Intrinsic Carrier Concentration in Semiconductors

Derivation

Intrinsic Carrier Concentration in Semiconductors

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Statement

For a nondegenerate semiconductor with parabolic conduction and valence bands, integrating the density of states weighted by the Fermi-Dirac occupation gives thermal equilibrium carrier densities \( n = N_c\, e^{-(E_c-\mu)/k_BT} \) and \( p = N_v\, e^{-(\mu-E_v)/k_BT} \). Their product obeys the law of mass action \( np = N_cN_v\,e^{-E_g/k_BT} \), independent of the chemical potential \(\mu\), and the intrinsic density is \( n_i = \sqrt{N_cN_v}\,e^{-E_g/2k_BT} \propto T^{3/2}e^{-E_g/2k_BT} \), with the effective densities of states \( N_{c,v}=2\left(m^*_{e,h}k_BT/2\pi\hbar^2\right)^{3/2} \).

Why it matters

The law of mass action is the single most-used relation in semiconductor physics: it fixes the equilibrium electron and hole populations from three inputs only — the gap \(E_g\), the temperature \(T\), and the band-edge effective masses — regardless of how the crystal is doped. Every diode, transistor, LED and solar cell is designed around the resulting \(np\) product.

The exponential factor \(e^{-E_g/2k_BT}\) is why a room-temperature semiconductor conducts far less than a metal yet vastly more than an insulator, why silicon (\(E_g=1.12\,\text{eV}\)) and germanium (\(0.66\,\text{eV}\)) differ by four orders of magnitude in \(n_i\), and why leakage currents roughly double every \(8\)–\(10\,\text{K}\). The \(T^{3/2}\) prefactor and the \(E_g/2\) activation slope are directly measurable, turning this derivation into a spectroscopy of the band gap.

Assumptions

Parabolic, isotropic bands near the edges.If the bands are non-parabolic or strongly anisotropic, \(g(E)\) is not \(\propto\sqrt{E-E_c}\); one must replace \(m^*\) by an energy-averaged density-of-states mass or integrate the true dispersion, and the clean \(N_c\propto T^{3/2}\) scaling is lost.

Nondegenerate statistics: \(E_c-\mu \gg k_BT\) and \(\mu-E_v \gg k_BT\).If the Fermi level enters within a few \(k_BT\) of a band edge (heavy doping, low \(T\)), the Boltzmann tail \(f\approx e^{-(E-\mu)/k_BT}\) fails; the Fermi-Dirac integral \(F_{1/2}\) must be kept and \(np\) is no longer a pure exponential of \(\mu\)-independent form.

Temperature-independent, well-defined gap \(E_g=E_c-E_v\).Real gaps shrink with \(T\) (Varshni law); ignoring this overestimates \(n_i\) at high \(T\) because the true activation energy is smaller than the \(0\,\text{K}\) gap.

Band edges are the only relevant states; no gap states, excitonic, or many-body corrections.Deep-level impurities, band-gap renormalization at high carrier density, or bound electron-hole pairs add channels the single-particle integral omits, shifting the effective \(N_c,N_v\) and the activation energy.

Equilibrium and a single global chemical potential.Under illumination or bias the electron and hole populations acquire separate quasi-Fermi levels \(\mu_n\neq\mu_p\); then \(np=n_i^2\,e^{(\mu_n-\mu_p)/k_BT}\neq n_i^2\) and the mass-action law no longer holds.

Derivation

\[ n=\int_{E_c}^{\infty} g_c(E)\,f(E)\,dE,\qquad p=\int_{-\infty}^{E_v} g_v(E)\,\bigl[1-f(E)\bigr]\,dE \]

Definition: carrier density is the density of states summed against occupation; holes are unoccupied valence states, weight \(1-f\). A

\[ g_c(E)=\frac{1}{2\pi^2}\left(\frac{2m_e^*}{\hbar^2}\right)^{3/2}\sqrt{E-E_c},\qquad g_v(E)=\frac{1}{2\pi^2}\left(\frac{2m_h^*}{\hbar^2}\right)^{3/2}\sqrt{E_v-E} \]

