Charge carrier density

This article is about the number of individual electrons or holes per volume. For the electric charge per volume, see charge density. For the number of possible states per energy range, see density of states.

Charge carrier density, also known as carrier concentration, denotes the number of charge carriers per volume. In SI units, it is measured in m⁻³. As with any density, in principle it can depend on position. However, usually carrier concentration is given as a single number, and represents the average carrier density over the whole material.

Charge carrier densities involve equations concerning the electrical conductivity, related phenomena like the thermal conductivity, and chemicals bonds like covalent bond.

Calculation

The carrier density is usually obtained theoretically by integrating the density of states over the energy range of charge carriers in the material (e.g. integrating over the conduction band for electrons, integrating over the valence band for holes).

If the total number of charge carriers is known, the carrier density can be found by simply dividing by the volume. To show this mathematically, charge carrier density is a particle density, so integrating it over a volume ${\displaystyle V}$ gives the number of charge carriers ${\displaystyle N}$ in that volume $${\displaystyle N=\int _{V}n(\mathbf {r} )\,dV.}$$ where ${\displaystyle n(\mathbf {r} )}$ is the position-dependent charge carrier density.

If the density does not depend on position and is instead equal to a constant ${\displaystyle n_{0}}$ this equation simplifies to $${\displaystyle N=V\cdot n_{0}.}$$

Semiconductors

The carrier density is important for semiconductors, where it is an important quantity for the process of chemical doping. Using band theory, the electron density,${\displaystyle n_{0}}$ is number of electrons per unit volume in the conduction band. For holes, ${\displaystyle p_{0}}$ is the number of holes per unit volume in the valence band. To calculate this number for electrons, we start with the idea that the total density of conduction-band electrons, ${\displaystyle n_{0}}$, is just adding up the conduction electron density across the different energies in the band, from the bottom of the band ${\displaystyle E_{c}}$ to the top of the band ${\displaystyle E_{\text{top}}}$.

$${\displaystyle n_{0}=\int _{E_{c}}^{E_{\text{top}}}N(E)\,dE}$$

Because electrons are fermions, the density of conduction electrons at any particular energy, ${\displaystyle N(E)}$ is the product of the density of states, ${\displaystyle g(E)}$ or how many conducting states are possible, with the Fermi–Dirac distribution, ${\displaystyle f(E)}$ which tells us the portion of those states which will actually have electrons in them $${\displaystyle N(E)=g(E)f(E)}$$

In order to simplify the calculation, instead of treating the electrons as fermions, according to the Fermi–Dirac distribution, we instead treat them as a classical non-interacting gas, which is given by the Maxwell–Boltzmann distribution. This approximation has negligible effects when the magnitude ${\displaystyle |E-E_{f}|\gg k_{\text{B}}T}$, which is true for semiconductors near room temperature. This approximation is invalid at very low temperatures or an extremely small band-gap.

$${\displaystyle f(E)={\frac {1}{1+e^{\frac {E-E_{f}}{k_{\text{B}}T}}}}\approx e^{-{\frac {E-E_{f}}{k_{\text{B}}T}}}}$$

The three-dimensional density of states is:$${\displaystyle g(E)={\frac {1}{2\pi ^{2}}}\left({\frac {2m^{*}}{\hbar ^{2}}}\right)^{\frac {3}{2}}{\sqrt {E-E_{0}}}}$$

After combination and simplification, these expressions lead to:

$${\displaystyle n_{0}=2\left({\frac {m^{*}k_{\text{B}}T}{2\pi \hbar ^{2}}}\right)^{3/2}e^{-{\frac {E_{c}-E_{f}}{k_{\text{B}}T}}}}$$

Here ${\displaystyle m^{*}}$ is the effective mass of the electrons in that particular semiconductor, and the quantity ${\displaystyle E_{c}-E_{f}}$ is the difference in energy between the conduction band and the Fermi level, which is half the band gap, ${\displaystyle E_{g}}$:

$${\displaystyle E_{g}=2(E_{c}-E_{f})}$$

A similar expression can be derived for holes. The carrier concentration can be calculated by treating electrons moving back and forth across the bandgap just like the equilibrium of a reversible reaction from chemistry, leading to an electronic mass action law. The mass action law defines a quantity ${\displaystyle n_{i}}$ called the intrinsic carrier concentration, which for undoped materials:

$${\displaystyle n_{i}=n_{0}=p_{0}}$$

The following table lists a few values of the intrinsic carrier concentration for intrinsic semiconductors, in order of increasing band gap.

