Unveiling Current Flow: Drift vs. Diffusion in Semiconductors

Demonstrative Video

Variation of Semiconductor parameters with temperature

Variation of semiconductor parameters with temperature:
- Intrinsic concentration $\left(n_i\right)$
- mobility $\left(\mu\right)$
- conductivity $\left(\sigma\right)$
- Energy gap $\left(E_G = E_C-E_V\right)$
$\begin{aligned} \text{electron conc.} ~ n & = N_c \cdot \mathrm{e}^{{-(E_C-E_F)/kT}} \\ \text{Hole conc.}~p & = N_V \cdot \mathrm{e}^{-(E_F-E_V)/kT}\\ n_i^2 & = n\cdot p = N_CN_V \cdot \mathrm{e}^{-(E_C-E_V)/kT} \\ & = N_CN_V \cdot \mathrm{e}^{-E_G/kT} \end{aligned}$
Note: $n_i$ is independent of Fermi level but depends on $T$ and $E_G$

$n_i$ is ver sensitive to $T$ and is given by: $n_i^2 = A_oT^3\mathrm{e}^{-E_{Go}/kT}$

$\begin{aligned} A_o&= \text{a constant, independent of temperature}\\ T&= \text{temperature}~ \left(\right. \left.{ }^{\circ} \mathrm{K}\right)\\ E_{G O}&= \text{forbidden energy gap at}~ { }^{\circ} \mathrm{K} (\text{in eV}) \\ k&= \text{Boltzmann constant}~ \left( \mathrm{eV} /{ }^{\circ} \mathrm{K}\right) \end{aligned}$
$T$ Effect on Extrinsic semiconductor:
- $T\uparrow~\Rightarrow~n_i^2\uparrow ~\Rightarrow$ effect charge density
- N-type: $n$ does not change appreciably but $p \uparrow$
- P-type: $p$ constant, $n\uparrow$

Mobility ( $\mu$ ) $\propto T^{-m} \quad T~\text{varies from 100 to 400}^\circ K$

$m = \left.\begin{cases} \text{Silicon} \Rightarrow & \text{electrons} (2.5), ~\text{holes}(2.7) \\ \text{Germanium} \Rightarrow & \text{electrons} (1.66), ~\text{holes}(2.33) \end{cases} \right\}$
Intrinsic semiconductor: $T\uparrow \Rightarrow \mu \downarrow$
$\mu$ is a function of Electric field intensity (E [V/m])

$\mu = \left.\begin{cases} E < 10^{3} & \text{constant} \\ 10^{3} < E < 10^{4} & \propto E^{-0.5} \\ \text{higher field} & \propto 1/E \end{cases} \right\}$

Conductivity( $\sigma$ )
- depends on number of electron-hole pairs and mobility
- $T\uparrow$ number of e-p pairs $\uparrow$ and $\mu~\downarrow$
- number of e-p pairs $>~\mu$
- At T $^\circ{K}$ $\sigma = \sigma_0\left[1+\alpha(T-T_0)\right]$ where $\alpha$ is temperature coefficient
- Intrinsic semiconductor: T $\uparrow~\Rightarrow~\sigma~\uparrow$
- Extrinsic semiconductor: T $\uparrow~\Rightarrow~\sigma~\downarrow$ as the number of majority carriers is constant and $\mu~\downarrow$
Energy Gap

$E_G(T) = E_{G0} - \beta \cdot T \qquad T~\uparrow \Rightarrow E_G(T)~\downarrow$ $\begin{aligned} \beta & = \text{constant} = \begin{cases} \text{Si} & 3.6 \times 10^{-4} \\ \text{Ge} & 2.23 \times 10^{-4} \end{cases}\\ E_{G0} & = \text{Energy gap at}~ 0^{\circ}\mathrm{K} \quad = \begin{cases} \text{Si} & = 1.21~ \mathrm{eV}\\ \text{Ge} & = 0.785~ \mathrm{eV} \end{cases} \end{aligned}$

Drift and Diffusion Currents

Flow of charge (current) through a semiconductor or PN junction diode has two components:
1. Drift current
2. Diffusion current
Drift current:
- Drift current arises from the movement of carriers in response to an applied electric field.
- Positive carriers (holes) move in the same direction as the electric field
- Negative carriers (electrons) move in the opposite direction.
- The net motion of charged particles generates a drift current that is in the same direction as the applied electric field.

$\text{Drift current density,}~J ~\text{A/cm\textsuperscript{2}} = \begin{cases} J_n = qn\mu_n E ~\Rightarrow \text{due to free electrons} \\ J_p = qp\mu_p E ~\Rightarrow \text{due to holes} \\ \end{cases}$

$\begin{aligned} n &= \text{number of free electrons per cubic centimetre} \\ p & = \text{number of holes per cubic centimetre} \\ \mu_n & = \text{mobility of electrons in} ~\mathrm{cm}^2 / \mathrm{V}-s \\ \mu_p & = \text{mobility of holes in} ~\mathrm{cm}^2 / \mathrm{V}-\mathrm{s} \\ E & = \text{applied electric field intensity in}~ \mathrm{V} / \mathrm{cm} \\ q & = \text{charge of an electron} =1.6 \times 10^{-19} ~\text{coulomb} \\ \end{aligned}$

Diffusion current :
- Electric current can flow even in the absence of applied voltage provided a concentration gradient exists
- When the number of either electrons or holes is greater in one region than that of other region
- Charge carrier move from higher to lower concentration of same type of charge carrier
- Movement of charge carrier resulting in a current called diffusion current

$\text{Diffusion current density}~J~\text{A/cm\textsuperscript{2}} = \begin{cases} J_p = -qD_p\dfrac{dp}{dx} \Rightarrow \text{hole}\\ \\ J_n = -qD_n\dfrac{dn}{dx} \Rightarrow \text{electron} \end{cases}$

hole density $p(x)$ decreases with increasing $x$ , hence negative sign
$J_p$ is positive in $+x$ direction
$dn/dx$ and $dp/dx$ are concentration gradients
$D_n$ and $D_p$ are diffusion coefficients in cm²/s
Total current : $= \begin{cases} J_p=q p \mu_p E-q D_p \dfrac{\mathrm{d} p}{\mathrm{~d} x} \\ \\ J_n=q n \mu_n E-q D_n \dfrac{\mathrm{d} n}{\mathrm{~d} x} \end{cases}$

Einstein Relationship :
- Relationship between mobility and diffusion coefficient of a particular type of charge carrier in the same semiconductor $\frac{D_p}{\mu_p}=\dfrac{D_n}{\mu_n}=\dfrac{k T}{q}=V_T$
- Higher the charge carrier mobility, greater will be its tendency to diffuse
- Used to determine $D_{p,n}$ by experimentally measuring $\mu_{p,n}$