MM 765: Semiconductor Devices

Recap: The Ideal Diode Equation

We derived the ideal diode equation using strict assumptions: abrupt depletion approximation, Maxwell–Boltzmann statistics, low injection, and constant total current through the structure.

Under these assumptions, the current is carried entirely by minority-carrier diffusion outside the space-charge region. This ideal current varies as:

I = I_s \left[ \exp\left(\frac{eV_a}{kT}\right) - 1 \right]

Real diodes deviate from this at low forward bias, at high forward bias, and under reverse bias.

Assumptions to Break:

Constant total current through the depletion region (assuming no generation/recombination).
Low-level injection (assuming injected minority carriers are fewer than majority carriers).
Ohmic contacts and zero bulk resistance.

Ideal I-V Characteristic

Generation & Recombination Currents

Reverse-Biased Generation Current

Under reverse bias, mobile carriers are swept out of the space-charge region (\( n \approx 0 \), \( p \approx 0 \)). Thermal energy continually generates electron-hole pairs via mid-gap defect states.

Because the region is depleted, these newly generated pairs do not recombine. The strong electric field sweeps them apart.

R = -\frac{n_i}{\tau_{p0} + \tau_{n0}} \equiv -G \implies J_{gen} = \frac{e n_i W}{2 \tau_0}

Total Reverse Current: \(J_R = J_s + J_{gen}\)

Silicon: \(n_i\) is small. \(J_{gen} \propto n_i\) dominates over \(J_s \propto n_i^2\).

Germanium: \(n_i\) is large. \(J_s \propto n_i^2\) overtakes \(J_{gen}\) and dominates.

Thermal Generation in Depletion Region

Relative Magnitude at 300K

Forward-Bias Recombination Current

Under forward bias, electrons and holes are injected into the space-charge region. Some fall into mid-gap traps and recombine before crossing.

This lost charge must be replenished by the external circuit, creating the forward-bias recombination current (\(J_{rec}\)).

J_{rec} = \int_0^W eR\,dx \approx J_{r0} \exp\left(\frac{eV_a}{2kT}\right)

Recombination in Depletion Region

Combined Generation & Recombination Simulator

Slide left to apply Reverse Bias (Generation) and right for Forward Bias (Recombination).

Applied Voltage \( V_a \): 0.00 V

Reverse Bias Equilibrium Forward Bias

Junction Physics

SRH Recombination Rate

Relative SRH recombination rate across space-charge region

J-V Plot

Ideality Factor & High-Level Injection

Total forward current is combined with an ideality factor (\(n\)):

\[ I = I_s \left[ \exp\left(\frac{eV_a}{nkT}\right) - 1 \right] \]

\(n=2\): Recombination dominates (low \(V_a\)).
\(n=1\): Diffusion dominates (higher \(V_a\)).

At very high forward bias, injected minority carriers exceed majority carriers (\(n \approx p\)). Space-charge neutrality requires majority carriers to increase. The slope returns to \(e/2kT\).

Bias \(V_a\):

High-Level Injection Carrier Distribution

Injection Level:

Complete Forward I-V Characteristic

Small-Signal Model

Diffusion Resistance & Capacitance

Superimposing a small AC signal on a DC bias yields a linear admittance model: \( Y = g_d + j\omega C_d \).

r_d = \frac{V_t}{I_{DQ}}, \qquad C_d = \frac{1}{2V_t} \left( I_{p0}\tau_{p0} + I_{n0}\tau_{n0} \right)

Both \(r_d\) and \(C_d\) are defined by the DC operating point \(I_{DQ}\).

DC Bias \(V_{DQ}\): 0.60V

AC Charge Fluctuation (Diffusion Capacitance)

Linearity of \(C_d\) vs \(I_{DQ}\)

Effect of Series Resistance & Circuit Model

Complete model includes Junction Capacitance (\(C_j\)) and Series Resistance (\(r_s\)). Series resistance bends the I-V curve horizontally at high currents since \( V_{app} = V_a + I \cdot r_s \).

\(r_s\) (\(\Omega\)): 10

Switching Transients & Charge Storage

Switching is not instantaneous: stored minority charge must be swept out before the junction can support reverse voltage.

t_s \approx \tau_{p0}\ln\left(1+\frac{I_F}{I_R}\right)

Ratio \(I_F / I_R\):

Charge Sweep-Out Animation

Turn-On Transient

Tunnel Diode (Degenerate Doping)

Degenerate doping moves Fermi levels into the bands. A very narrow depletion region allows quantum mechanical tunneling, producing negative differential resistance. The curve shows a characteristic peak and a smooth valley caused by the combination of direct tunneling, trap-assisted excess current, and thermal diffusion.

