Semi-Conductors

Reverse Bias

• Negative potential to P region
• SCR and electrostatic potential step increases
• Net current decreases
• Minority carrier concentrations at the edges of the SCR essentially become 0

Carrier Concentrations
\( \frac{p_p}{p_n} = \frac{n_n}{n_p} = exp(\frac{kT}{q}V_0)\)

Si dielectric constant
\(\varepsilon = \varepsilon_0 * \varepsilon_{Si} = 8.854*10^{-14}Fcm^{-1}*11.7 \)

Contact Potential from width of depletion region
\( V_0 = -\frac{1}{2}\epsilon_0W = \frac{1}{2}\frac{q}{\epsilon}N_Dx_{n0}W\)

Width of SCR
\( W = [\frac{2\epsilon V_0}{q}(\frac{N_a+N_d}{N_aN_d})]^\frac{1}{2} = [\frac{2\epsilon V_0}{q}(\frac{1}{N_a}+\frac{1}{N_d})]^\frac{1}{2} \)

Penetration of SCR into material
\( x_{n0} = \frac{WN_a}{N_a+N_d} = [\frac{2\epsilon V_0}{q}(\frac{N_a}{N_d(N_a + N_d)})]^\frac{1}{2} \)
\(x_{p0} = \frac{WN_d}{N_a+N_d} = [\frac{2\epsilon V_0}{q}(\frac{N_d}{N_a(N_a + N_d)})]^\frac{1}{2} \)

Forward Bias

\( V = V_0 - V_f \)

• Applied Field in opposite direction of built-in field
• SCR decreases due to smaller field
• Allows many more charge carriers to diffuse across the barrier increasing diffusion current

Drift current across junction

• Drift current is largely unaffected by the potential barrier

• Changes in bias do not affect drift current

• Minority carriers generated by thermal excitation within one diffusion length it can diffuse to SCR and be swept across barrier

• Reverse Saturation Current

Total Current

• Sum of drift and diffusion components
• Reverse Bias -> only saturation Current
• Forward Bias -> Increases chance of diffusion by factor \(e^{\frac{qV}{kT}} \)
  \( I = I_0(e^{\frac{qV}{kT}}-1 )\)
• \(I_0 \) = saturation current magnitude b/c diffusion and gen current equal and opposite at equilibrium

Quasi Fermi Levels

• When a forward bias is applied to the junction the equilibrium Fermi Level is split into two quasi Fermi Levels
• These are separated within SCR by qV
• The minority carrier quasi Fermi Level varies the most, majority Fermi level doesn't change much
  \( pn = n_i^2 e^{\frac{F_n - F_p}{kT}} \)

Important note is that the current at the junction is dominated by the injection of carriers from the heavily doped side into the side with lesser doping

Majority Carrier Current on either side of the junction is always the total current in the diode minus the minority carrier diffusion current at that point
\(I_n(x_n) = I - I_p(x_n) \)

n varies from 3 - 6 depending on type of material used for junction

Switching Diodes

• The switching speed of a diode can be improved with two methods:

1. Adding recombination centres into the bulk material
2. By using short-base diodes (lightly doped base shorter than minority carrier diffusion length)

Voltage across junction does not become 0 until all excess charges have recombined
\(v(t) = \frac{kT}{q}ln(\frac{I\tau_p}{qAL_pp_n}e^{\frac{-t}{\tau_p}} + 1)\)

Junction Capacitance

• Dominant under Reverse Bias
  \( C_j = \frac{\epsilon A}{W} \)
• If doping is not equal C depends on concentration of lesser doped side
  \( C_j = \frac{A}{2}[\frac{2q\epsilon}{V_0-V}N_d]^\frac{1}{2} \) for p+-n

Charge Storage Capacitance

• Dominant under Forward Bias
• Diffusion Capacitance results from changes in the injected minority carrier charges. Only happens in short-base p-n junctions
  \( C_s = \frac{dQ_p}{dV}\)

Rectifying Junctions

• Current in both forward and reverse bias is due to injection of majority carriers from SC into metal
• Schottky M-S junctions are majority carrier devices
• Very fast switching characteristics
  \( I = I_0(e^{\frac{qV}{kT}}-1)\)

To ensure that almost all injected minority carriers are swept across the reverse biased junction, the base should be short compared to diffusion length and carrier lifetime should be long
\(W_b << L_p or L_n\)

The Emitter injects minority carriers into the Base. These carriers are swept across the reverse biased junction to the Collector.

To ensure more carriers are injected into the base than the emitter the emitter must be doped heavily compared to the base

Base Current \(I_B\)

• Current flowing out of the base and not into the collector
• Good transistors will limit this current

1. There must be some recombination of injected holes in the base, electrons lost to recombination must be supplied through base current
2. Some electrons will be injected into emitter from base, these are supplied by \(I_B\)
3. Some electrons are swept into base from B-C due to thermal generation in collector. This is small and reduces \(I_B\)
   

\(B\) Base Transport Factor: Fraction of injected carriers that make it to the collector

• From this we can solve for pinch off voltage:
  \(V_P = \frac{qa^2N_d}{2\varepsilon}\)
  \(V_P = -V_{GD}(pinch off) = -V_G + V_D\)

Drain Current

• Depends on differential volume of neutral channel material, resistance of the channel, and differential voltage change with distance along channel
  \(I_D = \frac{Z2h(x)}{\rho}\frac{dV_x}{dx}\)

Transconductance in Saturation region

\(g_m = \frac{\delta I_{D(sat)}}{\delta V_G} = G_0[1-(\frac{-V_G}{V_P})^{\frac{1}{2}}]\)

Saturation Current

Assume saturation current remains constant at pinch-off

• Greatest at \(V_G = 0\)
  \(I_D(sat.) = G_0V_P[\frac{V_G}{V_P} +\frac{2}{3}(-\frac{V_G}{V_P})^{\frac{3}{2}}+\frac{1}{3}]\)
• Conductance of channel \(G_0 \equiv \frac{2aZ}{\rho L}\)