Materials Science and Engineering/Doctoral review questions/Daily Discussion Topics/01132008

Carrier Model Equations

Density of States and Fermi Function

$g_{c}(E)={\frac {m_{n}*{\sqrt {2m_{n}*(E-E_{c})}}}{\pi ^{2}\hbar ^{3}}},E\geq E_{c}$

$g_{v}(E)={\frac {m_{p}*{\sqrt {2m_{p}*(E_{v}-E)}}}{\pi ^{2}\hbar ^{3}}},E\leq E_{c}$

 $f(E)={\frac {1}{1+e^{(E-E_{F})/kT}}}$

Carrier Concentration Relationships

$n=N_{C}{\frac {2}{\sqrt {\pi }}}F_{1/2}(\eta _{c})$

$p=N_{V}{\frac {2}{\sqrt {\pi }}}F_{1/2}(\eta _{c})$

$N_{C}=2\left[{\frac {m_{n}*kT}{2\pi \hbar ^{2}}}\right]^{3/2}$

$N_{V}=2\left[{\frac {m_{p}*kT}{2\pi \hbar ^{2}}}\right]^{3/2}$

$n=N_{C}e^{(E_{F}-E_{C})/kT}$

$p=N_{v}e^{E_{V}-E_{F})/kT}$

$n=n_{i}e^{(E_{F}-E_{i})/kT}$

$p=n_{i}e^{(E_{i}-E_{F})/kT}$

ni, np-Product, and Charge Neutrality

$n_{i}={\sqrt {N_{c}N_{v}}}e^{-E_{G}/2kT}$

$np=n_{i}^{2}$

$p-n+N_{D}-N_{A}\;$

n,p, and Fermi Level Computational Relationships

$n={\frac {N_{D}-N_{A}}{2}}+\left[\left({\frac {N_{D}-N_{A}}{2}}\right)^{2}+n_{i}^{2}\right]^{1/2}$

$n\approx N_{D}$

$p\approx n_{i}^{2}/N_{D}$

$p\approx N_{A}$

$N_{A}\gg N_{D},N_{A}\gg n_{i}$

$n\approx n_{i}^{2}/N_{A}$

 $E_{i}={\frac {E_{c}+E_{v}}{2}}+{\frac {3}{4}}kT\ln \left({\frac {m_{p}*}{m_{n}*}}\right)$

$E_{F}-E_{i}=kT\ln(n/n_{i})=-kT\ln(p/n_{i})\;$

$N_{D}\gg N_{A},N_{D}\gg n_{i}$

$E_{F}-E_{i}=kT\ln(N_{D}/n_{i})\;$

$N_{A}\gg N_{D},N_{A}\gg n_{i}$

$E_{i}-E_{F}=kT\ln(N_{A}/n_{i})\;$

Einstein Relationship

The Einstein relationship connects $D$ and $\mu$

The net carrier current needs to be zero in the case of equilibrium conditions

$J_{N|drift}+J_{N|diff}=q\mu nE+qD_{N}{\frac {dn}{dx}}=0$

$E={\frac {1}{q}}{\frac {dE_{i}}{dx}}$

$n=n_{i}e^{(E_{F}-E_{i})/kT}$

The change of the Fermi energy with respect to position is 0, $dE_{F}/dx=0$

${\frac {dn}{dx}}=-{\frac {n_{i}}{kT}}e^{(E_{F}-E_{i})/kT}{\frac {dE_{i}}{dx}}=-{\frac {q}{kT}}nE$

$(qnE)\mu _{n}-(qnE){\frac {q}{kT}}D_{N}=0$

The electric field, $E$, does not equal zero

 ${\frac {D_{N}}{\mu _{n}}}={\frac {kT}{q}}$
 ${\frac {D_{P}}{\mu _{p}}}={\frac {kT}{q}}$

The Built-in Potential (Vbi)

The electric field is related to the derivative of the potential with position

$E=-{\frac {dV}{dx}}$

$-\int _{-x_{p}}^{x_{n}}Edx=\int _{V(-x_{p})}^{V(x_{n})}dV=V(x_{n})-V(-x_{p})=V_{bi}$

The following is true under equilibrium

$J_{N}=q\mu _{n}nE+qD_{N}{\frac {dn}{dx}}$

Use the Einstein relationship

$E=-{\frac {D_{N}}{\mu _{n}}}{\frac {dn/dx}{n}}$

$E=-{\frac {kT}{q}}{\frac {dn/dx}{n}}$

$V_{bi}=-\int _{-x_{p}}^{x_{n}}Edx$

$V_{bi}={\frac {kT}{q}}\int _{n(-x_{p})}^{n(x_{n})}{\frac {dn}{n}}$

$V_{bi}={\frac {kT}{q}}\ln \left[{\frac {n(x_{n})}{n(-x_{p})}}\right]$

Consider nondegenerately doped step junction with $N_{D}$ and $N_{A}$ as the n- and p-side doping concentrations.

$n(x_{n})=N_{D}\;$

$n(-x_{p})={\frac {n_{i}^{2}}{N_{A}}}$

 $V_{bi}={\frac {kT}{q}}\ln \left({\frac {N_{A}N_{D}}{n_{i}^{2}}}\right)$

LCAO and Tight-Binding

LCAO

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 [1]. In quantum mechanics, electron configurations of atoms are described as wavefunctions. In mathematical sense, these wave functions are the basis set of functions, the basis functions, which describe the electrons of a given atom. In chemical reactions, orbital wavefunctions are modified, i.e. the electron cloud shape is changed, according to the type of atoms participating in the chemical bond.