Forward Bias:

Dynamic I-V Characteristic

Reverse Bias Zener Tunneling

Under reverse bias, filled valence-band states on the p-side line up with empty conduction-band states on the n-side. Tunneling occurs immediately with large reverse current.

Threshold Voltage & Material Comparison

The "Knee" and Built-In Potential

If the I-V curve mathematically passes precisely through the origin (0, 0), why do we discuss a threshold voltage (\(V_{th}\))? The answer lies in the exponential nature of the Shockley equation and the built-in potential barrier (\(V_{bi}\)).

Because the current grows exponentially, it remains microscopically small until the applied forward bias closely approaches \(V_{bi}\). At this point, the current "blows up," creating a sharp upward bend or "knee" on a linear graph. This practical turn-on point is defined as \(V_{th}\).

Key Physical Dependencies:

Bandgap (\(E_g\)): Materials with wider bandgaps have exponentially smaller intrinsic carrier concentrations (\(n_i\)), resulting in a much larger \(V_{bi}\) and consequently a higher \(V_{th}\).
Temperature: As temperature rises, thermal generation increases \(n_i\), which lowers \(V_{bi}\). For standard Silicon, \(V_{th}\) drops by approximately -2 mV/°C.

Beyond Silicon: Wide Bandgap (WBG)

While Germanium (Ge) offers a low turn-on voltage, its narrow bandgap makes it highly "leaky" and highly susceptible to thermal breakdown. Silicon (Si) acts as the perfectly balanced workhorse, and Gallium Arsenide (GaAs) is preferred for optoelectronics and high-speed RF applications.

However, modern power electronics are aggressively shifting towards Gallium Nitride (GaN). Its wide bandgap (3.4 eV) creates an enormous \(V_{bi}\), pushing \(V_{th}\) up to ~3.0V, but it grants the material near-zero leakage and the ability to block thousands of volts without undergoing avalanche breakdown.

Note: Ultra-Wide Bandgap (UWBG) oxide semiconductors like \(\beta\text{-Ga}_2\text{O}_3\) (4.8 eV) are currently being heavily researched to push these extreme high-voltage blocking boundaries even further.

Standard Semiconductors

Wide Bandgap (WBG) Scale

Parameter	Germanium (Ge)	Silicon (Si)	Gallium Arsenide (GaAs)	Gallium Nitride (GaN)
Bandgap (\(E_g\))	0.66 eV	1.12 eV	1.42 eV	3.40 eV
Threshold Voltage (\(V_{th}\))	~0.2 V to 0.3 V	~0.6 V to 0.7 V	~1.0 V to 1.2 V	~2.8 V to 3.4 V
Reverse Leakage (\(I_s\))	High (Microamps)	Low (Nano/Picoamps)	Very Low	Virtually Zero (Femtoamps)
Breakdown Voltage (\(V_{br}\))	Lowest	High	Highest (Standard)	Extreme (Thousands of Volts)
Primary Applications	RF detection, vintage audio	General electronics, computing	Optoelectronics, RF/Microwave	Blue LEDs, Power Grids, EV Inverters

Ideal vs. Real Diode: A Comprehensive Comparison

Feature / Parameter	Ideal Diode	Real Diode
Key Assumptions	Abrupt depletion boundary. Strict Maxwell-Boltzmann statistics. Strictly low-level injection. No generation/recombination inside space charge region. Zero bulk/contact series resistance (\(r_s = 0\)). No junction breakdown mechanisms.	Recombination & generation occurs inside depletion region (dominating at low \(V_a\) and reverse bias). High-level injection occurs at large \(V_a\) (\(n \approx p\)). Finite bulk/contact series resistance (\(r_s > 0\)). Zener and Avalanche breakdown at large reverse bias.
Forward I-V Equation	\[ I = I_s \left( \exp\left(\frac{eV_a}{kT}\right) - 1 \right) \]	\[ I \approx I_s \exp\left(\frac{eV_a}{nkT}\right) \] (where ideality factor \(1 \le n \le 2\))
I-V Characteristic Plot
AC Small-Signal Circuit

Further Study

Diode Applications

Explore practical uses: Rectifiers, Tunnel Oscillators, PIN RF switches, and specialty devices.

Explore Applications

Practice Assessment

Lecture 9 Knowledge Check

Test your understanding of Transients, Non-Idealities, Small-Signal Model & Tunneling.

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