Surface states
Encyclopedia
Surface states are electronic state
s found at the surface
of materials. They are formed due to the sharp transition from solid material that ends with a surface and are found only at the atom layers closest to the surface. The termination of a material with a surface leads to a change of the electronic band structure
from the bulk material to the vacuum
. In the weakened potential at the surface, new electronic states can be formed, so called surface states.
As stated by Bloch's theorem
, eigenstates of the single-electron Schrödinger equation with a perfectly periodic potential, a crystal, are Bloch waves
Here
is a function with the same periodicity as the crystal, n is the band index and k is the wave number. The allowed wave numbers for a given potential are found by applying the usual Born–von Karman cyclic boundary conditions . The termination of a crystal, i.e. the formation of a surface, obviously causes deviation from perfect periodicity. Consequently, if the cyclic boundary conditions are abandoned in the direction normal to the surface the behavior of electrons will deviate from the behavior in the bulk and some modifications of the electronic structure has to be expected.
A simplified model of the crystal potential in one dimension can be sketched as shown in figure 1 . In the crystal, the potential has the periodicity, a, of the lattice while close to the surface it has to somehow attain the value of the vacuum level. The step potential (solid line) shown in figure 1 is an oversimplification which is mostly convenient for simple model calculations. At a real surface the potential is influenced by image charges and the formation of surface dipoles and it rather looks as indicated by the dashed line.
Given the potential in figure 1, it can be shown that the one-dimensional single-electron Schrödinger equation gives two qualitatively different types of solutions..
The first type of solution can be obtained for both metal
s and semiconductor
s. In semiconductors though, the associated eigenenergies have to belong to one of the allowed energy bands. The second type of solution exists in forbidden energy gap
of semiconductors as well as in local gaps of the projected band structure of metals. It can be shown that the energies of these states all lie within the band gap. As a consequence, in the crystal these states are characterized by an imaginary wavenumber
leading to an exponential decay into the bulk.
, figure 1. Within the crystal the potential is assumed periodic with the periodicity a of the lattice.
The Shockley states are then found as solutions to the one-dimensional single electron Schrödinger equation
with the periodic potential
where l is an integer.
The solution must be obtained independently for the two domains z<0 and z>0, where at the domain boundary (z=0) the usual conditions on continuity of the wave function and its derivatives is applied. Since the potential is periodic deep inside the crystal the electronic wave functions must be Bloch wave
s here. The solution in the crystal is then a linear combination of an incoming and a wave reflected from the surface. For z>0 the solution will be required to decrease exponentially into the vacuum
The wave function for a state at a metal surface is qualitatively shown in figure 2. It is an extended Bloch wave within the crystal with an exponentially decaying tail outside the surface. The consequence of the tail is a deficiency of negative charge density
just inside the crystal and an increased negative charge density just outside the surface, leading to the formation of a dipole double layer
. The dipole perturbs the potential at the surface leading, for example, to a change of the metal work function
.
The nearly free electron approximation can be used to derive the basic properties of surface states for narrow gap semiconductors. The semi-infinite linear chain model is also useful in this case. However, now the potential along the atomic chain is assumed to vary as a cosine function
whereas at the surface the potential is modeled as a step function of height V0.
The solutions to the Schrödinger equation must be obtained separately for the two domains z < 0 and z > 0. In the sense of the nearly free electron approximation, the solutions obtained for z < 0 will have plain wave character for wave vectors away from the Brillouin zone boundary
, where the dispersion relation will be parabolic, as shown in figure 4.
At the Brillouin zone boundaries, Bragg reflection occurs resulting in a standing wave
consisting of a wave with wave vector
and wave vector 
.
Here
is a lattice vector of the reciprocal lattice
(see figure 4).
Since the solutions of interest are close to the Brillouin zone boundary, we set
, where κ is a small quantity. The arbitrary constants A,B are found by substitution into the Schrödinger equation. This leads to the following eigenvalues
demonstrating the band splitting at the edges of the Brillouin zone
, where the width of the forbidden gap is given by 2V. The electronic wave functions deep inside the crystal, attributed to the different bands are given by
Where C is a normalization constant.
Near the surface at z = 0,
the bulk solution has to be fitted to an exponentially decaying solution, which is compatible with the constant potential V0.
It can be shown that the matching conditions can be fulfilled for every possible energy eigenvalue which lies in the allowed band. As in the case for metals, this type of solution represents standing Bloch waves extending into the crystal which spill over into the vacuum
at the surface. A qualitative plot of the wave function is shown in figure 2.
