p-n Junction

A p-n junction contains p-type and n-type semiconductor in the same crystal. On the sides are metal contacts, with a positive anode and negative cathode forming a diode.

Depletion Region

Holes diffuse from the p-region to n-region, and electrons diffuse from the n-region to the p-region. The majority carriers form a total current $I_{D}$ from the p-region to n-region. The majority carriers are depleted due to recombination near the junction interface, which thus creates a depletion region. The decrease in p-region holes results in a negative net charge, and the decrease in n-region electrons results in a positive net charge. This creates an electric field from the n-region to the p-region, which opposes the charge carrier diffusion. The electric field grows until its force prevents further diffusion, putting the depletion region in equilibrium. The voltage across the depletion region, $V_{0}$ , is a barrier which carriers must overcome to diffuse. $I_{D}$ is strongly dependent on $V_{0}$ .

Then, the thermally-generated holes in the n-region are sent to the p-region, and the thermally-generated electrons in the p-region are sent to the n-region. The movement of these minority carriers form a drift current $I_{s}$

Built-in voltage(no external bias)
$V_{0} = V_{T} ln (\frac{N _{a} N _{D}}{n _{i}^{2}})$ (around 0.6 - 0.9V for room temperature Si).
However, the voltage across the junction is still 0V.

Under equilibrium with no external current, $I_{D} = I_{S}$ .

Depletion Region Width

When $N_{A} > N_{D}$ , the depletion region goes further into the n-region to balance out the greater acceptor charge. When $N_{D} > N_{A}$ , the depletion region goes further into the p-region to balance out the greater donor charge. The charges of both sides must equal each other, so $A q N_{A} x_{p} = A q N_{D} x_{n}$ . Therefore, $\frac{x _{n}}{x _{p}} = \frac{N _{A}}{N _{D}}$ .

Total depletion region width
Tends to be within 0.1-1 $\upmu$ m, and can be calculated using electrostatics $W = x_{n} + x_{p} = \frac{2 ϵ _{s}}{q} (\frac{1}{N _{A}} + \frac{1}{N _{D}}) V_{0}$ ,
Si permittivity: $ϵ_{s} = 11.7 ϵ_{0} = 1.04 \times 1 0^{- 12} \frac{F}{c m}$ .

**Values in terms of $W$ and $V_{0}$
$x_{n} = W \frac{N _{A}}{N _{A} + N _{D}}$
$x_{p} = W \frac{N _{D}}{N _{A} + N _{D}}$
$∣ Q_{+} ∣ = ∣ Q_{-} ∣ = Q_{J} = A q (\frac{N _{A} N _{D}}{N _{A} + N _{D}}) W = A 2 ϵ_{s} q (\frac{N _{A} N _{D}}{N _{A} + N _{D}}) V_{0}$

Reverse Bias

Adding a reverse bias $V_{R}$ significantly reduces diffusion current so that $I_{D} ≅ 0$
Thus, $I = I_{D} - I_{S} = - I_{S}$ . This current is very small and strongly depends on temperature. The barrier voltage increases to $V_{0} + V_{R}$ . The depletion region width and junction charge are also greater due to an increase in uncovered fixed charges.

Depletion region width under reverse bias
$W = x_{n} + x_{p} = \frac{2 ϵ _{s}}{q} (\frac{1}{N _{A}} + \frac{1}{N _{D}}) (V_{0} + V_{R})$

Junction charge under reverse bias
$Q_{J} = A 2 ϵ_{s} q (\frac{N _{A} N _{D}}{N _{A} + N _{D}}) (V_{0} + V_{R})$

Forward Bias

Adding a forward bias $V_{F}$ significantly increases diffusion current, which has the equation $I = I_{D} - I_{S}$ . Additionally, the barrier voltage is reduced to $V_{0} - V_{F}$ .

Total hole concentration under forward bias
$p_{n} (x) = p_{n 0} + p_{n 0} (e^{V / V_{T}} - 1) e^{- (x - x_{n}) / L_{p}}$ for $x > x_{n}$

Hole diffusion current density
$J_{p} (x) = q (\frac{D _{p}}{L _{p}}) p_{n 0} (e^{V / V_{T}} - 1) e^{- (x - x_{n}) / L_{p}}$
For $x > x_{n}$ , electrons are externally injected and balance out recombination, making the total diffusion current density constant.

Electron diffusion current density
$J_{n} (x) = q (\frac{D _{n}}{L _{n}}) n_{p 0} (e^{V / V_{T}} - 1) e^{- (x - x_{p}) / L_{n}}$
For $x < x_{p}$ , electrons are externally injected and balance out recombination, making the total diffusion current density constant.

Total diffusion current
$I = A (J_{p} + J_{n}) = A q (\frac{D _{p}}{L _{p}} p_{n 0} + \frac{D _{n}}{L _{n}} n_{p 0}) (e^{V / V_{T}} - 1)$
$= A q n_{i}^{2} (\frac{D _{p}}{L _{p} N _{D}} + \frac{D _{n}}{L _{n} N _{A}}) (e^{V / V_{T}} - 1)$
$= I_{S} (e^{V / V_{T}} - 1)$

Saturation/Scale Current
$I_{S} = A q n_{i}^{2} (\frac{D _{p}}{L _{p} N _{D}} + \frac{D _{n}}{L _{n} N _{A}})$
Ranges from $1 0^{- 18}$ to $1 0^{- 12}$

Reverse Breakdown

Junction breakdown: The significant increase in reversed current due to a large reverse bias. However, breakdown is non-destructive.

Zener breakdown: High E field breaks covalent bonds, $V_{BR} < 5 V$

Avalanche breakdown: Carriers with high kinetic energy break bonds, $V_{BR} > 8 V)$

Capacitance

$C_{j 0} = A (\frac{ϵ _{s} q}{2}) (\frac{N _{A} N _{D}}{N _{A} + N _{D}}) (\frac{1}{V _{0}})$
$C_{j} = \frac{C _{j 0}}{1 + \frac{V _{R}}{V _{0}}}$

Graded junction: Carrier concentration gradually changes from p-type to n-type
$C_{j} = \frac{C _{j 0}}{( 1 + \frac{V _{R}}{V _{0}} ) ^{m}}$
$m$ - grading coefficient, between $1/3$ to $1/2$

Excess hole charge in n-type region
$Q_{p} = A q (p_{n} (x_{n}) - p_{n 0}) L_{p}$
$= \frac{L _{p}^{2}}{D _{p}} I_{p} = τ_{p} I_{p}$

Excess electron charge in p-type region
$Q_{n} = A q (p_{p} (x_{p}) - n_{p 0}) L_{n}$
$= \frac{L _{n}^{2}}{D _{n}} I_{n} = τ_{p} I_{n}$

Excess minority carrier lifetime: Average time for minority carrier to recombine
$τ_{p} = \frac{L _{p}^{2}}{D _{p}}$
$τ_{n} = \frac{L _{n}^{2}}{D _{n}}$

Total charge
$Q = τ_{p} I_{p} + τ_{n} I_{n} = τ_{T} I$

$τ_{T}$ - mean transit time
When $N_{A} ≫ N_{D}$ , $τ_{T} ≅ τ_{p}$
When $N_{D} ≫ N_{A}$ , $τ_{T} ≅ τ_{n}$

Incremental diffusion capacitance
$C_{d} = \frac{d Q}{d V} = τ_{T} \frac{d I}{d V}$

When approximating $I \approx I_{S} e^{V / V_{T}}$ , $C_{d} = (\frac{τ _{T}}{V _{T}}) I$

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