to an electric current. The electrons at $k_F$ can be scattered back to lower lying states at $-k_F$ with a probability proportional to the inverse relaxation time. (c) and (d) Corresponding situation for a completely filled band without and with electric field. All electrons are merely shifted into states which had been occupied already in the field-free case.

Semiconductors

Charge neutrality and the position of the chemical potential in an intrinsic semiconductor. (a) Schematic density of states for a semiconductor. (b) Fermi-Dirac distribution. (c) Occupied density of states (grey area) for the chemical potential just above the valence band maximum. (d) Occupied density of states for the chemical potential close to the middle of the gap. Note that the temperature in (c) and (d) is much higher than room temperature to make the presence of excited carriers visible.

Transport of charge in an electric field for a partially valence band. The process can be interpreted as a motion of electrons towards the positive terminal or as a motion of a hole towards the negative terminal.

(a) Sketch of the valence band and conduction band in the vicinity of the gap. The bands are described as parabolas with different curvatures (effective masses). (b) Even simpler picture in which valence band and conduction band are represented by single energy levels.

Set-up for the measurement of cyclotron resonance.

Non-ionised dopant atoms in a Si lattice. (a) donor (b) acceptor.

Energy levels for dopant atoms. (a) The donor levels are placed shortly below the conduction band edge. Ionisation of a donor corresponds to the transfer of the extra electron into the conduction band. (b) The acceptor levels are placed just above the valence band. Ionisation of an acceptor corresponds to accepting (taking) an electron from the valence band and thereby generating a mobile hole.

Electron density and position of the chemical potential for an $n$ doped semiconductor. (a) Qualitative picture. At very low temperatures, almost none of the donors are ionised. The chemical potential is found between the donor energy and the conduction band. At high temperatures all the donors are ionised and the chemical potential lies in the middle of the gap. (b) Quantitative picture. The position of the chemical potential and the logarithmic carrier density are shown as a function of inverse temperature.

The $pn$ junction. (a) Energy levels and carrier densities in separate $p$ and $n$ semiconductors. Ionised donors (acceptors) are symbolised by plus (minus) signs. The chemical potential is very close to the VB and the CB for $p$ and $n$ doping, respectively. (b) Formation of a depletion zone when the the junction is established. (c) A position-independent chemical potential requires band bending over the depletion zone.

Idealised model of the depletion zone solved using the Poisson equation. (a) charge density in the depletion zone. (b) electric field (c) electrostatic potential.

Definition of the energies in the $pn$ junction. The VBM on the $n$ side is taken as an energy zero but this choice does not affect the densities.

The $pn$ junction as a diode (only considering the electrons, not the holes). (a) without any bias voltage (b) forward bias (c) reverse bias.

Characteristic I/V curve for a diode showing rectifying behaviour.

Design and working principle of a MOSFET transistor. (a) without applied voltage (b) with a small positive gate voltage (c) with a gate voltage large enough to generate an inversion layer.

Generation of an inversion layer in the MOSFET. The positive gate voltage leads to a band bending which is strong enough to turn electrons from being minority carriers to being majority carriers.

Optoelectronic devices. (a) A light emitting diode works because of carrier recombination. (b) A photodetector or solar cell is based on carrier separation in the depletion zone.

Magnetism

(a) Precession (b) Possible orientations for the magnetic moments in field direction for H with $l=2$. (c) Possible orientations for the magnetic moments in field direction for Cr$^{3+}$.

Paramagnetic susceptibility of a solid with localised magnetic moments. The limit of Curie's law is indicated as a dashed line.

(a) Density of states for free electrons split up into electrons having their magnetic moment parallel or antiparallel to an external field but the field is almost zero. (b) When the field is increased to the value $B_0$, the energy of the electrons is increased or reduced by $\mu_B B_0$, depending on the orientation of their magnetic moments. (c) The electrons with a magnetic moment antiparallel to the field can go to a lower energy state by flipping their spin. In this way, a constant Fermi level is maintained.

Types of magnetic ordering. The arrows denote direction and size of the localised magnetic moments.

(a) Density of states in a transition metal (here Fe), separated in two spin directions but without a net magnetisation. (b) In the case of a spontaneous magnetisation the electrons with one spin direction (here the 'down' direction) will be in the majority. This is described by a relative shift of the $d$ bands of different spin directions.

Magnetisation of a ferromagnetic material below the Curie temperature $\Theta_C$.

