Hirzebruch surface

Surface states are electronic states 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.

Origin of surface states at condensed matter interfaces

Figure 1. Simplified one-dimensional model of a periodic crystal potential terminating at an ideal surface. At the surface, the model potential jumps abruptly to the vacuum level (solid line). The dashed line represents a more realistic picture, where the potential reaches the vacuum level over some distance.

Figure 2. Real part of the type of solution to the one-dimensional Schrödinger equation, which correspond to the bulk states. These states have Bloch character in the bulk, while decaying exponentially into the vacuum.

Figure 3. Real part of the type of solution to the one-dimensional Schrödinger equation, which correspond to surface states. These states decay into both the vacuum and the bulk crystal and thus represent states localized at the crystal surface.

As stated by Bloch's theorem, eigenstates of the single-electron Schrödinger equation with a perfectly periodic potential, a crystal, are Bloch waves

$\begin{array}{cc}{\Psi }_{n𝒌}& ={\mathrm{e}}^{i𝒌\cdot 𝒓}{u}_{n𝒌}(𝒓).\end{array}$

Here $u_{n𝒌}(𝒓)$ 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 metals and semiconductors. 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.

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 and the Russian physicist Igor Tamm. 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 equation 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. 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 metals and wide gap semiconductors.

Topological surface states

DTZ's public sale group in Singapore auctions all forms of residential, workplace and retail properties, outlets, homes, lodges, boarding homes, industrial buildings and development websites. Auctions are at present held as soon as a month.

We will not only get you a property at a rock-backside price but also in an space that you've got longed for. You simply must chill out back after giving us the accountability. We will assure you 100% satisfaction. Since we now have been working in the Singapore actual property market for a very long time, we know the place you may get the best property at the right price. You will also be extremely benefited by choosing us, as we may even let you know about the precise time to invest in the Singapore actual property market.

The Hexacube is offering new ec launch singapore business property for sale Singapore investors want to contemplate. Residents of the realm will likely appreciate that they'll customize the business area that they wish to purchase as properly. This venture represents one of the crucial expansive buildings offered in Singapore up to now. Many investors will possible want to try how they will customise the property that they do determine to buy by means of here. This location has offered folks the prospect that they should understand extra about how this course of can work as well.

Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.

is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease

In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. 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 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.

SEE http://rmp.aps.org/abstract/RMP/v82/i4/p3045_1

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 V₀ of the vacuum in the form of a step function, 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

$\begin{array}{ccc}[-\frac{{\hslash }^2}{2m}\frac{d^2}{d{z}^2}+V(z)]\Psi (z)& =& E\Psi (z),\end{array}$

with the periodic potential

$\begin{array}{c}V(z)=\{\begin{array}{cc}V(z+la),& \text{for}z<0\\ V_0,& \text{for}z>0\end{array},\end{array}$

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 waves 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

$\begin{array}{ccc}\Psi (z)& =& \{\begin{array}{cc}B{u}_{-k}{e}^{-ikz}+C{u}_k{e}^{ikz},& \text{for}z<0\\ A\exp [-\sqrt{2m(V_0-E)}\frac{z}{\hslash }],& \text{for}z>0\end{array},\end{array}$

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.

Surface states in semiconductors

Figure 4. Electronic band structure in the nearly free electron picture. Away from the Brillouin zone boundary the electron wave function has plane wave character and the dispersion relation is parabolic. At the Brillouin zone boundary the wave function is a standing wave composed of an incoming and a Bragg-reflected wave. This ultimately leads to the creation of a band gap.

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

$\begin{array}{cc}V(z)& =V[\exp \left(i\frac{2\pi z}{a}\right)+\exp (-i\frac{2\pi z}{a})]\\ & =2V\cos \left(\frac{2\pi z}{a}\right),\\ \end{array}$

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 plane wave character for wave vectors away from the Brillouin zone boundary $k=\pm \pi /a$, 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 $k=\pi /a$ and wave vector $k=-\pi /a$.

