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Semi-Conductors (Bipolar Junction Transistors (Fundamentals of Operation …
Semi-Conductors
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Field Effect Transistors
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JFET
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- Majority Carrier device
- Current between source and drain is controlled by Voltage on the gate
- The voltage-variable depletion region width of a junction is used to control the effective cross-sectional area of a conducting channel
- Width of the depletion channel
\(W(x=L) = [\frac{2\varepsilon (-V_{GD})}{q}(\frac{1}{N_d})]^{\frac{1}{2}} = a\)
- if we define \(-V_{GD}\) at pinch off as \(V_P\)
- 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 \(G_0 \equiv \frac{2aZ}{\rho L}\)
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Junctions
Current
- Current is constant throughout the junction, therefore to find the total current can find current on either side of the junction
Space Charge Region
Built in Potential
- The built in potential is NOT the same as the bias applied to a junction
- The built in potential can be solved using only dopant concentrations
Width
- \(\varepsilon = 8.854 * 10^{-14} * 11.8\)
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Excess Carriers
Diffusion of Carriers
Diffusivity
\(D_n = \mu_n * \frac{kT}{q} \frac{cm^2}{s}\)
- Boltzman Constant: \(k = 1.38*10^{-23} \frac{J}{K} = 8.617*10^{-5} \frac{eV}{K}\)
Junctions
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Equilibrium conditions
Four components of particle flow in p-n junction
- electron drift
- electron diffusion
- hole drift
- hole diffusion
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Equilibrium Fermi Levels
- At equilibrium Fermi Levels must equal
\( E_{Fn} - E_{Fp} = 0\)
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Space Charge Region
- Charge must equal on either side so extends more into lower doped region
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Charge Neutrality in step p-n junction
- ie. total charge on either side
\( qAx_{p0}N_A = qAx_{n0}N_D\)
Poisson's Equation
- Solves for electric field distribution within SCR
\( \frac{d\epsilon(x)}{dx} = \frac{q}{\varepsilon}(p-n+N_D^+-N_A^-)\)
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
\( x_{n0} = \frac{WN_a}{N_a+N_d} = [\frac{2\epsilon V_0}{q}(\frac{N_a}{N_d(N_a + N_d)})]^\frac{1}{2} \)
\(x_{p0} = \frac{WN_d}{N_a+N_d} = [\frac{2\epsilon V_0}{q}(\frac{N_d}{N_a(N_a + N_d)})]^\frac{1}{2} \)
Metal-Semiconductor Junctions
- M-S contacts can either have ohmic (linear) or rectifying (diode-like) behaviour
- When a metal is brought into contact with SC charge transfer occurs until Fermi levels align
Schottky Barriers
n-type Semiconductor
- Creates a depletion of electrons at surface, which leads to a contact potential and barrier potential
- Barrier potential is gap for electrons from metal into SC Conduction band
\(\phi_m > \phi_s\)
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p-type Semiconductor
- Creates a depletion of holes at the surface
- V prevents further net hole diffusion
- Barrier potential for hole injection from metal into valence band of semiconductor
\(\phi_m < \phi_s\)
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Forward Biased M-S Junction
- Applying forward bias reduces contact potential \(V_0=(V_0-V)\)
- Allows charge carriers to diffuse to metal and creating current within M-S Junction
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Reverse Biased M-S Junction
- Applying reverse bias increases contact potential to \(V_0=(V_0+V_r)\)
- Makes charge carrier flow from SC to M negligible
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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
\( I = I_0(e^{\frac{qV}{kT}}-1)\)
Ohmic Junctions
n-type
- Energy bands are bent down
- Accumulation of electrons at interface
- No depletion layer
\(\phi_m < \phi_s\)
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p-type
- Energy bands are bent up
- Leads to accumulation of holes at interface
- no depletion layer
\(\phi_m > \phi_s\)
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Reverse-Bias Breakdown
- After critical reverse bias is reached reverse breakdown causes large reverse current through diode
Zener Breakdown
- When heavily doped junction is reverse biased the energy bands can become crossed at low voltages
- Creates a large electric field within SCR which tears electrons from covalent bonds and tunnels them across the band gap from p+ to n+
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Avalanche Breakdown
- Low doped junction, impact ionization
- electron from p side enters SCR and collides with lattice creating EHP, this new electron also collides with lattice and creates another EHP ... etc
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Carrier Multiplication
- Relationship between carrier multiplication and the probability of an electron entering SCR having an ionizing impact with the lattice
\( M = \frac{1}{1-(\frac{V}{V_{br}})^n}\)
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