(a) Domains of different magnetisation in a ferromagnetic solid. (b) detailed picture of the magnetisation rotation in a Bloch wall between two domains.

(a) $\mathbf{B}$ field inside and outside a piece of magnetic material. (a) Material with a single domain leading to a strong external field. (b) The introduction of a few domains greatly reduces the external field. (c) The material can be magnetised by the movement of domain walls caused by an external field $\mathbf{B}_0$.

Magnetisation of a ferromagnetic sample as a function of externally applied field $B_0$. The starting point of the curve is the origin.

Dielectrics

A plane plate capacitor. (a) Charges on the plates of the capacitor with no dielectric present between the plates. (b) Polarisation of the dielectric material between the plates. (c) The net effect of the polarisation are surface charge densities of the dielectric at the plate-dielectric interface.

Mechanisms leading to microscopic electric polarisation. (a) The electric field polarises all atoms in the solid. (b) In ionic solids, like NaCl, the lattice can be polarised, giving rise to local electric dipoles. The dashed grid gives the position of the ions without an applied field. (c) If there are permanent dipoles in the solid and these are free to be rotated, they orient themselves parallel to the field. A molecule with a permanent dipole is e.g. water.

The local field on microscopic polarisable units. Left: Polarised microscopic dipoles in a dielectric placed in an external field. The electric field $\mathbf{E}$ is the average internal field in the dielectric. Right: The local field felt by every single dipole is not just $\mathbf{E}$ but $\mathbf{E}+\mathbf{E}'$ because the surrounding dipoles lead to a field increase.

Dielectric function for the damped, driven harmonic oscillator close to the resonance angular frequency $\omega_0$. The left side shows the real part and the right side the imaginary part of $\epsilon$. $\epsilon_{\mathrm{stat}}$ and $\epsilon_{\mathrm{opt}}$ stand for the static and optical value of the real part of $\epsilon$, respectively.

(a) Contributions to the imaginary part of $\epsilon$ by transitions between occupied and unoccupied states. The grey lines denote possible transitions. The density of such transitions is high at $\mathbf{k}=0$ and $\mathbf{k}'$ because valence band and conduction band are parallel there. (b) $\epsilon_i$ resulting from this band structure.

Upper part: The unit cell of barium titanate BaTiO$_3$.Lower part: In the ferroelectric state, the (negative) oxygen sub-lattice is displaced from the sub-lattice containing the (positive) Ba and Ti ions.

(a) Exposing a piezoelectric material to mechanical stress results in a macroscopic electric polarisation. (b) Conversely, an electric field across the sample leads to a mechanical strain. (c) This is caused by the deformation of microscopic units in the crystal. Such a unit is shown without applied field or stress on the left. The unit consists of three dipole moments which sum up to a total moment of zero. On the right, stress is applied. This results in the distortion of the unit and the net dipole moment is no longer zero.

Superconductivity

Typical temperature dependent resistivity for a normal metal and for a superconductor.

Periodic table of the elements with the superconducting elements in bold large letters. Light grey letters indicates that the elements do only become superconducting in a high pressure modification.

(a) Combined effect of a magnetic field and finite temperature on a superconductor. In the region below the curve, i.e. for low temperatures and small external fields, the solid is superconducting. Above the curve it is in its normal state. (b) The same but including the effect of a finite current density.

The Meissner effect is not merely a consequence of zero resistivity. (a) shows the behaviour of a 'perfect conductor', i.e. a material which merely has zero resistivity below $T_\mathrm{C}$. For this material the magnetic field in the specimen below $T_\mathrm{C}$ depends on the presence and size of a magnetic field before cooling down below $T_\mathrm{C}$. (b) shows the situation for a genuine superconductor which displays the Meissner effect. The interior of the specimen is field free below $T_\mathrm{C}$, independent of the sample's history.

Illustration of the isotope effect. The graph shows the critical temperature as a function of isotope mass as a log-log plot. The data point lie on a straight line suggesting a power low behaviour with $T_\mathrm{C} \propto M^{-1/2}$. Data taken from E. Maxwell, Phys. Rev. 86, 235 (1952) and B. Serin et al., Phys. Rev. B 86 162 (1952)).

Exponential damping of an external magnetic field near the surface of a superconductor.

Electrons in the crystal can locally deform the lattice via the electrostatic interaction with the ions. The locally increased positive charge, in turn, is attractive for other electrons. Situation for (a) a very slow or static electron and (b) an electron near the Fermi level of a metal.

