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:
Drift current
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 \(\mu_{p,n}\)