Prior result (density of states, 3D parabolic band) with band-edge effective masses \(m_e^*,m_h^*\). A

\[ f(E)=\frac{1}{e^{(E-\mu)/k_BT}+1}\;\xrightarrow[\;E-\mu\gg k_BT\;]{}\;e^{-(E-\mu)/k_BT} \]

Nondegenerate limit: in the conduction band \(E-\mu\gg k_BT\), so the \(+1\) is negligible and Fermi-Dirac reduces to Maxwell-Boltzmann. B

\[ n=\frac{1}{2\pi^2}\left(\frac{2m_e^*}{\hbar^2}\right)^{3/2}e^{-(E_c-\mu)/k_BT}\int_{E_c}^{\infty}\sqrt{E-E_c}\;e^{-(E-E_c)/k_BT}\,dE \]

Substitute steps 2–3 into step 1 and factor the constant Boltzmann weight \(e^{(\mu-E_c)/k_BT}\) out of the integral. B

\[ x\equiv\frac{E-E_c}{k_BT}\;\Rightarrow\;\int_{E_c}^{\infty}\sqrt{E-E_c}\,e^{-(E-E_c)/k_BT}dE=(k_BT)^{3/2}\int_0^\infty x^{1/2}e^{-x}\,dx \]

Dimensionless change of variable; pulls the temperature scaling out cleanly. A

\[ \int_0^\infty x^{1/2}e^{-x}\,dx=\Gamma\!\left(\tfrac{3}{2}\right)=\frac{\sqrt{\pi}}{2} \]

Standard Gamma integral; \(\Gamma(3/2)=\tfrac12\Gamma(1/2)=\tfrac12\sqrt\pi\). C

\[ n=\underbrace{\frac{\sqrt\pi}{4\pi^2}\left(\frac{2m_e^*k_BT}{\hbar^2}\right)^{3/2}}_{\displaystyle N_c}\,e^{-(E_c-\mu)/k_BT},\qquad N_c=2\left(\frac{m_e^*k_BT}{2\pi\hbar^2}\right)^{3/2} \]

Collect constants from steps 4–6; algebraic identity \(\frac{\sqrt\pi}{4\pi^2}2^{3/2}=\frac{2}{(2\pi)^{3/2}}\) gives the compact effective density of states. B

\[ p=N_v\,e^{-(\mu-E_v)/k_BT},\qquad N_v=2\left(\frac{m_h^*k_BT}{2\pi\hbar^2}\right)^{3/2} \]

Repeat steps 3–7 for the valence integral with \(1-f\approx e^{-(\mu-E)/k_BT}\) (valid when \(\mu-E_v\gg k_BT\)) and \(g_v\). B

\[ np=N_cN_v\,e^{-(E_c-\mu)/k_BT}\,e^{-(\mu-E_v)/k_BT}=N_cN_v\,e^{-(E_c-E_v)/k_BT}=N_cN_v\,e^{-E_g/k_BT} \]

Multiply steps 7 and 8: the chemical potential cancels exactly, leaving the law of mass action with \(E_g=E_c-E_v\). A

\[ n=p\equiv n_i\;\Rightarrow\;n_i=\sqrt{np}=\sqrt{N_cN_v}\,e^{-E_g/2k_BT}=2\left(\frac{k_BT}{2\pi\hbar^2}\right)^{3/2}(m_e^*m_h^*)^{3/4}\,e^{-E_g/2k_BT} \]

Intrinsic (undoped) condition sets \(n=p\); the mass-action product is a perfect square, so \(n_i=\sqrt{np}\) is doping-independent. A

\[ N_c e^{-(E_c-\mu_i)/k_BT}=N_v e^{-(\mu_i-E_v)/k_BT}\;\Rightarrow\;\mu_i=\frac{E_c+E_v}{2}+\frac{3}{4}k_BT\ln\!\frac{m_h^*}{m_e^*} \]

Setting \(n=p\) and solving for \(\mu\) locates the intrinsic Fermi level; using \(N_v/N_c=(m_h^*/m_e^*)^{3/2}\) gives the offset from midgap. C

