Ambipolar Transport
If a localized pulse of excess electrons and holes is created, the applied electric field naturally tries to rip them apart (electrons drift opposite to the field, holes drift with it). However, massive internal electrostatic (Coulomb) forces immediately develop between the separating charges. This internal field binds the electrons and holes together, forcing them to move and spread as a single unified packet.
The Ambipolar Transport Equation
Ambipolar Diffusion Co-eff (\( D' \)):
\[ D' = \frac{\mu_n n D_p + \mu_p p D_n}{\mu_n n + \mu_p p} \]
Ambipolar Mobility (\( \mu' \)):
\[ \mu' = \frac{\mu_n \mu_p (p - n)}{\mu_n n + \mu_p p} \]
The Minority Carrier Simplification
The complex ambipolar equation vastly simplifies under low-level injection in an extrinsic semiconductor. Because the vast sea of majority carriers dominates the denominator terms, the combined ambipolar parameters effectively collapse down to strictly the minority carrier properties!
The Haynes-Shockley Experiment
The Haynes-Shockley experiment (1951) was a landmark physical setup that allowed scientists to directly measure minority carrier mobility (\( \mu \)), diffusion coefficient (\( D \)), and lifetime (\( \tau \)) all from a single experimental pass.
The Setup: An electric field \( E_0 \) is established across a bar of n-type semiconductor. A sharp laser or voltage pulse injects a tight cluster of excess holes at Contact A.
The Journey: This pulse immediately begins to:
- Drift down the bar due to \( E_0 \).
- Diffuse and spread out wide due to concentration gradients.
- Recombine and decay in amplitude due to minority lifetime.
The Readout: A reverse-biased Contact B, located a distance \( d \) away, sweeps up the surviving holes, plotting a voltage curve over time. By analyzing the peak arrival time (\( t_0 \)) and pulse spread (\( \Delta t \)), the transport properties are mathematically extracted.
Parameter Extraction
- 1. Mobility (\( \mu \)): Determined from the peak arrival time (\( t_0 \)). The pulse drifts a distance \( d \) under the electric field \( E_0 \):
\( \mu = \frac{d}{E_0 t_0} \)
- 2. Diffusion Coefficient (\( D \)): Determined from the temporal spread of the pulse (\( \Delta t \)). A wider pulse indicates higher diffusion:
\( D = \frac{(\mu E_0 \Delta t)^2}{16 t_0} \)
- 3. Lifetime (\( \tau \)): Extracted by repeating the experiment at different fields (\( E_0 \)). The peak amplitude decays exponentially with longer transit times (\( t_0 \)):
\( \text{Amplitude} \propto e^{-t_0 / \tau} \)
Interactive Experiment Simulator
Lower the E-Field to see the pulse arrive later, spread wider, and decay more.
Quasi-Fermi Energy Levels
When excess carriers are pumped into a semiconductor, the system is no longer in thermal equilibrium. Because the fundamental definition of the Fermi Level (\( E_F \)) assumes equilibrium, a single \( E_F \) can no longer accurately describe both the electron and hole concentrations simultaneously.
Important Insight: Under non-equilibrium conditions, the majority carrier concentration (e.g., electrons in n-type) doesn't change much proportionally, so the Quasi-Fermi level for majority carriers barely shifts from the equilibrium \( E_F \). But for minority carriers, the concentration change is huge, resulting in a dramatic shift in the Quasi-Fermi level for those minority carriers!
To preserve our elegant exponential formulas, we define two pseudo-levels: the Quasi-Fermi Level for Electrons (\( E_{Fn} \)) and the Quasi-Fermi Level for Holes (\( E_{Fp} \)).
The separation between \( E_{Fn} \) and \( E_{Fp} \) is a direct measure of how far the system has been pushed out of equilibrium.
Carrier Injection Bending
Quasi-Fermi Calculations
Calculation Example
Consider an n-type semiconductor at 300 K.
- Intrinsic density: \( n_i = 10^{10} \text{ cm}^{-3} \)
- Equilibrium electrons: \( n_0 = 10^{15} \text{ cm}^{-3} \)
- Equilibrium holes: \( p_0 = \frac{n_i^2}{n_0} = 10^{5} \text{ cm}^{-3} \)
Inject Excess Carriers: Use the slider to increase \( \delta n \) and \( \delta p \). Notice how \( E_{Fn} \) barely moves (since electrons are the majority), while \( E_{Fp} \) shifts dramatically!
Quasi-Fermi Band Diagram
Shockley-Read-Hall (SRH) Recombination
Direct band-to-band recombination (an electron directly dropping into a hole) is rare in indirect bandgap semiconductors like Silicon. Instead, recombination usually happens via trap states—allowed energy levels (\( E_t \)) created by defects or impurities right in the middle of the forbidden bandgap.
"Think of a trap state as a resting bench halfway down a steep cliff. It is much easier for a carrier to jump to the bench first, and then to the bottom, rather than leaping the entire way in one go."
The Four Fundamental Processes
According to the SRH theory, four distinct events can occur at a single trap center. We assume the trap is an acceptor type (neutral when empty, negatively charged when filled with an electron).