If imaginary values of κ are considered, i.e. κ = - i·q for z ≤ 0 and one defines
one obtains solutions with a decaying amplitude into the crystal
The energy eigenvalues are given by
E is real for large negative z, as required. Also in the range
all energies of the surface states fall into the forbidden gap. The complete solution is again found by matching the bulk solution to the exponentially decaying vacuum solution. The result is a state localized at the surface decaying both into the crystal and the vacuum. A qualitative plot is shown in figure 3.
The results for surface states of a monoatomic linear chain
can readily be generalized to the case of a three-dimensional crystal. Because of the two-dimensional periodicity of the surface lattice Bloch's theorem must hold for translations parallel to the surface. As a result, the surface states can be written as the product of a Bloch waves with k-values
parallel to the surface and a function representing a one-dimensional surface state
The energy of this state is increased by a term
so that we have
where m* is the effective mass of the electron. The matching conditions at the crystal surface, i.e. at z=0, have to be satisfied for each
separately and for each 
a single, but generally different energy level for the surface state is obtained.
and its wave vector 
parallel to the surface, while a bulk state is characterized by both 
and 
wave numbers. In the two-dimensional Brillouin zone
of the surface, for each value of
therefore a rod of 
is extending into the three-dimensional Brillouin zone of the Bulk. Bulk energy bands that are being cut by these rods allow states that penetrate deep into the crystal.
One therefore generally distinguishes between true surface states and surface resonances. True surface states are characterized by energy bands that are not degenerate with bulk energy bands. These state are existing in the forbidden energy gap
only and are therefore localized at the surface, similar to the picture given in figure 3. At energies where a surface and bulk state are degenerate surface and the bulk state can mix, forming a surface resonance. Such state can propagate deep into the bulk similar to Bloch wave
s, while retaining an enhanced amplitude close to the surface.
(LCAO), see figure 5. In this picture, it is easy to comprehend that the existence of a surface will give rise to surface states with energies different from the energies of the bulk states: Since the atoms residing in the topmost surface layer are missing their bonding partners on one side their orbitals have less overlap with the orbitals of neighboring atoms. The splitting and shifting of energy levels of the atoms forming the crystal is therefore smaller at the surface than in the bulk.
If a particular orbital
is responsible for the chemical bonding, e.g. the sp3 hybrid in Si or Ge, it is strongly affected by the presence of the surface, bonds are broken, and the remaining lobes of the orbital stick out from the surface. They are called dangling bond
s. The energy levels of such states are expected to significantly shift from the bulk values.
In contrast to the nearly free electron model used to describe the Shockley states, the Tamm states are suitable to describe also transition metal
s and wide bandgap semiconductors
.
Extrinsic surface states are usually defined as states not originating from a clean and well ordered surface. Surfaces that are fit into the category extrinsic are :
Generally for extrinsic surface states is that they cannot easily be characterized in terms of their chemical, physical or structural properties.
) or angle resolved ultraviolet photoelectron spectroscopy
(ARUPS).
Electronic state
Electronic state is a quantum state of a system consisting of electrons . The state with lowest energy is called ground state, states with higher energy are excited states.See Energy level....
s found at the surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
of materials. They are formed due to the sharp transition from solid material that ends with a surface and are found only at the atom layers closest to the surface. The termination of a material with a surface leads to a change of the electronic band structure
Electronic band structure
In solid-state physics, the electronic band structure of a solid describes those ranges of energy an electron is "forbidden" or "allowed" to have. Band structure derives from the diffraction of the quantum mechanical electron waves in a periodic crystal lattice with a specific crystal system and...
from the bulk material to the vacuum
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...
. In the weakened potential at the surface, new electronic states can be formed, so called surface states.
Origin of surface states at condensed matter interfaces
Bloch's theorem
* For the theorem named after Felix Bloch on wave functions of a particle in a periodic potential, see Bloch wave.* For the theorem in complex variables named after André Bloch, see Bloch's theorem ....