Result

\[ np=N_cN_v\,e^{-E_g/k_BT},\qquad n_i=\sqrt{N_cN_v}\,e^{-E_g/2k_BT}=2\left(\frac{k_BT}{2\pi\hbar^2}\right)^{3/2}(m_e^*m_h^*)^{3/4}e^{-E_g/2k_BT} \]

Reading. In equilibrium the product of electron and hole densities depends only on the gap and temperature, not on doping or the Fermi level's position — add donors and \(n\) rises exactly as \(p\) falls so that \(np\) is pinned. The intrinsic density carries the tell-tale \(T^{3/2}\) prefactor from the phase-space of thermally accessible band states and an Arrhenius factor with activation energy \(E_g/2\), half the gap, because a single thermal event creates one electron and one hole sharing the cost \(E_g\).

Units check. \(\left(k_BT/\hbar^2\right)^{3/2}\) times a mass\(^{3/2}\) gives \(\left(\text{J}\cdot\text{kg}/(\text{J}\cdot\text{s})^2\right)^{3/2}=\left(\text{kg}\,\text{J}/\text{J}^2\text{s}^2\right)^{3/2}=\left(\text{kg}/(\text{J}\,\text{s}^2)\right)^{3/2}\). With \(\text{J}=\text{kg}\,\text{m}^2\text{s}^{-2}\), the bracket is \(\text{kg}/(\text{kg}\,\text{m}^2)=\text{m}^{-2}\), so \(N_c\sim(\text{m}^{-2})^{3/2}=\text{m}^{-3}\), a number density. The exponentials are dimensionless since \(E_g,k_BT\) share units of energy.

Limiting cases

High temperature, \(k_BT\gtrsim E_g\): the exponential saturates toward unity and \(n_i\to\sqrt{N_cN_v}\); the material becomes an intrinsic conductor as carriers flood both bands (but the nondegenerate assumption then breaks).
Low temperature, \(k_BT\ll E_g\): \(n_i\) collapses exponentially; the semiconductor freezes toward insulating behavior and, if doped, extrinsic carriers from shallow dopants dominate over \(n_i\).
Symmetric masses, \(m_e^*=m_h^*\): \(N_c=N_v\) and \(\mu_i\) sits exactly at midgap, \(\mu_i=(E_c+E_v)/2\).
Wide gap (\(E_g\to\infty\)): \(n_i\to0\); a perfect insulator has no thermally generated intrinsic carriers.
Doped but non-degenerate: the law of mass action still holds, so the minority density is \(n_i^2/(\text{majority density})\) — the workhorse formula for minority-carrier engineering.

Breaks when

Degenerate doping or low temperature. When \(\mu\) lies within \(\sim2k_BT\) of a band edge (heavy doping, \(n\gtrsim10^{19}\,\text{cm}^{-3}\) in Si), the Boltzmann approximation of step 3 fails. The full Fermi-Dirac integral \(F_{1/2}(\eta)\) must replace the exponential, \(np\) is no longer a clean \(\mu\)-independent product, and Pauli blocking flattens the population.
Non-parabolic or multi-valley bands. Narrow-gap materials (InSb, InAs) have strongly non-parabolic conduction bands, so \(g(E)\not\propto\sqrt{E-E_c}\) and \(N_c\) is no longer simply \(\propto T^{3/2}\); the effective mass itself becomes energy- and temperature-dependent.
Temperature-dependent gap. Because \(E_g(T)\) shrinks (Varshni), an Arrhenius fit of \(\ln(n_i/T^{3/2})\) yields the extrapolated \(E_g(0)\), not the room-temperature gap; treating \(E_g\) as constant misassigns the activation energy at high \(T\).
Non-equilibrium (illumination, bias, high injection). Split quasi-Fermi levels make \(np=n_i^2 e^{(\mu_n-\mu_p)/k_BT}\); under strong injection or optical pumping \(np\) can exceed \(n_i^2\) by many orders, and mass action as an equality is void.

