Surface states are
electronic state
A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical particles, which can have any amount of energy. The t ...
s found at the
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
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
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or ' ...
from the bulk material to the
vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often di ...
. In the weakened potential at the surface, new electronic states can be formed, so called surface states.
Origin at condensed matter interfaces
![Solution2](https://upload.wikimedia.org/wikipedia/commons/1/1e/Solution2.PNG)
As stated by
Bloch's theorem
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who d ...
, eigenstates of the single-electron Schrödinger equation with a perfectly periodic potential, a crystal, are
Bloch waves
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who di ...
:
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 surface states with wave functions localized close to the crystal surface.
The first type of solution can be obtained for both
metal
A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typicall ...
s and
semiconductor
A semiconductor is a material which has an electrical conductivity value falling between that of a conductor, such as copper, and an insulator, such as glass. Its resistivity falls as its temperature rises; metals behave in the opposite way. ...
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
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
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,
named after the American physicist
William Shockley
William Bradford Shockley Jr. (February 13, 1910 – August 12, 1989) was an American physicist and inventor. He was the manager of a research group at Bell Labs that included John Bardeen and Walter Brattain. The three scientists were jointl ...
and the Russian physicist
Igor Tamm
Igor Yevgenyevich Tamm ( rus, И́горь Евге́ньевич Тамм , p=ˈiɡərʲ jɪvˈɡʲenʲjɪvitɕ ˈtam , a=Ru-Igor Yevgenyevich Tamm.ogg; 8 July 1895 – 12 April 1971) was a Soviet physicist who received the 1958 Nobel Prize in ...
. There is no strict physical distinction between the two types of states, but the qualitative character and the mathematical approach used in describing them is different.
* Historically, surface states that arise as solutions to the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
in the framework of the
nearly free electron 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. Figure 3 shows an example of a Shockley state, derived using the nearly free electron approximation. Within the crystal, Shockley states resemble exponentially-decaying Bloch waves.
* Surface states that are calculated in the framework of a
tight-binding model
In solid-state physics, the tight-binding model (or TB model) is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon Quantum superposition, superposition of wave functions for isolated ...
are often called Tamm states. In the tight binding approach, the electronic
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
s are usually expressed as
linear combinations of atomic orbitals
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 wave ...
(LCAO). In contrast to the nearly free electron model used to describe the Shockley states, the Tamm states are suitable to describe also
transition metal
In chemistry, a transition metal (or transition element) is a chemical element in the d-block of the periodic table (groups 3 to 12), though the elements of group 12 (and less often group 3) are sometimes excluded. They are the elements that ca ...
s and
wide gap semiconductors
Wide-bandgap semiconductors (also known as WBG semiconductors or WBGSs) are semiconductor materials which have a larger band gap than conventional semiconductors. Conventional semiconductors like silicon have a bandgap in the range of 0.6 ...
.
Qualitatively, Tamm states resemble localized atomic or molecular orbitals at the surface.
Topological surface states
All materials can be classified by a single number, a topological invariant; this is constructed out of the bulk electronic wave functions, which are integrated in over the Brillouin zone, in a similar way that the
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
is calculated in
geometric topology. In certain materials the topological invariant can be changed when certain bulk energy bands invert due to strong spin-orbital coupling. At the interface between an insulator with non-trivial topology, a so-called topological insulator, and one with a trivial topology, the interface must become metallic. More over, the surface state must have linear Dirac-like dispersion with a crossing point which is protected by time reversal symmetry. Such a state is predicted to be robust under disorder, and therefore cannot be easily localized.
Shockley states
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 V
0 of the vacuum in the form of a
step 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. Informally speaking, a step function is a piecewise constant function having onl ...
, 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, and ''P'' is the normalization factor.
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 are applied. Since the potential is periodic deep inside the crystal, the electronic
wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
s must be
Bloch wave
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who d ...
s here. The solution in the crystal is then a linear combination of an incoming wave 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
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in ...
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
In solid-state physics, the work function (sometimes spelt workfunction) is the minimum thermodynamic work (i.e., energy) needed to remove an electron from a solid to a point in the vacuum immediately outside the solid surface. Here "immediately" ...
.
Surface states in semiconductors
![BandstructureNFE](https://upload.wikimedia.org/wikipedia/commons/4/4a/BandstructureNFE.PNG)
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 V
0.
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 plane 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
In physics, a wave vector (or wavevector) is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave (inversely proportional to the wavelength), ...
and wave vector
.
:
Here
is a
lattice vector
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an or ...
of the
reciprocal lattice
In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...
(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
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal 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 ''V
0''.
:
It can be shown that the matching conditions can be fulfilled for every possible energy
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
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
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or " void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often di ...
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
![Orbitals](https://upload.wikimedia.org/wikipedia/commons/b/be/Orbitals.PNG)
The results for surface states of a
monatomic 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.
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 zone
In mathematics and solid state physics, the first Brillouin zone is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal 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 band
In solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called ''band gaps'' or ...
s 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 states exist 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 a bulk state are degenerate, the surface and the bulk state can mix, forming a
surface resonance. Such a state can propagate deep into the bulk, similar to
Bloch wave
In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who d ...