, eigenstates of the single-electron Schrödinger equation with a perfectly periodic potential, a crystal, are Bloch waves
Here
is a function with the same periodicity as the crystal, n is the band index and k is the wave number. The allowed wave numbers for a given potential are found by applying the usual Born–von Karman cyclic boundary conditions . The termination of a crystal, i.e. the formation of a surface, obviously causes deviation from perfect periodicity. Consequently, if the cyclic boundary conditions are abandoned in the direction normal to the surface the behavior of electrons will deviate from the behavior in the bulk and some modifications of the electronic structure has to be expected.A simplified model of the crystal potential in one dimension can be sketched as shown in figure 1 . In the crystal, the potential has the periodicity, a, of the lattice while close to the surface it has to somehow attain the value of the vacuum level. The step potential (solid line) shown in figure 1 is an oversimplification which is mostly convenient for simple model calculations. At a real surface the potential is influenced by image charges and the formation of surface dipoles and it rather looks as indicated by the dashed line.
Given the potential in figure 1, it can be shown that the one-dimensional single-electron Schrödinger equation gives two qualitatively different types of solutions..
- The first type of states (see figure 2) extends into the crystal and has Bloch character there. These type of solutions correspond to bulk states which terminate in an exponentially decaying tail reaching into the vacuum.
- The second type of states (see figure 3) decays exponentially both into the vacuum and the bulk crystal. These type of solutions correspond to states, with wave functions localized close to the crystal surface.
The first type of solution can be obtained for both metal
Metal
A metal , is an element, compound, or alloy that is a good conductor of both electricity and heat. Metals are usually malleable and shiny, that is they reflect most of incident light...
s and semiconductor
Semiconductor
A semiconductor is a material with electrical conductivity due to electron flow intermediate in magnitude between that of a conductor and an insulator. This means a conductivity roughly in the range of 103 to 10−8 siemens per centimeter...
s. In semiconductors though, the associated eigenenergies have to belong to one of the allowed energy bands. The second type of solution exists in forbidden energy gap
Band gap
In solid state physics, a band gap, also called an energy gap or bandgap, is an energy range in a solid where no electron states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference between the top of the valence band and the...
of semiconductors as well as in local gaps of the projected band structure of metals. It can be shown that the energies of these states all lie within the band gap. As a consequence, in the crystal these states are characterized by an imaginary wavenumber
Wavenumber
In the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...
leading to an exponential decay into the bulk.
Shockley states and Tamm states
In the discussion of surface states, one generally distinguishes between Shockley states and Tamm states. However there is no real physical distinction between the two terms, only the mathematical approach in describing surface states is different.- Historically, surface states that arise as solutions to the Schrödinger equationSchrödinger equationThe Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
in the framework of the nearly-free electron approximationNearly-free electron modelIn solid-state physics, the nearly-free electron model is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model is closely related to the more conceptual Empty Lattice Approximation...
for clean and ideal surfaces, are called Shockley states. Shockley states are thus states that arise due to the change in the electron potential associated solely with the crystal termination. This approach is suited to describe normal metals and some narrow gap semiconductors. Figures 1 and 2 are examples of Shockley states, derived using the nearly free electron approximation.
- Surface states that are calculated in the framework of a tight-binding model are often called Tamm states. In the tight binding approach, the electronic wave functions are usually expressed as linear combinations of atomic orbitals (LCAO). In contrast to the nearly free electron model used to describe the Shockley states, the Tamm states are suitable to describe also transition metalTransition metalThe term transition metal has two possible meanings:*The IUPAC definition states that a transition metal is "an element whose atom has an incomplete d sub-shell, or which can give rise to cations with an incomplete d sub-shell." Group 12 elements are not transition metals in this definition.*Some...
s and wide gap semiconductors.
Surface states in metals
A simple model for the derivation of the basic properties of states at a metal surface is a semi-infinite periodic chain of identical atoms. In this model, the termination of the chain represents the surface, where the potential attains the value V0 of the vacuum in the form of a step functionStep function
In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals...
, figure 1. Within the crystal the potential is assumed periodic with the periodicity a of the lattice.
The Shockley states are then found as solutions to the one-dimensional single electron Schrödinger equation
with the periodic potential
where l is an integer.
The solution must be obtained independently for the two domains z<0 and z>0, where at the domain boundary (z=0) the usual conditions on continuity of the wave function and its derivatives is applied. Since the potential is periodic deep inside the crystal the electronic wave functions must be Bloch wave
Bloch wave
A Bloch wave or Bloch state, named after Felix Bloch, is the wavefunction of a particle placed in a periodic potential...
s here. The solution in the crystal is then a linear combination of an incoming and a wave reflected from the surface. For z>0 the solution will be required to decrease exponentially into the vacuum
The wave function for a state at a metal surface is qualitatively shown in figure 2. It is an extended Bloch wave within the crystal with an exponentially decaying tail outside the surface. The consequence of the tail is a deficiency of negative charge density
Charge density
The linear, surface, or volume charge density is the amount of electric charge in a line, surface, or volume, respectively. It is measured in coulombs per meter , square meter , or cubic meter , respectively, and represented by the lowercase Greek letter Rho . Since there are positive as well as...
just inside the crystal and an increased negative charge density just outside the surface, leading to the formation of a dipole double layer
Double layer
Double layer may refer to:* Double layer , a structure in a plasma and consists of two parallel layers with opposite electrical charge* Double layer , a structure that appears on the surface of an object when it is placed into a liquid...
