Velocity Saturation
The total velocity of a charge carrier is the sum of its random thermal velocity and its drift velocity.
- At \( T = 300 \) K, the average random thermal energy yields a thermal velocity \( v_{th} \approx 10^7 \) cm/s for an electron in Si.
- In low-doped Si, assuming \( \mu_n = 1350 \) cm²/Vs, an applied electric field of 75 V/cm gives a drift velocity \( v_{drf} \approx 10^5 \) cm/s (only ~1% of \( v_{th} \)). The applied field doesn't appreciably change the total energy of the electron.
- At high fields, the drift velocity hits a saturation value (\( v_{sat} \approx 10^7 \) cm/s) due to increased scattering with optical phonons.
Carrier Drift Velocity vs. E-Field
At low fields, slope is mobility. At high fields, \( v_{drf} \) saturates.
Carrier Diffusion
Besides drift, diffusion is another mechanism which can induce a current. Diffusion is the flow of charge carriers from a region of high concentration to a region of low concentration due to random thermal motion.
Deriving Diffusion Flux
Consider the net flow of electrons per unit time per unit area crossing the plane at \( x = 0 \). If \( l \) is the mean free path (\( v_{th}\tau_{cn} \)), half the electrons at \( x = -l \) travel right, and half at \( x = +l \) travel left. The net flow \( F_n \) in the positive x-direction is:
Expanding the electron concentration in a Taylor series about \( x = 0 \) and keeping only the first two terms:
Diffusion Current Density
Since each electron has a charge \( -e \), the current is \( J = +e v_{th} l \frac{dn}{dx} \). We group the constants into a diffusion coefficient \( D_n \) (units of cm²/s).
\[ J_{nx|dif} = e D_n \frac{dn}{dx} \]
For holes, the charge is \( +e \), but the particles flow down the gradient (negative sign), resulting in a current in the negative x-direction.
\[ J_{px|dif} = -e D_p \frac{dp}{dx} \]
1D Particle Diffusion
Simulating 2,000 particles spreading from a high-concentration step.
Diffusion Currents: Electrons vs. Holes
Because electrons carry a negative charge, their diffusion current vector points in the opposite direction of their physical flow. Holes, being positively charged, have a current vector pointing in the same direction as their flow.
Electrons (e⁻)
Gradient: \( \frac{dn}{dx} < 0 \)
\( J_{nx|dif} = e D_n \frac{dn}{dx} \)
Holes (h⁺)
Gradient: \( \frac{dp}{dx} < 0 \)
\( J_{px|dif} = -e D_p \frac{dp}{dx} \)
Total Current Density
The total current density is the sum of the drift and diffusion components for both electrons and holes.
1D Total Current Density
3D Generalization
Note: Mobility and diffusion coefficients are not independent parameters (linked by the Einstein Relation).
Gradient in Doping Concentration
What happens when a semiconductor is non-uniformly doped? Consider a semiconductor doped with donor impurities where the concentration \( N_d(x) \) decreases with increasing \( x \).
- In thermal equilibrium, the Fermi level \( E_F \) must be constant throughout the crystal.
- The concentration gradient causes electrons to diffuse in the positive x-direction.
- This flow leaves behind fixed positive donor ions, establishing an internal, built-in electric field.
- At equilibrium, this induced electric field causes a drift current that perfectly opposes the diffusion current, halting further separation of charge.
Induced Electric Field Derivation
The electric potential \( \phi \) is related to electron potential energy. The electric field is:
Assuming quasi-neutrality (\( n_0 \approx N_d(x) \)):
Since \( E_F \) is constant w.r.t x, taking the derivative yields the built-in field:
Non-Uniform Doping Band Diagram
Why Study Non-Equilibrium?
"A semiconductor in perfect thermal equilibrium is mathematically elegant, but entirely useless for modern computing."
Up until this point, we have primarily focused on semiconductors resting peacefully in the dark at a constant temperature. We mapped out their Fermi levels and calculated carrier concentrations using \( n_0 \) and \( p_0 \).
However, true device operation—whether it is a solar cell absorbing sunlight, a light-emitting diode (LED) emitting photons, or a Bipolar Junction Transistor (BJT) switching currents—requires us to purposefully push the material out of equilibrium.
The Need for Advanced Theory
- Excess Carriers: Applying a voltage or shining light creates excess electrons (\( \delta n \)) and excess holes (\( \delta p \)) that break the \( n_0 p_0 = n_i^2 \) rule.
- Coupled Motion: Electrons and holes do not wander independently. The immense Coulomb attraction between them forces them to move together as a single wave—a phenomenon known as Ambipolar Transport.
