PHY.K02UF Molecular and Solid State Physics

Semiconductors

A pure semiconductor is a material with poor electrical conductivity. However the conductivity can be increased by order of magnitude by the addition of a small concentration of dopant atoms or by applying an electric field. Semiconductors are important for computation, communication, power electronics, solar cells, lighting, and display technologies.

For semiconductors, the chemical potential lies in a band gap and the band gap is less than about 3 eV. To calculate the properties of a semiconductor, first the crystal structure must be determined. This can be done by x-ray diffraction.

 Some common crystal  structures for semiconductors Diamond C, Si, Ge, α-Sn Zincblende  GaAs, InP, GaP, InAs, AlAs Wurzite ZnO, GaN, AlN, CdSe
• Electron states in a semiconductor

You should be able to look at a bandstructure diagram for a semiconductor and be able to identify the conduction band, the valence band, the energy gap, whether the semiconductor is direct or in direct, and to be able to determine the effective masses of electrons and holes.

Some example electronic bandstructures are: Si, GaAs, GaN, GaP, Ge, InAs, 6H SiC.

Some calculated electron density of states: Si diamond, GaN, 6H SiC, GaAs, GaP, Ge, InAs.

Many propertities of semiconductors related to the electron density of states can be calcuated using the Boltzmann approximation. There is also a table summarizing the thermodynamic properties of semiconductors in the Boltzmann approximation.

Important results of the Boltzmann approximation are:

$$\text{electron concentration in the conduction band}\qquad n = N_c(T)\exp\left(\frac{\mu-E_c}{k_BT}\right)$$ $$\text{hole concentration in the valence band}\qquad p = N_v(T)\exp\left(\frac{E_v-\mu}{k_BT}\right)$$ $$\text{law of mass action}\qquad n_i^2 = pn = N_c(T)N_v(T)\exp\left(\frac{-E_g}{k_BT}\right)$$
• Phonon states

From the change in energy as a function of lattice constant, an effective spring constant between the atoms can be determined and this can be used to calculate the phonon dispersion relation and the phonon density of states. The phonon density of states can be used to calculate the phonon contribution to the thermodynamic properties. At room temperature, the phonon contribution to the thermodynamic properties usually dominates over the electron contribution of semiconductors.

Phonon densities of states: Si, α-Sn, ZnO (rocksalt), ZnO (zincblende), ZnO (wurtzite).

From the phonon densities of states, the following properties can be calculated: Energy spectral density u(ω,T), Internal energy density u(T), Specific heat cv(T), Helmholtz free energy density f(T), Entropy density s(T).

• Intrinsic semiconductors are undoped.
$$\text{carrier densities}\qquad n = p = \sqrt{N_cN_v}\exp\left(\frac{-E_g}{2k_BT}\right)$$ $$\text{chemical potential}\qquad\mu = \frac{E_c+E_v}{2}+\frac{k_BT}{2}\ln \left(\frac{N_v}{N_c}\right)$$
• Extrinsic semiconductors are have been doped with a donor concentration $N_d$ or an acceptor concentration $N_a$ or they have been doped with both donors and acceptors.

The temperature dependence of the carriers concentrations of extrinsic semiconductors can be divided into three regimes: freeze out, the extrinsic regime, and the intrinsic regime.

In the extrinsic regime:

 n-type p-type electron concentration $n=N_d-N_a$ $n=\frac{n_i^2}{N_a-N_d}$ hole concentration $p=\frac{n_i^2}{N_d-N_a}$ $p=N_a-N_d$ chemical potential $\mu= E_c - k_BT \ln\left(\frac{N_c}{N_d - N_a}\right)$ $\mu= E_v + k_BT \ln\left(\frac{N_v}{N_a - N_d}\right)$

• You should able to describe what a pn-junction is and how a light emitting diode and a solar cell work.

 Properties Si Ge GaAs Bandgap Eg $1.166-\frac{4.73\times 10^{-4}T^2}{T+636}$ eV (indirect) $0.7437-\frac{4.77\times 10^{-4}T^2}{T+235}$ eV (indirect) $1.519-\frac{5.41\times 10^{-4}T^2}{T+204}$ eV (direct) Effective density of states in conduction band (300 K) Nc 2.78 × 1025 m-3 1.04 × 1025 m-3 4.45 × 1023 m-3 Effective density of states in valence band (300 K) Nv 9.84 × 1024 m-3 6.0 × 1024 m-3 7.72 × 1024 m-3 Effective mass electrons m*/m0 ml* = 0.98 mt* = 0.19 ml* = 1.64 mt* = 0.082 m* = 0.067 Effective mass holes m*/m0 mlh* = 0.16 mhh* = 0.49 mlh* = 0.044 mhh* = 0.28 mlh* = 0.082 mhh* = 0.45 Crystal structure diamond diamond zincblende Density 2.328 g/cm³ 5.3267 g/cm³ 5.32 g/cm³ Atoms/m³ 5.0 × 1028 4.42 × 1028 4.42 × 1028