Interactive Trap Dynamics
Click the buttons below to trigger the specific SRH processes manually and observe the carrier transitions.
The Net Recombination Rate
By balancing the rates of these four processes (using capture cross-sections, trap density \( N_t \), and the Fermi probability function), we derive the full SRH Recombination Rate (\( R \)):
Where \( n', p' \) are the hypothetical carrier concentrations if \( E_F \) were exactly at the trap level \( E_t \).
Simplification: Low-Level Injection
Consider an n-type material (\( n_0 \gg p_0 \)) under low-level injection. Assuming the trap energy is near midgap, the formula simplifies massively. The recombination rate becomes purely a function of the minority carrier (hole) lifetime:
Insight: If the concentration of defects/traps (\( N_t \)) increases, the probability of excess carrier recombination increases, leading to a much shorter minority carrier lifetime.
The PN Junction
A PN junction is formed by intimately joining a p-type semiconductor (rich in holes) and an n-type semiconductor (rich in electrons). This metallurgical junction is the fundamental building block of diodes, transistors, and solar cells.
1. The Diffusion Force
Because the p-side has massive amounts of holes and the n-side has massive amounts of electrons, a sharp concentration gradient exists at the junction.
Majority carriers want to diffuse: Electrons spill over into the p-side, and holes spill over into the n-side.
2. The Space Charge Region
As electrons leave the n-side, they leave behind positively charged donor ions (\( N_d^+ \)). As holes leave the p-side, they leave negatively charged acceptor ions (\( N_a^- \)).
These fixed ions create a Depletion Region (devoid of mobile charge) and build up a strong internal Electric Field that points from n to p, stopping further diffusion.
Junction Formation Animation
Watch isolated p-type and n-type crystals join, diffuse, and settle into equilibrium.
Derivation of Built-in Potential Barrier (\( V_{bi} \))
Assuming no voltage is applied and \(E_F\) is constant, electrons in the n-type region experience a barrier \(V_{bi}\) to cross over into the p-type region. The intrinsic Fermi level \(E_{Fi}\) is equidistant from the conduction band, so \(V_{bi}\) is determined by the difference in \(E_{Fi}\) between regions: \(V_{bi} = \phi_{Fn} + \phi_{Fp}\).
In the n-region, the mobile carrier concentration is: \( n_0 = n_i e^{(E_F - E_{Fi})/kT} \). With \( e\phi_{Fn} = E_{Fi} - E_F \), we get \( n_0 = n_i e^{e\phi_{Fn}/kT} \). Assuming \( n_0 = N_d \), solving for \(\phi_{Fn}\) gives:
Similarly, for the p-region, \( p_0 = N_a = n_i e^{(E_{Fi} - E_F)/kT} \) and \( e\phi_{Fp} = E_{Fi} - E_F \), yielding:
Summing these potentials provides the built-in potential barrier:
Electric Field & Potential Across the Space Charge Region
Assuming the space charge region abruptly ends at \( x = +x_n \) and \( x = -x_p \). From Poisson's equation for a 1D case:
Where \(\rho(x) = -eN_a\) for \(-x_p < x < 0\) and \(\rho(x) = eN_d\) for \(0 < x < x_n\). The electric field in the p-region (\(E = 0\) at \(x = -x_p\)) is:
Similarly, the electric field in the n-region (\(E = 0\) at \(x = x_n\)) is:
Since the electric field is continuous at the metallurgical junction (\(x = 0\)), we get \(N_a x_p = N_d x_n\).
Integrating the electric field gives the potential. Arbitrarily setting \(\phi(-x_p) = 0\), the potential in the p-region is:
By ensuring potential is continuous at \(x = 0\), the potential in the n-region becomes:
The magnitude of the potential at \(x = x_n\) is the built-in potential barrier \(V_{bi}\):
Energy Band Diagram at Equilibrium
To keep \( E_F \) flat across the junction, \( E_c \), \( E_v \), and \( E_{Fi} \) all bend by the same total amount, \( eV_{bi} \). Drag the slider to see how the doping asymmetry shifts where the flat \( E_F \) sits relative to each band edge — splitting the bending into \( e\phi_{Fp} \) on the p-side and \( e\phi_{Fn} \) on the n-side.
Interactive PN Junction Profiler
By solving Poisson's Equation (\( \frac{d^2\phi}{dx^2} = -\frac{\rho(x)}{\varepsilon_s} \)) across the step junction, we can determine the Charge Density (\( \rho \)), Electric Field (\( E \)), and Potential (\( \phi \)) as a function of position. Adjust the doping concentrations below to see how the Depletion Width (\( W = x_n + x_p \)) and the profiles change.
Calculated Values
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Note: The depletion region extends further into the more lightly doped side. (\( N_a x_p = N_d x_n \))
\[ W = x_n + x_p = \sqrt{ \frac{2 \varepsilon_s V_{bi}}{e} \left( \frac{N_a + N_d}{N_a N_d} \right) } \]
Lecture 7 Knowledge Check
Put your understanding of the Haynes-Shockley experiment, Quasi-Fermi levels, SRH Recombination, and the PN Junction to the test.