Material	Carrier density (1/cm3) at 300K
Germanium[1]	2.33×1013
Silicon[2]	9.65×109
Gallium Arsenide[3]	2.1×106
3C-SiC[4]	10
6H-SiC[4]	2.3×10−6
4H-SiC[4]	8.2×10−9
Gallium nitride[4]	1.9×10−10
Diamond[4]	1.6×10−27

These carrier concentrations will change if these materials are doped. For example, doping pure silicon with a small amount of phosphorus will increase the carrier density of electrons, n. Then, since n > p, the doped silicon will be a n-type extrinsic semiconductor. Doping pure silicon with a small amount of boron will increase the carrier density of holes, so then p > n, and it will be a p-type extrinsic semiconductor.

Metals

The carrier density is also applicable to metals, where it can be estimated from the simple Drude model. In this case, the carrier density (in this context, also called the free electron density) can be estimated by:^[5]

$${\displaystyle n={\frac {N_{\text{A}}Z\rho _{m}}{m_{a}}}}$$

Where ${\displaystyle N_{\text{A}}}$ is the Avogadro constant, Z is the number of valence electrons, ${\displaystyle \rho _{m}}$ is the density of the material, and ${\displaystyle m_{a}}$ is the atomic mass. Since metals can display multiple oxidation numbers, the exact definition of how many "valence electrons" an element should have in elemental form is somewhat arbitrary, but the following table lists the free electron densities given in Ashcroft and Mermin, which were calculated using the formula above based on reasonable assumptions about valence, ${\displaystyle Z}$, and with mass densities, ${\displaystyle \rho _{m}}$ calculated from experimental crystallography data.^[5]

Material	Number of valence electrons	Carrier density (1/cm3) at 300K
Copper	1	8.47×1022
Silver	1	5.86×1022
Gold	1	5.90×1022
Beryllium	2	2.47×1023
Magnesium	2	8.61×1022
Calcium	2	4.61×1022
Strontium	2	3.55×1022
Barium	2	3.15×1022
Niobium	1	5.56×1022
Iron	2	1.70×1023
Manganese	2	1.65×1023
Zinc	2	1.32×1023
Cadmium	2	9.27×1022
Aluminum	3	1.81×1023
Gallium	3	1.54×1023
Indium	3	1.15×1023
Thallium	3	1.05×1023
Tin	4	1.48×1023
Lead	4	1.32×1023
Bismuth	5	1.41×1023
Antimony	5	1.65×1023

The values for n among metals inferred for example by the Hall effect are often on the same orders of magnitude, but this simple model cannot predict carrier density to very high accuracy.

Measurement

The density of charge carriers can be determined in many cases using the Hall effect,^[6] the voltage of which depends inversely on the carrier density.

References

1. O. Madelung, U. Rössler, M. Schulz (2002). "Germanium (Ge), intrinsic carrier concentration". Group IV Elements, IV-IV and III-V Compounds. Part b – Electronic, Transport, Optical and Other Properties. Landolt-Börnstein – Group III Condensed Matter. pp. 1–3. doi:10.1007/10832182_503. ISBN 978-3-540-42876-3.{{cite book}}: CS1 maint: multiple names: authors list (link)
2. Pietro P. Altermatt, Andreas Schenk, Frank Geelhaar, Gernot Heiser (2003). "Reassessment of the intrinsic carrier density in crystalline silicon in view of band-gap narrowing". Journal of Applied Physics. 93 (3): 1598. Bibcode:2003JAP....93.1598A. doi:10.1063/1.1529297.{{cite journal}}: CS1 maint: multiple names: authors list (link)
3. Rössler, U. (2002). "Gallium arsenide (GaAs), intrinsic carrier concentration, electrical and thermal conductivity". Group IV Elements, IV-IV and III-V Compounds. Part b – Electronic, Transport, Optical and Other Properties. Landolt-Börnstein – Group III Condensed Matter. pp. 1–8. doi:10.1007/10832182_196. ISBN 978-3-540-42876-3.
4. Gachovska, Tanya K.; Hudgins, Jerry L. (2018). "SiC and GaN Power Semiconductor Devices". Power Electronics Handbook. Elsevier. p. 98. doi:10.1016/b978-0-12-811407-0.00005-2. ISBN 9780128114070.
5. Ashcroft, Mermin. Solid State Physics. pp. 4–5.
6. Edwin Hall (1879). "On a New Action of the Magnet on Electric Currents". American Journal of Mathematics. 2 (3): 287–92. doi:10.2307/2369245. JSTOR 2369245. S2CID 107500183. Archived from the original on 27 July 2011.