Failure modes

Using \(E_g\) instead of \(E_g/2\) in the intrinsic exponent. The activation energy for \(n_i\) is half the gap; \(E_g\) is the exponent for the product \(np\), not for \(n_i\) itself.
Confusing the free-electron mass with the effective mass. \(N_c\) uses the density-of-states effective mass (which folds in valley degeneracy), not \(m_0\); for silicon the six equivalent valleys make \(m_{dos}^*\) noticeably larger than any single \(m_l^*,m_t^*\).
Forgetting the valley/band degeneracy factor. Multi-valley conduction bands multiply \(N_c\) by the number of equivalent minima; omitting it underestimates \(n\) by that integer factor.
Assuming \(\mu_i\) is exactly at midgap. It sits at midgap only if \(m_e^*=m_h^*\); the \(\tfrac34 k_BT\ln(m_h^*/m_e^*)\) shift is small but sign-carrying.
Applying mass action out of equilibrium. Students plug \(np=n_i^2\) into an illuminated or forward-biased device; there the correct statement uses quasi-Fermi levels.
Mixing \(k_BT\) in eV and energies in joules. A units mismatch in the exponent throws \(n_i\) off by many decades because it is exponentiated.

Discussion

The deepest content of this derivation is the cancellation in step 9: the chemical potential \(\mu\), which encodes the entire doping state of the crystal, disappears from the product \(np\). Physically, \(np\) counts electron-hole generation-recombination balance, a two-body process whose equilibrium constant is set by the reservoir temperature and the "reaction energy" \(E_g\) alone — exactly the structure of a chemical mass-action law \(K_{eq}(T)\). Electrons in the conduction band and holes in the valence band behave like the products of a dissociation reaction from the filled valence "ground state," and \(N_cN_v e^{-E_g/k_BT}\) is its equilibrium constant.

The \(T^{3/2}\) prefactor is not incidental: it is the thermal de Broglie phase-space volume, \(N_c=2/\lambda_{th}^3\) with \(\lambda_{th}=(2\pi\hbar^2/m^*k_BT)^{1/2}\) the thermal wavelength and the \(2\) from spin. Thus \(N_c\) is literally the quantum concentration — the density at which the average interparticle spacing equals the thermal wavelength and quantum degeneracy sets in. When \(n\) approaches \(N_c\), Boltzmann statistics must give way to full Fermi-Dirac, which is precisely the degenerate-doping breakdown listed above.

Experimentally, plotting \(\ln(n_i T^{-3/2})\) against \(1/T\) yields a straight line of slope \(-E_g/2k_B\): this Arrhenius construction is a primary method of measuring band gaps from conductivity or Hall data, and its success across silicon, germanium and III–V compounds is one of the cleanest confirmations of band theory. The same \(e^{-E_g/2k_BT}\) governs the reverse-saturation current of a \(p\)-\(n\) junction, tying this equilibrium result directly to device leakage and its steep temperature sensitivity.

A subtler point concerns the effective density-of-states mass. For an ellipsoidal, multi-valley band the correct \(m^*\) in \(N_c\) is \(m_{dos}^*=\nu^{2/3}(m_1m_2m_3)^{1/3}\), the geometric mean of the three principal masses scaled by the number of equivalent valleys \(\nu\) to the \(2/3\) power. This is why silicon's \(N_c\) uses \(m_{dos}^*\approx1.08\,m_0\) even though its longitudinal and transverse masses are \(0.98\,m_0\) and \(0.19\,m_0\): the valley count \(\nu=6\) and the anisotropy conspire into a single scalar. The derivation's parabolic-isotropic form survives only because these geometric factors can be absorbed into one number; when they cannot (non-parabolicity), the whole \(T^{3/2}\) law degrades.

Common misconceptions. Doping does not change \(n_i\) — it changes \(n\) and \(p\) individually while their product stays \(n_i^2\); \(n_i\) is a property of the host material and temperature only. And \(n_i\) is not "the number of carriers you get from doping": it is the intrinsic thermal-generation floor, typically dwarfed by dopant carriers at room temperature but decisive for junction leakage and high-temperature operation.