It was introduced in 1929 by Sir John Lennard-Jones with the description of bonding in the diatomic molecules of the first main row of the periodic table, but had been used earlier by Pauling for H2+.

A mathematical description is

$\ \phi _{i}=c_{1}\chi _{1}+c_{2}\chi _{2}+c_{3}\chi _{3}+\cdots +c_{n}\chi _{n}$

or

$\ \phi _{i}=\sum _{r}c_{ri}\chi _{r}$

where $\phi _{i}$ is a molecular orbital represented as the sum of n atomic orbitals $\chi _{r}$ (chi), each multiplied by a corresponding coefficient $c_{r}$. The coefficients are the weights of the contributions of the n atomic orbitals to the molecular orbital.

The orbitals are thus expressed as linear combinations of basis functions, and the basis functions are one-electron functions centered on nuclei of the component atoms of the molecule. The atomic orbitals used are typically those of hydrogen-like atoms since these are known analytically i.e. Slater-type orbitals but other choices are possible like Gaussian functions from standard basis sets.

Tight-Binding

In the tight binding model, it is assumed that the full Hamiltonian $H$ of the system may be approximated by the Hamiltonian of an isolated atom centred at each lattice point. The atomic orbitals $\psi _{n}$, which are eigenfunctions of the single atom Hamiltonian $H_{at}$, are assumed to be very small at distances exceeding the lattice constant. This is what is meant by tight-binding. It is further assumed that any corrections to the atomic potential $\Delta U$, which are required to obtain the full Hamiltonian $H$ of the system, are appreciable only when the atomic orbitals are small. A solution to the time-independent single electron Schrödinger equation $\Phi$ is then assumed to be a linear combination of atomic orbitals $\ {\psi }_{n}$

$\Phi ({\vec {r}})=\sum _{n,{\vec {R}}}b_{n,{\vec {R}}}\ \psi _{n}({\vec {r}}-{\vec {R}})$

where n refers to the n-th atomic energy level and $^{\vec {R}}$ is an atomic site in the crystal lattice.

Using this approximate form for the wavefunction, and assuming only the m-th atomic energy level is important for the m-th energy band, the Bloch energies $\varepsilon _{m}$ are of the form

$\varepsilon _{m}({\vec {k}})=E_{m}-{\beta _{m}+\sum _{{\vec {R}}\neq 0}\gamma _{m}({\vec {R}})e^{i{\vec {k}}\cdot {\vec {R}}} \over {b_{m,{\vec {R}}}}+\sum _{{\vec {R}}\neq 0}\alpha _{m}({\vec {R}})e^{i{\vec {k}}\cdot {\vec {R}}}}$,

where $E_{m}$ is the energy of the $m$th atomic level,

$\beta _{m}=-\int \psi _{m}^{*}({\vec {r}})\Delta U({\vec {r}})\Phi ({\vec {r}})d{\vec {r}}$,

$\alpha _{m}({\vec {R}})=\int \psi _{m}^{*}({\vec {r}})\Phi ({\vec {r}}-{\vec {R}})d{\vec {r}}$,

and

$\gamma _{m}({\vec {R}})=-\int \psi _{m}^{*}({\vec {r}})\Delta U({\vec {r}})\Phi ({\vec {r}}-{\vec {R}})d{\vec {r}}$

are the overlap integrals.

The tight-binding model is typically used for calculations of electronic band structure and energy gaps in the static regime. However, in combination with other methods such as the random phase approximation (RPA) model, the dynamic response of systems may also be studied.

Phase Diagram

The Iron–Iron Carbide (Fe–Fe₃C) phase diagram. The percentage of carbon present and the temperature define the phase of the iron carbon alloy and therefore its physical characteristics and mechanical properties. The percentage of carbon determines the type of the ferrous alloy: iron, steel or cast iron

A phase diagram for a binary system displaying a eutectic point.

Complex types of phase diagrams can be constructed, particularly when more than one pure component is present. In that case concentration becomes an important variable. Phase diagrams with more than two dimensions can be constructed that show the effect of more than two variables on the phase of a substance. Phase diagrams can use other variables in addition to or in place of temperature and pressure and composition, for example the strength of an applied electrical or magnetic field and they can also involve substances that take on more than just three states of matter.

One type of phase diagram plots temperature against the relative concentrations of two substances in a binary mixture called a binary phase diagram, as shown at right. Such a mixture can be either a solid solution, eutectic or peritectic, among others. These two types of mixtures result in very different graphs. A textbook example of a eutectic phase diagram is that of the olivine (forsterite and fayalite) system.

Another type of binary phase diagram is a boiling point diagram for a mixture of two components, i. e. chemical compounds. For two particular volatile components at a certain pressure such as atmospheric pressure, a boiling point diagram shows what vapor (gas) compositions are in equilibrium with given liquid compositions depending on temperature. In a typical binary boiling point diagram, temperature is plotted on a vertical axis and mixture composition on a horizontal axis.

A simple example diagram with hypothetical components 1 and 2 in a non-azeotropic mixture is shown at right. The fact that there are two separate curved lines joining the boiling points of the pure components means that the vapor composition is usually not the same as the liquid composition the vapor is in equilibrium with. See Vapor-Liquid Equilibrium for a fuller discussion.

In addition to the above mentioned types of phase diagrams, there are thousands of other possible combinations. Some of the major features of phase diagrams include congruent points, where a solid phase transforms directly into a liquid. There is also the peritectoid, a point where two solid phases combine into one solid phase during heating. The inverse of this, when one solid phase transforms into two solid phases during heating, is called the eutectoid.

External Link

• A collection of about 150 alloy phase diagrams and some PT diagrams