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
In solid-state physics, the tight-binding model (or TB model) is an approach to the calculation of electronic band structure using an approximate set of wave functions based upon Quantum superposition, superposition of wave functions for isolated ...
are often called Tamm states. In the tight binding approach, the electronic wave functions are usually expressed as a
linear combination of atomic orbitals
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 wavefun ...
(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 ''sp
3'' 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
In chemistry, a dangling bond is an unsatisfied valence on an immobilized atom. An atom with a dangling bond is also referred to as an immobilized free radical or an immobilized radical, a reference to its structural and chemical similarity to a f ...
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
In chemistry, a transition metal (or transition element) is a chemical element in the d-block of the periodic table (groups 3 to 12), though the elements of group 12 (and less often group 3) are sometimes excluded. They are the elements that ca ...
s and
wide-bandgap semiconductor
Wide-bandgap semiconductors (also known as WBG semiconductors or WBGSs) are semiconductor materials which have a larger band gap than conventional semiconductors. Conventional semiconductors like silicon have a bandgap in the range of 0.6 ...
s.
Extrinsic surface states
Surface states originating from clean and well ordered surfaces are usually called ''
intrinsic
In science and engineering, an intrinsic property is a property of a specified subject that exists itself or within the subject. An extrinsic property is not essential or inherent to the subject that is being characterized. For example, mass ...
''. 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
In science and engineering, an intrinsic property is a property of a specified subject that exists itself or within the subject. An extrinsic property is not essential or inherent to the subject that is being characterized. For example, mass ...
'' surface states are usually defined as states not originating from a clean and well ordered surface. Surfaces that fit into the category ''extrinsic'' are:
# Surfaces with defects, where the translational symmetry of the surface is broken.
# Surfaces with adsorbates
# Interfaces between two materials, such as a semiconductor-oxide or semiconductor-metal interface
# Interfaces between solid and liquid phases.
Generally, ''extrinsic'' surface states cannot easily be characterized in terms of their chemical, physical or structural properties.
Experimental observation
Angle resolved photoemission spectroscopy
An experimental technique to measure the dispersion of surface states is angle resolved photoemission spectroscopy (
ARPES
Angle-resolved photoemission spectroscopy (ARPES) is an experimental technique used in condensed matter physics to probe the allowed energies and momenta of the electrons in a material, usually a crystalline solid. It is based on the photoelec ...
) or angle resolved
ultraviolet photoelectron spectroscopy
Ultraviolet photoelectron spectroscopy (UPS) refers to the measurement of kinetic energy spectra of photoelectrons emitted by molecules which have absorbed ultraviolet photons, in order to determine molecular orbital energies in the valence regio ...
(ARUPS).
Scanning tunneling microscopy
The surface state dispersion can be measured using a
scanning tunneling microscope
A scanning tunneling microscope (STM) is a type of microscope used for imaging surfaces at the atomic level. Its development in 1981 earned its inventors, Gerd Binnig and Heinrich Rohrer, then at IBM Zürich, the Nobel Prize in Physics in 1986 ...
; in these experiments, periodic modulations in the surface state density, which arise from scattering off of surface impurities or step edges, are measured by an STM tip at a given bias voltage. The wavevector versus bias (energy) of the surface state electrons can be fit to a free-electron model with effective mass and surface state onset energy.
A recent new theory
A recent new theory
[
]
[
]
[
]
tries to understand the electronic states in ideal crystals of finite size based
on the mathematical theory of periodic differential equations.
This theory provides some fundamental new understandings of surface states.
A one-dimensional finite crystal with two ends at
and
(
: potential period,
:positive integer)
''always has one and only one state whose energy and properties depend on
but not
for each band gap.'' This state is either a band-edge state or a surface state in the band gap.
Therefore:
#Truncation of the lattice's periodic potential may or may not leads to a surface state in a band gap.
#An ideal one-dimensional crystal of finite length
with two ends can have at most ''only one surface state at one end'' in each band gap.
#The fundamental property of a surface state is that its existence and properties depend on the location of the periodicity truncation.
An ideal simple three-dimensional finite crystal may have vertex-like, edge-like, surface-like, and bulk-like states.
The properties and energy of each vertex-like state depend on three
(boundary in the
direction), but neither one
(size in the
direction);
The properties and energy of each edge-like state depend on two
and the other
, but neither one of the two corresponding
nor the other
;
The properties and energy of each surface-like state depend on one
and the other two
, but neither the corresponding
nor the other two
;
The properties and energy of each bulk-like state depend on three
, but neither one
.
For such a three-dimensional finite crystals, among these states from the same bulk energy band, the following general relations exist:
The energy of every vertex-like state >
The energy of every edge-like state >
The energy of every relevant surface-like state >
The energy of every relevant bulk-like state.
Therefore, a surface state is always in a band gap is ''only valid for one-dimensional cases.''
Subsequent investigations have confirmed many fundamental understandings of the new theory.
References
{{reflist, 2
Materials science
Electronic band structures
Semiconductor structures