Worked examples

Example 1 — Intrinsic carrier density of silicon at 300 K. Given \(E_g=1.12\,\text{eV}\), density-of-states masses \(m_e^*=1.08\,m_0\), \(m_h^*=0.81\,m_0\), and \(k_BT=0.02585\,\text{eV}\).

\[ N_c=2\left(\frac{m_e^*k_BT}{2\pi\hbar^2}\right)^{3/2}=2.51\times10^{19}\,\text{cm}^{-3}\times\left(\frac{m_e^*}{m_0}\right)^{3/2} \]

Use the tabulated reference value \(2\,(m_0k_BT/2\pi\hbar^2)^{3/2}=2.51\times10^{19}\,\text{cm}^{-3}\) at 300 K, then scale by \((m^*/m_0)^{3/2}\). A

\[ N_c=2.51\times10^{19}\times1.08^{3/2}=2.82\times10^{19}\,\text{cm}^{-3},\quad N_v=2.51\times10^{19}\times0.81^{3/2}=1.83\times10^{19}\,\text{cm}^{-3} \]

Insert the two masses; \(1.08^{3/2}=1.122\), \(0.81^{3/2}=0.729\). A

\[ \frac{E_g}{2k_BT}=\frac{1.12}{2\times0.02585}=21.66,\qquad e^{-21.66}=3.92\times10^{-10} \]

Compute the dimensionless exponent (both energies in eV) and exponentiate. A

\[ n_i=\sqrt{N_cN_v}\,e^{-E_g/2k_BT}=\sqrt{2.82\times10^{19}\cdot1.83\times10^{19}}\times3.92\times10^{-10} \]

Assemble the master formula; \(\sqrt{N_cN_v}=2.27\times10^{19}\,\text{cm}^{-3}\). A

\[ n_i\approx8.9\times10^{9}\,\text{cm}^{-3} \]

Reading. Within a factor of order unity of the accepted \(\sim1\times10^{10}\,\text{cm}^{-3}\) for silicon (differences trace to the exact effective-mass and gap values used). Compared with \(N_c\sim10^{19}\), the exponential suppression of ten orders of magnitude is what makes intrinsic silicon a poor conductor.

Units check. \(\sqrt{\text{cm}^{-3}\cdot\text{cm}^{-3}}=\text{cm}^{-3}\), times a dimensionless exponential, gives a number density in \(\text{cm}^{-3}\).

Example 2 — Intrinsic density of germanium at 300 K. Given \(E_g=0.66\,\text{eV}\), \(m_e^*=0.55\,m_0\), \(m_h^*=0.37\,m_0\), \(k_BT=0.02585\,\text{eV}\).

\[ N_c=2.51\times10^{19}\times0.55^{3/2}=1.02\times10^{19}\,\text{cm}^{-3},\quad N_v=2.51\times10^{19}\times0.37^{3/2}=5.65\times10^{18}\,\text{cm}^{-3} \]

Same reference density scaled by the germanium masses; \(0.55^{3/2}=0.408\), \(0.37^{3/2}=0.225\). A

\[ \frac{E_g}{2k_BT}=\frac{0.66}{0.05170}=12.77,\qquad e^{-12.77}=2.85\times10^{-6} \]

The much smaller gap gives a far weaker exponential suppression than silicon. A

\[ n_i=\sqrt{1.02\times10^{19}\cdot5.65\times10^{18}}\times2.85\times10^{-6}=7.59\times10^{18}\times2.85\times10^{-6} \]

Combine effective densities of states with the Arrhenius factor. A

\[ n_i\approx2.2\times10^{13}\,\text{cm}^{-3} \]

Reading. Close to the accepted \(\sim2.4\times10^{13}\,\text{cm}^{-3}\). Germanium's intrinsic density exceeds silicon's by \(\sim2500\times\) despite similar \(N_c,N_v\) — the entire difference is the smaller gap in the exponent, illustrating the exponential sensitivity to \(E_g\).

Units check. \(\text{cm}^{-3}\times\) (dimensionless) \(=\text{cm}^{-3}\), a number density.