. The dipole perturbs the potential at the surface leading, for example, to a change of the metal work function
Work function
In solid-state physics, the work function is the minimum energy needed to remove an electron from a solid to a point immediately outside the solid surface...
.
Surface states in semiconductors
whereas at the surface the potential is modeled as a step function of height V0.
The solutions to the Schrödinger equation must be obtained separately for the two domains z < 0 and z > 0. In the sense of the nearly free electron approximation, the solutions obtained for z < 0 will have plain wave character for wave vectors away from the Brillouin zone boundary
, where the dispersion relation will be parabolic, as shown in figure 4.At the Brillouin zone boundaries, Bragg reflection occurs resulting in a standing wave
Standing wave
In physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling...
consisting of a wave with wave vector
Wave vector
In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave , and its direction is ordinarily the direction of wave propagation In...
and wave vector .Here
is a lattice vector of the reciprocal latticeReciprocal lattice
In physics, the reciprocal lattice of a lattice is the lattice in which the Fourier transform of the spatial function of the original lattice is represented. This space is also known as momentum space or less commonly k-space, due to the relationship between the Pontryagin duals momentum and...
(see figure 4).
Since the solutions of interest are close to the Brillouin zone boundary, we set
, where κ is a small quantity. The arbitrary constants A,B are found by substitution into the Schrödinger equation. This leads to the following eigenvaluesdemonstrating the band splitting at the edges of the Brillouin zone
Brillouin zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. The boundaries of this cell are given by planes related to points on the reciprocal lattice. It is found by the same method as for the Wigner–Seitz cell in the Bravais lattice...
, where the width of the forbidden gap is given by 2V. The electronic wave functions deep inside the crystal, attributed to the different bands are given by
Where C is a normalization constant.
Near the surface at z = 0,
the bulk solution has to be fitted to an exponentially decaying solution, which is compatible with the constant potential V0.
It can be shown that the matching conditions can be fulfilled for every possible energy eigenvalue which lies in the allowed band. As in the case for metals, this type of solution represents standing Bloch waves extending into the crystal which spill over into the vacuum
Vacuum
In everyday usage, vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than atmospheric pressure. The word comes from the Latin term for "empty". A perfect vacuum would be one with no particles in it at all, which is impossible to achieve in...
at the surface. A qualitative plot of the wave function is shown in figure 2.
If imaginary values of κ are considered, i.e. κ = - i·q for z ≤ 0 and one defines
one obtains solutions with a decaying amplitude into the crystal
The energy eigenvalues are given by
E is real for large negative z, as required. Also in the range
all energies of the surface states fall into the forbidden gap. The complete solution is again found by matching the bulk solution to the exponentially decaying vacuum solution. The result is a state localized at the surface decaying both into the crystal and the vacuum. A qualitative plot is shown in figure 3.Surface states of a three-dimensional crystal
Quantum wire
In condensed matter physics, a quantum wire is an electrically conducting wire, in which quantum effects are affecting transport properties. Due to the quantum confinement of conduction electrons in the transverse direction of the wire, their transverse energy is quantized into a series of...
can readily be generalized to the case of a three-dimensional crystal. Because of the two-dimensional periodicity of the surface lattice Bloch's theorem must hold for translations parallel to the surface. As a result, the surface states can be written as the product of a Bloch waves with k-values
parallel to the surface and a function representing a one-dimensional surface stateThe energy of this state is increased by a term
so that we havewhere m* is the effective mass of the electron. The matching conditions at the crystal surface, i.e. at z=0, have to be satisfied for each
separately and for each a single, but generally different energy level for the surface state is obtained.True surface states and surface resonances
A surface state is described by the energy and its wave vector parallel to the surface, while a bulk state is characterized by both and wave numbers. In the two-dimensional Brillouin zoneBrillouin zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. The boundaries of this cell are given by planes related to points on the reciprocal lattice. It is found by the same method as for the Wigner–Seitz cell in the Bravais lattice...