Occupation of single-electron levels at zero temperature in (a) a normal metal and (b) a superconductor. For the superconductor, the electrons close to $E_F$ are bound in Cooper pairs and these occupy a single many body state, the BCS ground state, which cannot be shown in this figure of single-particle levels. In order to excite single electrons out of this ground state, a Cooper pair has to be broken. This costs an energy of $\Delta$ per electron and creates two un-paired electrons in the lowest possible unoccupied single particle states.

Gap size for a superconductor in the BCS model as a function of temperature. At the transition temperature $T_\mathrm{C}$ the gap is closed.

Tunnelling experiment between a superconductor and a normal metal. The two are separated by a thin insulating oxide. Only elastic single-electron tunnelling is considered. (a) Situation without applied voltage. (b) For a small tunnelling voltage no tunnelling is possible because of the lack of available states in the gap. (c) As the tunnelling voltage exceeds $\Delta / e$, single-electron tunnelling becomes possible. (d) Thick line: tunnelling current vs. voltage for the present junction, dashed line: corresponding curve for tunnelling between two metals.

Qualitative low temperature heat capacity of a superconductor in both the superconducting and the normal state. A normal state below $T_\mathrm{C}$ can be realised by applying a magnetic field.

(a) A superconducting ring enclosing a magnetic flux. The magnetic flux through such a ring is quantised in multiples of $h/2e$ (b) A superconducting quantum interference device (SQID).

Resistivity, internal magnetic field and magnetisation as a function of an external magnetic field for type I and type II superconductors. The temperature is assumed to be zero Kelvin, such that the materials are in their superconducting state when no magnetic field is applied. The type I material has only one critical field $B_\mathrm{C}$ whereas the type two material has two different critical fields $B_\mathrm{C1}$, $B_\mathrm{C2}$.

Magnetic flux in a type II superconductor. The field penetrates through thin filaments of normal materials while the rest of the sample remains superconducting. The filaments are surrounded by currents of superconducting current which keep the rest of the sample field free.

Increase of the critical temperature $T_\mathrm{C}$ of known superconductors as a function of time. The dashed horizontal line is the boiling temperature of liquid nitrogen.

Finite Solids and Nanostructures

(a) A thin metal film on a semiconducting or insulating substrate. (b) Modelling the metal film as a potential well. The dashed lines indicate the simplification of an infinite potential well.

(a) Matching of a bulk electronic state (a Bloch wave) to an exponential decay outside the surface. (b) An electronic surface state which is decaying as the distance from the surface is increased, both outside and inside the solid.

Surface Science

Ultra High Vacuum (UHV)

Pumping of a UHV system.

A oil-sealed rotary vane pump. Principle of operation: (a) gas from the vacuum system is expanded into the pump and (b) the gas is pushed through the pump exhaust.

A turbomolecular pump. Picture: Leybold Vacuum GmbH, used with permisssion.

An ion pump.

An ion gauge.

A quadrupole mass spectrometer

A typical mass spectrum.

Electron spectroscopy

An EELS experiment. The momentum transfer parallel to the surface is determined by the electron energy and the scattering geometry.

The mean free part of the electrons in solid. The dots are measurements the dashed curve is a calculation. After Ref. [#!universalcurve!#].

An electron gun.

A hemispherical electron analyser with a lens system.

A simple picture for a plasma oscillation

Charge and field distribution for a surface plasmon

EELS spectra from the (010) surface of $\alpha$-Ga. The surface and bulk plasmon losses can be identified. The difference between the spectra is due to the experimental geometry (to be discussed in the exercises ).

Surface cleaning and chemical analysis

Sketch of the energies involved in photoelectron spectroscopy.

Decay of a core-hole by x-ray emission or the Auger process.

Nomenclature in the x-ray decay in Al and Mg.

Acceleration of electrons in a storage ring and emission of synchrotron radiation.

Spectral distribution of synchrotron radiation.

Radial part of the Ne $2p$ and Ar $3p$ wave functions together width the continuum wave function for the $l+1$ channel at zero kinetic energy. After U. Fano and J.W. Cooper, Rev. Mod. Phys. 40, 441 (1968).

XPS spectrum taken from an Al(001) surface with synchrotron radiation at $h\nu=600eV$.

