MM 765: Semiconductor Devices

The Schottky Barrier Diode

Unlike homojunctions, we now explore junctions formed between dissimilar materials. One of the earliest practical semiconductor devices was the metal-semiconductor diode, originally made by touching a metallic whisker to a crystal.

Depending on the materials and doping, these contacts can be:

Rectifying: Called a Schottky Barrier Diode. Conducts strongly in one direction, heavily blocks in the other.
Nonrectifying (Ohmic): Provides low-resistance conduction in both directions, crucial for connecting devices to the outside world.

Assume \(\Phi_m > \Phi_s\) for an n-type semiconductor.

Before contact, the Fermi level in the n-type semiconductor is higher than in the metal. When contact is made, electrons flow from the semiconductor into the metal until Fermi levels align, leaving behind a space charge (depletion) region.

\Phi_{B0} = \Phi_m - \chi \] \[ V_{bi} = \Phi_{B0} - \phi_n

Before Contact

Thermal Equilibrium

Ideal Junction Properties

Applying Bias

Applying a voltage \(V_a\) across the junction alters the barrier for electrons in the semiconductor.

Reverse Bias (\(V_a < 0\)): Barrier increases to \(e(V_{bi} + V_R)\). Negligible current flows.
Forward Bias (\(V_a > 0\)): Barrier decreases to \(e(V_{bi} - V_a)\). Electrons easily flow into the metal.

Note: The barrier \(e\Phi_{B0}\) for electrons in the metal remains essentially constant.

Electrostatics & Capacitance

The electrostatics are nearly identical to a one-sided \(p^+n\) junction. The electric field is linear, peaking at the interface (\(x=0\)).

W = x_n = \left[ \frac{2\epsilon_s(V_{bi} + V_R)}{eN_d} \right]^{1/2}

A capacitance exists due to the separation of charge in the depletion region. The relationship is linear when plotting \((1/C')^2\) vs \(V_R\):

\left(\frac{1}{C'}\right)^2 = \frac{2(V_{bi} + V_R)}{e\epsilon_s N_d}

The slope is inversely proportional to doping \(N_d\), and the intercept yields \(-V_{bi}\).

Bias Voltage: 0.00 V

Electrostatics Profile

\(N_d\): 1x

Nonideal Effects

Schottky Barrier Lowering

An electron in the semiconductor at distance \(x\) induces a positive image charge in the metal at \(-x\). When an external electric field \(E\) is present, the peak of the barrier is lowered.

\Delta\phi = \sqrt{\frac{eE}{4\pi\epsilon_s}} \quad \text{at} \quad x_m = \sqrt{\frac{e}{16\pi\epsilon_s E}}

This image-force-induced lowering effectively increases the reverse-bias saturation current as applied reverse voltage increases.

Image Charge Force

Electric Field:

Interface States & Fermi Pinning

The periodic crystal lattice is abruptly terminated at the interface, creating allowed electronic states within the bandgap. A thin oxide layer (\(\delta\)) typically exists, supporting a potential drop.

Fermi Level Pinning

If the density of interface states (\(D_{it}\)) is very high, shifting the Fermi level creates massive surface charge. This charge requires a huge voltage drop across \(\delta\), absorbing the work function difference.

The Fermi level becomes "pinned" at a neutral level (\(\phi_0\)). Thus, the barrier height becomes independent of the metal work function: \(\Phi_{B0} \approx E_g - e\phi_0\).

Interface States Animation

Thermionic Emission & Diode Comparison

Current transport in a Schottky diode is governed by Thermionic Emission Theory. Only majority carriers with sufficient kinetic energy to overcome the potential barrier can cross.

The flow of electrons from metal to semiconductor (\(J_{m \to s}\)) is constant. The flow from semiconductor to metal (\(J_{s \to m}\)) depends exponentially on applied bias.

                            \[ J = J_{sT} \left[ \exp\left(\frac{eV_a}{kT}\right) - 1 \right] \quad \text{where} \quad J_{sT} = A^* T^2 \exp\left(\frac{-e\Phi_{Bn}}{kT}\right) \]
                        

Note: At exactly zero bias, \(J_{s \to m} = J_{m \to s}\), resulting in perfect equilibrium.

Bias Voltage: 0.00 V

Comparing Ideal pn vs Schottky

I-V Comparison (Linear Scale)

The I-V equation has the same form, but the mechanisms differ wildly.

Schottky: Majority carrier thermionic emission. Reverse saturation current \(J_{sT}\) is orders of magnitude larger, yielding a much lower effective turn-on voltage (~0.3V).
pn Junction: Minority carrier diffusion. Requires higher turn-on voltage (~0.7V).

Charge Transport & Switching

Schottky: Electrons cross straight into the metal and are absorbed immediately. No minority carrier injection.
pn Junction: Carriers diffuse as minority carriers and must recombine.

Consequence: Because Schottky diodes lack minority carrier storage, they switch exceptionally fast (picoseconds) compared to pn junctions (nanoseconds).