of the surface, for each value of
therefore a rod of is extending into the three-dimensional Brillouin zone of the Bulk. Bulk energy bands that are being cut by these rods allow states that penetrate deep into the crystal.One therefore generally distinguishes between true surface states and surface resonances. True surface states are characterized by energy bands that are not degenerate with bulk energy bands. These state are existing in the forbidden energy gap
Band gap
In solid state physics, a band gap, also called an energy gap or bandgap, is an energy range in a solid where no electron states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference between the top of the valence band and the...
only and are therefore localized at the surface, similar to the picture given in figure 3. At energies where a surface and bulk state are degenerate surface and the bulk state can mix, forming a surface resonance. Such state can propagate deep into the bulk similar to Bloch wave
Bloch wave
A Bloch wave or Bloch state, named after Felix Bloch, is the wavefunction of a particle placed in a periodic potential...
s, while retaining an enhanced amplitude close to the surface.
Tamm states
Surface states that are calculated in the framework of a tight-binding model are often called Tamm states. In the tight binding approach, the electronic wave functions are usually expressed as a linear combinations of atomic orbitalsLinear combination of atomic orbitals molecular orbital method
A linear combination of atomic orbitals or LCAO is a quantum superposition of atomic orbitals and a technique for calculating molecular orbitals in quantum chemistry. In quantum mechanics, electron configurations of atoms are described as wavefunctions...
(LCAO), see figure 5. In this picture, it is easy to comprehend that the existence of a surface will give rise to surface states with energies different from the energies of the bulk states: Since the atoms residing in the topmost surface layer are missing their bonding partners on one side their orbitals have less overlap with the orbitals of neighboring atoms. The splitting and shifting of energy levels of the atoms forming the crystal is therefore smaller at the surface than in the bulk.
If a particular orbital
Atomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...
is responsible for the chemical bonding, e.g. the sp3 hybrid in Si or Ge, it is strongly affected by the presence of the surface, bonds are broken, and the remaining lobes of the orbital stick out from the surface. They are called dangling bond
Dangling bond
In chemistry, a dangling bond is an unsatisfied valence on an immobilised atom.In order to gain enough electrons to fill their valence shells , many atoms will form covalent bonds with other atoms. In the simplest case, that of a single bond, two atoms each contribute one unpaired electron, and the...
s. The energy levels of such states are expected to significantly shift from the bulk values.
In contrast to the nearly free electron model used to describe the Shockley states, the Tamm states are suitable to describe also transition metal
Transition metal
The term transition metal has two possible meanings:*The IUPAC definition states that a transition metal is "an element whose atom has an incomplete d sub-shell, or which can give rise to cations with an incomplete d sub-shell." Group 12 elements are not transition metals in this definition.*Some...
s and wide bandgap semiconductors
Wide bandgap semiconductors
Wide bandgap semiconductors are semiconductor materials with electronic band gaps larger than one or two electronvolts . The exact threshold of "wideness" often depends on the application, such as optoelectronic and power devices...
.
Extrinsic surface states
Surface states originating from clean and well ordered surfaces are usually called intrinsic. These states include states originating from reconstructed surfaces, where the two-dimensional translational symmetry gives rise to the band structure in the k space of the surface.Extrinsic surface states are usually defined as states not originating from a clean and well ordered surface. Surfaces that are fit into the category extrinsic are :
- Surfaces with defects, where the translational symmetry of the surface is broken.
- Surfaces with adsorbates
- Interfaces between two material such as a semiconductor-oxide or semiconductor-metal interfaces
- Interfaces between solid and liquid phases.
Generally for extrinsic surface states is that they cannot easily be characterized in terms of their chemical, physical or structural properties.
Angle resolved photoemission spectroscopy (ARPES)
An experimental technique to measure the dispersion of surface states is angle resolved photoemission spectroscopy (ARPESARPES
Angle-resolved photoemission spectroscopy , also known as ARUPS , is a direct experimental technique to observe the distribution of the electrons in the reciprocal space of solids...
) or angle resolved ultraviolet photoelectron spectroscopy
Ultraviolet photoelectron spectroscopy
Ultraviolet photoelectron spectroscopy refers to the measurement of kinetic energy spectra of photoelectrons emitted by molecules which have absorbed ultraviolet photons, in order to determine molecular energy levels in the valence region.-Basic Theory:...
(ARUPS).