Comparison between experimental and calculated (from Koopman's theorem ) C$1s$ binding energies. Note that the agreement is very good but only if one of the axes is shifted by 15 eV. The good agreement is underlined by the line of slope 1 After D.A. Shirley, Adv. Chem. Phys. 23, 85, (1973).

A Ru $3d_{5/2}$ core level spectrum from a clean Ru surface. Apart from the bulk peak two surface-related peaks are visible, one from the first and one from the second layer. After Ref. A. Baraldi et al., Phys. Rev. B. 61, 4534 (2000).

Surface core level shift caused by $d$-band narrowing and an electrostatic shift for transition metals with less and more than half filling of the $d$ shell.

Final state effects in the photoemission process.

Typical Auger spectrum from a Cu surface.

Adsorption, Desorption and Chemical Reactions

NH$_{3}$ synthesis on an iron surface.

A physisorption potential for He on Au. After E. Zaremba and W. Kohn, Phys. Rev. B 15, 1769 (1977).

Schematic energy level diagram for an adsorbate / substrate system in case of a simple metal.

Schematic energy level diagram for an adsorbate / substrate system in case of a transition metal.

Change in state density due to adsorption of Li, Si and Cl on jellium. After N.D. Lang and A.R. Williams Phys. Rev. B 18, 616 (1978).

Electron density contours for Li, Si and Cl on jellium. The upper panel displays the total charge density. The lower panel the total charge density minus the superposition of the individual charge densities. Solid lines correspond to a charge accumulation, dashed lines to a charge depletion. AfterAfter N.D. Lang and A.R. Williams Phys. Rev. B 18, 616 (1978).

Energy-level diagram for the molecular orbitals in CO.

Charge density contour plots for a layer of CO adsorbed on a Ni surface. (a) and (b) show the $5\sigma$ and $2\pi$ orbitals of the free molecule, respectively. (c) shows the difference in charge density between the actual adsorption system and the unsupported monolayer. Solid lines correspond to a charge density increase and dashed lines to a depletion. After E. Wimmer, C.L. Fu and A.J. Freeman, Phys. Rev. Lett. 55, 616 (1985).

Schematic diagram of the potential energy of a molecule and its constituents as a function of distance from the surface. Different scenarios are possible (a) molecular chemisorption (b) molecular physisorption and (c) dissociative chemisorption. After Ref.J.E. Lennard-Jones, Trans. Farad. Soc. 28, 33 (1932).

The Langmuir model for adsorption: (a) associative adsorption (first order process); (b) dissociative adsorption (second order process). (c) simple energy diagram with activation energies for adsorption and desorption ($E_{a}$ and $E_{d}$) and the heat of adsorption $H_{a}$.

Sticking probability of N$_{2}$ on tungsten as a function of coverage. After King and Wells, Proc. Roy. Soc. London A339, 245 (1974).

Desorption in the Langmuir model.

Thermal desorption curves for a linear heating rate: (a) a first order process ($n=1$) and (b) a second order process $n=2$.

Thermal desorption spectrum for H$_{2}$ from a tungsten surface. After Tamm and Schmidt, J. Chem. Phys. 54, 4775 (1971).

A single-crystal adsorption calorimeter.

Surface Structure

The Wulff construction for the determination of equilibrium crystal shape.

The (111) surface of an fcc crystal.

A few important truncated bulk surface structures.

Relaxation (left) and reconstruction (middle) and adsorbate superstructures (right) on surfaces.

The Finnis Heine model for inward relaxations on metal surfaces. After R. Smoluchowski, Phys. Rev. 60, 661 (1941), M. W. Finnis and V. Heine, J. Phys. F 4, L37 (1974)

The 5 possible two-dimensional Bravais lattices.

The Woods terminology for surface lattices.

Examples for structures described by the Woods terminology.

Three types of overlayers: (a) simply related to the substrate, (b) rationally related and (c) a incommensurate structure with no common periodicity between substrate and adsorbate lattice.

A LEED system.

The Ewald construction for the surface case.

Linear imaging of the reciprocal lattice by LEED.

LEED patterns of W(100) taken at a electron kinetic energy of 45 eV (left) and 145 eV (right), respectively.

Intensity of the (0,0) spot from Ni(100) as a function of electron kinetic energy. The arrows indicate the positions where maxima would be expected if the third Laue condition would be valid. After K. Christmann et al., Surf. Sci. 40, 61 (1973).