Problems

Effective density of states for GaAs. GaAs has \(m_e^*=0.067\,m_0\). Compute \(N_c\) at 300 K.
Solution
\(N_c=2.51\times10^{19}\times(0.067)^{3/2}\,\text{cm}^{-3}\). Now \((0.067)^{3/2}=0.067\times\sqrt{0.067}=0.067\times0.2588=0.01734\). Thus \(N_c=2.51\times10^{19}\times0.01734=4.4\times10^{17}\,\text{cm}^{-3}\), matching the standard GaAs value \(\approx4.7\times10^{17}\) (small differences from the exact reference constant). The very light electron mass makes \(N_c\) two orders of magnitude below silicon's.
Minority carriers under doping. Silicon at 300 K (\(n_i=1.0\times10^{10}\,\text{cm}^{-3}\)) is doped \(n\)-type with \(N_D=1\times10^{17}\,\text{cm}^{-3}\) fully ionized. Find the equilibrium hole density.
Solution
By charge neutrality \(n\approx N_D=1\times10^{17}\,\text{cm}^{-3}\) (since \(N_D\gg n_i\)). Mass action gives \(p=n_i^2/n=(1.0\times10^{10})^2/1\times10^{17}=1.0\times10^{20}/1\times10^{17}=1\times10^{3}\,\text{cm}^{-3}\). Holes are the minority carrier, suppressed by fourteen orders of magnitude relative to electrons — yet they dominate recombination and diode reverse current.
Temperature doubling of leakage. For silicon (\(E_g=1.12\,\text{eV}\)), by what factor does \(n_i\) increase from 300 K to 320 K? Keep \(E_g\) constant and include the \(T^{3/2}\) prefactor.
Solution
\(\frac{n_i(320)}{n_i(300)}=\left(\frac{320}{300}\right)^{3/2}\exp\!\left[-\frac{E_g}{2k_B}\left(\frac1{320}-\frac1{300}\right)\right]\). Prefactor: \((1.0667)^{3/2}=1.102\). Exponent: \(E_g/2k_B=1.12/(2\times8.617\times10^{-5})=6499\,\text{K}\); \((1/320-1/300)=-2.083\times10^{-4}\,\text{K}^{-1}\), so the argument is \(-6499\times(-2.083\times10^{-4})=1.354\), giving \(e^{1.354}=3.87\). Total factor \(1.102\times3.87\approx4.3\). Intrinsic density more than quadruples over a \(20\,\text{K}\) rise — the exponential dominates the mild prefactor.
Intrinsic Fermi level offset from midgap. Using silicon masses \(m_e^*=1.08\,m_0\), \(m_h^*=0.81\,m_0\) at 300 K, find \(\mu_i-\tfrac12(E_c+E_v)\) in meV.
Solution
\(\mu_i-E_{mid}=\frac34 k_BT\ln(m_h^*/m_e^*)=0.75\times0.02585\,\text{eV}\times\ln(0.81/1.08)\). Here \(\ln(0.75)=-0.2877\), so the offset \(=0.01939\times(-0.2877)=-5.6\times10^{-3}\,\text{eV}=-5.6\,\text{meV}\). The intrinsic level lies \(\approx5.6\,\text{meV}\) below midgap because holes are lighter (smaller \(N_v\)) here — a small but definite shift.
Band gap from an Arrhenius plot. Measurements give \(n_i=2.5\times10^{13}\,\text{cm}^{-3}\) at 300 K and \(n_i=3.0\times10^{15}\,\text{cm}^{-3}\) at 400 K for a semiconductor. Estimate \(E_g\), treating \(N_cN_v\propto T^3\).
Solution
Write \(n_i=C\,T^{3/2}e^{-E_g/2k_BT}\). Then \(\ln(n_i T^{-3/2})=\ln C-\frac{E_g}{2k_B}\frac1T\). Compute \(y=\ln(n_i/T^{3/2})\): at 300 K, \(T^{3/2}=5196\), \(n_i/T^{3/2}=4.81\times10^{9}\), \(y_1=22.30\); at 400 K, \(T^{3/2}=8000\), \(n_i/T^{3/2}=3.75\times10^{11}\), \(y_2=26.65\). Slope \(=-E_g/2k_B=(y_2-y_1)/(1/400-1/300)=(4.35)/(-8.333\times10^{-4})=-5220\,\text{K}\). Thus \(E_g=2k_B\times5220=2\times8.617\times10^{-5}\times5220=0.90\,\text{eV}\). The Arrhenius slope of \(\ln(n_iT^{-3/2})\) vs \(1/T\) recovers the gap directly.