CND-14

Intrinsic carrier density

n_i = √(N_c N_v) exp(−E_g / 2 k T). Semiconductor law of mass action.

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Electrons in solidsBand gap

Governing equation

n_i=\sqrt{N_c N_v}\,e^{-E_g/2kT}

$T \in \left[100, 600\right]$

where

E_g: Band gap (eV)
T: Temperature (K)
N_c: N_c (10²⁵/m³)
N_v: N_v (10²⁵/m³)
n_i: Intrinsic density (1/m³)

The figure, part by part

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①

Governing law — red

n_i=\sqrt{N_c N_v}\,e^{-E_g/2kT}

$T \in \left[100, 600\right]$

$T = 100$ $n_i = 0.001$

Along the curve

②

Domain of the variable — blue

T_{\min} = 100,\quad T_{\max} = 600

$T \in \left[100, 600\right]$

$T = 100$ $T = 100$

Along the curve

③

Response curve — green

n_i=\sqrt{N_c N_v}\,e^{-E_g/2kT}

$T \in \left[100, 600\right]$

$T = 100$ $n_i = 0.001$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}n_i}{\mathrm{d}T}

$T \in \left[100, 600\right]$

$T = 103.1$ $n_i^{\prime} = 0.007$

Along the curve

Lecture brief

Historical brief

Drude electrons, Bloch waves, BCS pairing (1957) and Wiedemann–Franz heat are the first solids-and-metals laws. The lab is conductivity, gap and phonon heat in closed form. This sheet (CND-14 — Intrinsic carrier density) is the form associated with Band gap. Working symbols: $E_g$ , $T$ , $N_c$ , $N_v$ $\rightarrow$ $n_i$ . Silicon E_g ≈ 1.12 eV. n_i(300 K) ≈ 10¹⁶ m⁻³. Exponential in the gap.

Purpose

Purpose: compute

n_i

from

E_g

T

N_c

N_v

in Condensed matter via

n_i=\sqrt{N_c N_v}\,e^{-E_g/2kT}

n_i = √(N_c N_v) exp(−E_g / 2 k T). Semiconductor law of mass action. Use it when a real condensed matter question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

E_g = 1.120\,\mathrm{eV}

T = 300.000\,\mathrm{K}

N_c = 2.800\,\mathrm{10^{2}⁵/m^{3}}

N_v = 1.000\,\mathrm{10^{2}⁵/m^{3}}

, the governing relation

n_i=\sqrt{N_c N_v}\,e^{-E_g/2kT}

yields

n_i = 6.546e+15\,\mathrm{1/m^{3}}

. A gap, two Boltzmann tails, an n_i. Move a slider: the numbers are this situation, not a canned story.

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Inputs

Band gap E_g1.12 eVTemperature T300 KN_c N_c2.8 10²⁵/m³N_v N_v1 10²⁵/m³

Outputs

Intrinsic density n_i6546257275033092.000 1/m³

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Narration of this film

A gap, two Boltzmann tails, an n_i.

Silicon E_g ≈ 1.12 eV. n_i(300 K) ≈ 10¹⁶ m⁻³. Exponential in the gap.

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