The LEED pattern shows the sum of the reciprocal lattices from substrate and overlayer plus all possible combinations between them. For a simple overlayer structure as in (a), this combination does not lead to any new spots. For a coincidence structure (b) it does (grey spots). The arrows indicate the size of the surface unit cell as a whole. When this unit cell is taken to calculate the reciprocal lattice, the ``extra'' spots appear in a natural way.

Three examples for overlayer structures and the LEED patterns produced by them. (a) a (4x2) structure, (b) a c(4x2) structure. In the LEED patterns the open circles are the (1x1) spots. The (1x1) unit cell in reciprocal space is also given.

LEED patterns resulting from two different domains of the structures shown in Fig. 2.5.

Agreement between measured and calculated LEED I-V data for different spots in case of the Al(111) surface. The full line is experimental data, the dashed line calculation (David Adams, private communication.)

Relative change in the distance between first and second layer of simple metal surfaces as a function of bulk interlayer distance (both normalized by the nearest neighbour distance in the bulk). After h. Hofmann et al., Phys. Rev. B 53, 13715 (1996).

Adsorption geometry for K on Al(111) at 100 K (top) and 300 K (bottom). Both structures give rise to a $(\sqrt{3} \times \sqrt{3})R30^{\circ})$ LEED pattern. After K. Stampfl et al., Phys. Rev. Lett. 69, 1532 (1992).

Adsorption geometry for (2x1)-O on Cu(110). Left: top view of the clean surface, Right: top view of the reconstruction. After S.R. Parkin et al., Phys. Rev. B 41, 5432 (1990).

Left: ideal Si(111) surface. Right: the Takayanagi model for the Si(111) (7x7) reconstruction. AfterTakayanagi et al., J. Vac. Sci. Techn. A3, 1502 (1985).

The unreconstructed Si(100) surface and the asymmetric dimer model.

Structure and surface unit cell of NiO (top view). Top-layer Ni ions are red, second layer Ni ions are pink, oxygen ions are black.

X-ray absorption of Cu in the vicinity of the K-edge.

Schematic illustration of the interference leading to the EXAFS oscillations

SEXAFS data from the oxygen K-edge of the co-adsorption system SO$_{2}$ + O on Cu(111). After M. Polcik et al., Phys. Rev. B 57, 1868 (1998).

Structure SO$_{2}$ + O on Cu(111) as determined by SEXAFS. After After M. Polcik et al., Phys. Rev. B 57, 1868 (1998).

Principle of photoelectron diffraction

Modulus of the electron scattering factor.

Symbols in single scattering theory.

A grazing emission forwards scattering experiment.

A forward scattering experiment to determine molecular orientation.

A forward scattering experiment to determine molecular orientation of CO on Ni(100). After D.A. Wesner, F.P. Coenen and H.P. Bonzel, Surf. Sci. 199, L419 (1988).

XPS spectra of clean and C$_{2}$H$_{2}$ covered Ni(111).

Group plot of many XPS spectra of the O $1s$ peak from CO on Cu(110). All the spectra are taken with different photon energies. One spectrum is magnified.

I(E) and the modulation function for the O $1s$ peak from CO on Cu(110).

O $1s$ diffraction data to determine the adsorption site of the acetate species on Cu(110). After K.-U. Weiss, R. Dippel, K.-M. Schindler, P. Gardner, V. Fritzsche, A.M. Bradshaw, A.L.D. Kilcoyne and D.P. Woodruff, Phys. Ref. Lett. 69, 3196 (1992).The R-factor has the same meaning as in a quantitative LEED analysis.

Modulation functions for the two chemically different C atoms in the acetate species on Cu(110). After K.-U. Weiss, R. Dippel, K.-M. Schindler, P. Gardner, V. Fritzsche, A.M. Bradshaw, A.L.D. Kilcoyne and D.P. Woodruff, Phys. Ref. Lett. 69, 3196 (1992).

(a) Exponential leakage of the wavefunctions from a conductor into the vacuum. (b) Application of a voltage and tunnelling between two conductors because of the overlap of the wavefunction tails. $\Phi$ is the workfunction.

Schematic construction principle of an STM.

Principle of the scanning force microscope.

The electronic structure of surfaces

Definition of the energies contributing to the workfunction.

Charge distribution at a closed packed (left) and at an open (right) surface.

Workfunction of W for different surface orientations. After K. Besocke, B. Kraul-Urban, H. Wagner, Surf. Sci. 68, 39 (1977).