$\begin{array}{ccc}\Psi (z)& =& A{e}^{ikz}+B{e}^{i[k-(2\pi /a)]z}.\end{array}$

Here $G=2\pi /a$ 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 $k_{\perp }=\pi /a+\kappa$, 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

$\begin{array}{ccc}E& =& \frac{{\hslash }^2}{2m}{(\frac{\pi }{a}+\kappa )}^2\pm |V|[-\frac{{\hslash }^2\pi \kappa }{ma|V|}\pm \sqrt{{\left(\frac{{\hslash }^2\pi \kappa }{ma|V|}\right)}^2+1}]\end{array}$

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

$\begin{array}{ccc}{\Psi }_i& =& C{e}^{i\kappa z}(e^{i\pi z/a}+[-\frac{{\hslash }^2\pi \kappa }{ma|V|}\pm \sqrt{{\left(\frac{{\hslash }^2\pi \kappa }{ma|V|}\right)}^2+1}]e^{-i\pi z/a})\end{array}$

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.

$\begin{array}{ccc}{\Psi }_0& =& D\exp [-\sqrt{\frac{2m}{{\hslash }^2}(V_0-E)z}]\end{array}$

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

$\begin{array}{ccc}i\sin (2\delta )& =& -i\frac{{\hslash }^2\pi q}{maV}\end{array}$

one obtains solutions with a decaying amplitude into the crystal

$\begin{array}{ccc}{\Psi }_i(z\le 0)& =& F{e}^{qz}[\exp \left[i(\frac{\pi }{a}z\pm \delta )\right]\pm \exp [-i(\frac{\pi }{a}z\pm \delta )]]e^{\mp i\delta }\end{array}$

The energy eigenvalues are given by

$\begin{array}{ccc}E& =& \frac{{\hslash }^2}{2m}[{\left(\frac{\pi }{a}\right)}^2-q^2]\pm V\sqrt{1-{\left(\frac{{\hslash }^2\pi q}{maV}\right)}^2}\end{array}$

E is real for large negative z, as required. Also in the range $0\le q\le q_{max}=maV{\hslash }^2\pi$ 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

Figure 5. Atomic like orbitals of a Pt-atom. The orbitals shown are part of the double-zeta basis set used in density functional calculations. The orbitals are indexed according to the usual quantum numbers (n,l,m).

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 ${\mathbf{\text{𝐤}}}_{||}=(k_x,k_y)$ parallel to the surface and a function representing a one-dimensional surface state

$\begin{array}{ccc}{\Psi }_0(\mathbf{\text{𝐫}})& =& {\psi }_0(z)u_{{\mathbf{\text{𝐤}}}_{||}}({\mathbf{\text{𝐫}}}_{||})e^{-i{\mathbf{\text{𝐫}}}_{||}\cdot {\mathbf{\text{𝐤}}}_{||}}\end{array}$

The energy of this state is increased by a term $E_{||}$ so that we have

$\begin{array}{c}{E}_s=E_0+\frac{{\hslash }^2{\mathbf{\text{𝐤}}}_{||}^2}{2m^{\ast }},\end{array}$

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 ${\mathbf{\text{𝐤}}}_{||}$ separately and for each ${\mathbf{\text{𝐤}}}_{||}$ 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 $E_s$ and its wave vector ${\mathbf{\text{𝐤}}}_{||}$ parallel to the surface, while a bulk state is characterized by both ${𝐤}_{||}$ and ${𝐤}_{\perp }$ wave numbers. In the two-dimensional Brillouin zone of the surface, for each value of ${𝐤}_{||}$ therefore a rod of ${𝐤}_{\perp }$ 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 waves, 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 orbitals (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 bonds. 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 metals and wide bandgap semiconductors.

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 :

1. Surfaces with defects, where the translational symmetry of the surface is broken.
2. Surfaces with adsorbates
3. Interfaces between two material such as a semiconductor-oxide or semiconductor-metal interfaces
4. Interfaces between solid and liquid phases.

Generally, extrinsic surface states 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 (ARPES) or angle resolved ultraviolet photoelectron spectroscopy (ARUPS).

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.