- Tracking Time & Space: To design fast transistors, we must track exactly how long these excess carriers survive (recombination lifetime) and how far they can travel (diffusion length) before disappearing.
In this lecture, we bridge the gap between static physics and dynamic device operation by developing the Continuity Equation and the Ambipolar Transport Model.
The Einstein Relation
Before diving into dynamics, we must establish a profound link between how carriers drift under an electric field (Mobility, \( \mu \)) and how they spread due to random thermal motion (Diffusion Co-efficient, \( D \)).
Derivation from Equilibrium
Consider a non-uniformly doped semiconductor (e.g., \( N_d(x) \) varying with position). In thermal equilibrium with no external connections, the net electron current must be zero.
Assuming quasi-neutrality (\( n \approx N_d(x) \)):
Substitute the built-in electric field \( E_x = -\frac{kT}{e} \frac{1}{N_d(x)} \frac{dN_d(x)}{dx} \) (derived in Lecture 5):
Canceling terms yields the fundamental relationship:
The Einstein relation connects mobility and diffusion via thermal energy.
This relation proves that drift and diffusion are essentially two sides of the same coin—both govern carrier scattering and collision mechanisms in the lattice.
Non-Equilibrium Excess Carriers
When external energy (like light) strikes a semiconductor, it can break covalent bonds, elevating an electron from the Valence Band to the Conduction Band. This creates an electron-hole pair.
Because they are generated in pairs (for direct band-to-band generation), the generation rates are equal: \( g_n' = g_p' \).
The total concentration of carriers is now the thermal equilibrium base plus the excess carriers:
- \( n(t) = n_0 + \delta n(t) \)
- \( p(t) = p_0 + \delta p(t) \)
- Note: \( np \neq n_i^2 \) during non-equilibrium!
When the external light is turned off, these excess carriers naturally "fall back down" and recombine. The rate of recombination is proportional to the concentration of the carriers.
Excess Carrier Decay
Observe how the excess minority carriers \( \delta n(t) \) decay exponentially over time after the light source is turned off.
Low-Level Injection Assumption
To keep the mathematical models solvable, we frequently invoke the low-level injection condition. This means the injected excess carriers are much smaller than the majority carrier concentration, but can be much larger than the minority carrier concentration.
For a p-type material (\( p_0 \gg n_0 \)), low-level injection implies: \( \delta n(t) \ll p_0 \).
Under this condition, the massive population of majority holes barely changes percentage-wise, allowing us to treat the recombination lifetime \( \tau_{n0} \) as a constant. The decay equation simplifies to:
\( \delta n(t) = \delta n(0) e^{-t / \tau_{n0}} \)
The Continuity Equation
To describe exactly how the concentration of charge carriers behaves in both time and space, we must account for every particle entering, leaving, being created, or being destroyed in a given volume. This is essentially an accounting ledger for particles.
Tracking Particle Flux
Consider a tiny 1D differential volume of width \( dx \). The change in hole concentration over time (\( \frac{\partial p}{\partial t} \)) depends on:
- The number of holes flowing in at \( x \).
- Minus the number of holes flowing out at \( x + dx \).
- Plus new holes generated inside the volume (\( g_p \)).
- Minus holes lost to recombination (\( -\frac{p}{\tau_{pt}} \)).
Applying a Taylor series expansion to the flux difference \( F_p(x) - F_p(x+dx) \) yields \( -\frac{\partial F_p}{\partial x} dx \).
\[ \frac{\partial p}{\partial t} = -\frac{\partial F_p}{\partial x} + g_p - \frac{p}{\tau_{pt}} \]
Differential Volume Dynamics
Adjust generation and recombination to see how the output flux \( F_p(x+dx) \) responds.
Time-Dependent Diffusion Equations
By substituting the full expressions for particle flux (which includes both drift and diffusion terms, \( F_p = \mu_p p E - D_p \frac{\partial p}{\partial x} \)) into the continuity equation, we derive the master equations that govern device operation.
\[ \frac{\partial p}{\partial t} = D_p \frac{\partial^2 p}{\partial x^2} - \mu_p \frac{\partial (pE)}{\partial x} + g_p - \frac{p}{\tau_{pt}} \]
\[ \frac{\partial n}{\partial t} = D_n \frac{\partial^2 n}{\partial x^2} + \mu_n \frac{\partial (nE)}{\partial x} + g_n - \frac{n}{\tau_{nt}} \]
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.
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.
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
Lecture 6 Knowledge Check
Put your understanding of non-equilibrium transport, excess carriers, the continuity equation, ambipolar transport, and quasi-Fermi levels to the test.