Workfunction change upon the adsorption of K on W(110). After R. Blaszczyszyn et al, Surf. Sci. 51, 396 (1975).

Real space structure and Brillouin zone of Be. The Be(0001) surface is the closed packed surface on top of the real-space hexagon. The surface Brillouin zone of Be(0001) is also shown. It is the projection of the bulk Brillouin zone in the (0001) direction.

Projected bulk band structure and electronic surface states for Be(0001). For the shaded areas there are bulk states with the same $\vec{k}_{\parallel}$ and energy for a $k_{\perp}$ somewhere in the bulk Brilloin zone. After E.V. Chulkov, V.M. Silkin and E.N. Shirykalov, Surf. Sci. 188,287 (1987).

Bulk band structure of Beryllium. After M.Y. Chou, P.K. Lam, M.L. Cohen, Phys. Rev. B 28,4179 (1983).

Photoemission from the CO $4\sigma$ orbital with polarized light. After E.W. Plummer and T. Gustafsson, Science 198, 165 (1977).

Momentum conservation in angle-resolved photoemission.

Refraction at the surface potential barrier.

Surface state dispersion on Cu(111). Left: EDCs close to normal emission. The dispersion of the surface state is clearly evident. Right: Grey-scale image of the same data.

Surface state dispersion on Cu(111). After S.D. Kevan, Phys. Rev. Lett. 50,526 (1983).

Normal emission spectra from Be(0001) taken with different photon energies. After E. Jensen, R.A. Bartynski, T. Gustafsson, E.W. Plummer, M.Y. Chou, M.L. Cohen and G.B. Hoflund, Phys. Rev. B 30,5500 (1984).

Band structure of Cu determined using photoemission and free electron final states. The lines are theory. After E.W. Plummer and W. Eberhardt, Adv. Chem. Phys. 49, 533 (1982).

When taking a spectrum with two different photon energies but for the same $\vec{k}_{\parallel}$, bulk-related peaks will in general show a dispersion in the spectrum while surface state peaks stay at a fixed binding energy. One can think of the surface state as being present for all values of $k_{\perp}$, similar as in the Ewald construction in Fig. 2.5. See also Fig. 2.6.

Spectra from well-ordered and sputtered Bi surface. After F. Patthey et al., Phys. Rev. B 49,11293 (1994).

Tamm-surface state on Cu(001). After D. Purdie et al., Surf. Sci. 407,L671 (1998).

Photoemission from CO on Ni(100). After R. J. Smith et al., Phys. Rev. Lett., 37, 1081 (1976).

Dispersion of the CO $4\sigma$ state for $(2\sqrt{3} \times 2\sqrt{3})R30^{\circ}$CO-Co(0001) and $(\sqrt{3} \times \sqrt{3})R30^{\circ}$CO-Co(0001). After F. Greuter, D. Heskett, E.W. Plummer and H.-J. Freund, Phys. Rev. B 27, 7117 (1983).

(2x1) reconstruction on Si(001).

Two possible models for the (2x1) reconstruction of Si(111) and their electronic surface state dispersion. Note that the energy zero is the valence band maximum and not the Fermi energy. Both surfaces are semiconducting. After K.C. Pandey, Phys. Rev. Lett. 47, 1913 (1981).

Comparison between experiment and theory for the $\pi$-bonded chain model of Si(111)-(2x1). After Ref. F.J. Himpsel, Appl. Phys. A38, 205 (1985).

Experimental surface band structure from Si(100)-(2x1) in the $\bar{\Gamma}-\bar{J}'$ direction. After G. V. Hansson and R. I. G. Uhrberg in Angle-Resolved Photoemission, Theory and Current Applications, ed. S.D. Kevan, Elsevier 1992 and references therein.

Calculated dispersion for the upper surface states of Si(100)-(2x1) for a symmetric and an asymmetric dimer configuration. After D.J. Chadi, Phys. Rev. Lett. 43, 43 (1979).

Left: Surface state bands in the bulk gap and bulk electronic structure of an n-doped semiconductor. The situation is unstable because of the non-constant Fermi level. Right: electrons from the bulk donors flow into the previously unoccupied surface states and leave a positive space charge layer behind.

Fermi level pinning for Si(111). The measured workfunction is almost independent of the doping level. The straight line in the middle illustrates the expected behaviour due to the doping. After Ref. F.J. Allend and G.W. Gobeli, Phys. Rev. 127, 152 (1962).