Charge Motion

Turn-Off Transient Comparison

Metal-Semiconductor Ohmic Contacts

Every semiconductor device requires connections that do not restrict current flow. An Ohmic Contact provides low-resistance conduction in both directions.

1. Ideal Nonrectifying Barrier

Achieved by choosing a metal with a specific work function relative to the semiconductor.

n-type Condition: \(\Phi_m < \Phi_s\). Electrons form an accumulation layer curving bands downwards.
p-type Condition: \(\Phi_m > \Phi_s\). Holes form an accumulation layer curving bands upwards.

There is no depletion region barrier for majority carriers.

\(\Phi_m - \Phi_s\): 0.00 eV

n-type Ideal Contact

p-type Ideal Contact

2. The Tunneling Barrier

Because surface states pin the Fermi level, finding a metal to form an ideal ohmic contact is virtually impossible. We must use tunneling.

By heavily doping the semiconductor near the interface (\(N_d \approx 10^{20} \text{ cm}^{-3}\)), the depletion width \(W \propto 1/\sqrt{N_d}\) becomes incredibly thin (~10 Å).

Electrons tunnel directly through the barrier horizontally.

R_c = \left( \frac{\partial J}{\partial V} \right)^{-1}_{V=0} \quad [\Omega \cdot \text{cm}^2]

For tunneling, Specific Contact Resistance scales as: \( R_c \propto \exp\left[ \frac{2\sqrt{\epsilon_s m_n^*}}{\hbar} \frac{\Phi_{Bn}}{\sqrt{N_d}} \right] \)

Doping \(N_d\): 10^16.0

\(R_c\) vs. Doping Concentration

Semiconductor Heterojunctions & 2-DEG

A heterojunction is formed between two different lattice-matched semiconductor materials (e.g., GaAs and AlGaAs).

Because they have different bandgaps, discontinuities (\(\Delta E_c\), \(\Delta E_v\)) form at the interface.

\Delta E_c = e(\chi_{narrow} - \chi_{wide}) \] \[ \Delta E_v = \Delta E_g - \Delta E_c

Anisotype (e.g., nP): Doping type changes. Shared depletion width, similar to homojunctions.
Isotype (e.g., nN): Doping type is identical. Leads to dramatic band bending effects.

Straddling Band Alignment

Thermal Equilibrium nP Junction

Two-Dimensional Electron Gas (2-DEG)

In an isotype nN junction (e.g., undoped GaAs and highly doped N-AlGaAs), electrons flow into the narrow-gap material to reach thermal equilibrium.

This creates a triangular potential well. The conduction band dips below the Fermi level, trapping electrons.

Electrons are confined perpendicularly but free to move laterally, forming a Two-Dimensional Electron Gas (2-DEG).

High Electron Mobility Transistor (HEMT)

2-DEG electrons reside in undoped GaAs, physically separated from ionized impurities. This eliminates impurity scattering, yielding exceptionally high electron mobility.

The well is so narrow (~10 nm) that electron energies are quantized into discrete subbands (\(E_0, E_1\)). Electrons predominantly occupy the ground state \(E_0\).

2-DEG Formation

Subbands & Wavefunctions

Lecture Summary

Schottky Barrier Diodes: Formed by a metal contact on a lightly doped semiconductor, creating a rectifying potential barrier \(\Phi_{B0}\). Current is governed by thermionic emission of majority carriers, leading to a much larger reverse saturation current, lower turn-on voltage, and vastly superior switching speeds compared to pn junctions due to the lack of minority carrier storage.
Nonideal Effects: Image-force-induced barrier lowering causes reverse current to increase with applied bias. High densities of interface states can cause Fermi level pinning, making the barrier height nearly independent of the metal work function.
Ohmic Contacts: Essential for connecting devices without restricting current flow. They can be created ideally by specific work function differences (\(\Phi_m < \Phi_s\) for n-type) to form an accumulation layer, or practically by heavily doping the surface to thin the barrier enough for tunneling to dominate, characterized by the Specific Contact Resistance \(R_c\).
Semiconductor Heterojunctions: Junctions combining lattice-matched materials with different bandgaps (e.g., GaAs/AlGaAs), resulting in conduction and valence band discontinuities (\(\Delta E_c\), \(\Delta E_v\)).
Two-Dimensional Electron Gas (2-DEG): In isotype (e.g., nN) heterojunctions, band bending can create a triangular quantum well that traps electrons, forming a 2-DEG. This allows for extremely high electron mobility by physically separating the conduction electrons from ionized impurities, a principle used in HEMTs.

Further Study

Advanced Heterostructures

Explore practical uses of 2-DEGs in High Electron Mobility Transistors (HEMTs) and quantum well lasers.

Explore Devices

Practice Assessment

Lecture 10 Knowledge Check

Test your understanding of Schottky diodes, Ohmic contacts, Thermionic Emission, and Heterojunctions.

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Interactive Problem Solving

Design & Application Problems

Apply the concepts from Lecture 10 to real-world engineering challenges involving dissimilar materials.

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