Projected band structure of Bi(110) and experimental data along several high symmetry lines of the surface Brilloin zone. The data are the logarithm of the photoemission intensity. The grey scale is defined such that black means low intensity and white high intensity. The features A,B,C,D and E are the electronic surface states. After S. Agergaard et al. New Journal of Physics 3, 15 (2001).

Optical properties of surfaces

(Intensity) reflection coefficients for light polarized parallel (p) and perpendicular (s) to the plane of incidence. The complex index of refraction is n=3,k=30.

Electric field components at the surface of the material with n=3,k=30. The plane of incidence is the x/z plane. The z-axis is perpendicular to the surface.

Dispersion of a surface plasmon polariton.

Setup for a RAS experiment. The complex difference in reflectance along two mutually perpendicular directions is measured. After Phil. Trans. R. Soc. Lond. A 344,453-467 (1993).

Surface electronic structure of Ag(110) in the vicinity of the $\bar{Y}$ point of the SBZ. One occupied and one unoccupied surface state can be found.

Photoemission and RAS spectra for the clean and oxygen-covered Ag(110) surface. The photoemission spectrum taken at the $\bar{Y}$ point shows the surface state right below the Fermi level. In the RAS spectrum both surface states give rise to a peak corresponding to an interband transition between them. After K. Stahrenberg, T. Herrmann, N. Esser, J. Sahm, W. Richter, S.V. Hoffmann and Ph. Hofmann Phys. Rev. 58, R10207 (1998).

SHG spectrum from clean Ag(110) and two different polarization directions. For one polarization direction the surface state transition contributes to the SHG signal. After L. E. Urbach et al., Phys. Rev. B 45,3769 (1992).

Calculated surface phonon dispersion for Be(0001) (line, from the bulk force constants) together with a measurement (markers). The continuum is the projected bulk phonon structure. After J. Hannon et al., Phys. Rev. B 53,2090 (1996).

Vibrational modes and corresponding energies (in meV) for CO on a two-fold bridge site. The free molecule has only one vibrational mode, the others are frustrated rotations and translations. After H. Ibach, D. L. Mills, Electron Energy Loss Spectroscopy and Surface Vibrations, Academic Press, New York 1982. The energies are only intended as an order of magnitude and are taken from Ref. N.V. Richardson and A.M. Bradshaw, Surf. Sci. 88, 255 (1979).

An EELS spectrometer.

EELS spectrum from Mg(0001). There are three peaks visible: the elastic peak, a loss peak from a creating a phonon and a gain peak from destroying a phonon (Ismail, private communication).

Electric field created by an electron approaching a metal surface (and its image charge).

EELS spectra for H on W for different scattering geometries. After W. Ho et al., Phys. Rev. Lett. 40, 1463 (1978).

EELS spectrum of CO adsorbed on Pt(111). After M. Schulze, PhD thesis, University of Hannover 1988.

EELS spectra for C$_{2}$H$_{2}$ and C$_{2}$D$_{2}$ showing the isotope effect for acetylene on Ni(111). After S. Lehwald and H. Ibach, Surf. Sci. 89, 425 (1979).

EELS spectrum showing a phonon losses from Be(0001) as a function of scattering geometry ($\vec{k}$-vector). After J. B. Hannon, E. J. Mele and E. W. Plummer, Phys. Rev. B 53,2090 (1996).

EELS data from Cu(110) in the specular geometry as a function of temperature. After A. P. Baddorf and E. W. Plummer, Phys. Rev. Lett. 66,2770 (1991).

FT-IRAS spectrometer. After H. Kuzmany, `Festkörperspektroskopie', Springer 1990.

FT-IRAS spectrometer in UHV. After Y. Chabal et al., J. Electron Spectr. Rel. Phen., 29, 35 (1983).

The C-O stretch frequency of CO on a stepped Pt surface as a function of coverage. After B.E. Hayden et al. Surf. Sci. 149, 394 (1985).

He-atom scattering apparatus. After J. P. Toennies, Physica Scripta Volume T19A, 39 (1987).

Low-energy vibrational modes for CO on Pt(111) investigated by He-scattering. After A.M. Lahee et al, Surf. Sci. 177, 147 (1986).

Vibrational spectra of acetylene and deuterated acetylene taken by STS from a single molecule on Cu(100). After B. C. Stipe et al., Science, 280, 1732 (1998).

back to homepage