Course instructor: Abhijeet L. Sangle
(alsangle@iitb.ac.in, +91 22 2159 6742)
Unlike the homojunctions studied earlier, 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One of the earliest practical semiconductor devices was the metal-semiconductor diode, originally made by touching a metallic whisker to a crystal (point contact diode).
Modern vacuum and semiconductor technology allows for reliable and reproducible metal-semiconductor contacts.
Depending on the materials and doping, these contacts can be:
We will focus heavily on rectifying contacts made on n-type semiconductors.
Consider an ideal metal and n-type semiconductor before they are brought into contact.
Assume \(\Phi_m > \Phi_s\).
The Fermi level in the n-type semiconductor is initially higher than the Fermi level in the metal.
When contact is made, electrons flow from the semiconductor (higher energy) into the metal (lower energy) until the Fermi levels align.
This leaves behind positively charged donor ions in the semiconductor, creating a space charge (depletion) region.
\(\Phi_{B0}\) is the ideal Schottky barrier height. It is the potential barrier seen by electrons in the metal trying to move into the semiconductor.
Here \(e\phi_n\) is the difference between \(E_c\) and \(E_{Fs}\), and \(V_{bi}\) is the built-in potential barrier seen by electrons in the semiconductor trying to move into the metal.
Applying a voltage \(V_a\) across the junction alters the barrier for electrons in the semiconductor.
Note: The barrier \(e\Phi_{B0}\) for electrons in the metal remains essentially constant in the ideal case.
The electrostatics are nearly identical to a one-sided \(p^+n\) junction. Using Poisson's equation for a uniformly doped n-region:
The electric field is linear, peaking at the interface (\(x=0\)).
The space charge width \(W\) increases with reverse bias \(V_R\).
A capacitance exists due to the separation of charge in the depletion region. The space charge density per unit area is \(Q' = e N_d x_n\). Substituting \(x_n\), we get:
The capacitance per unit area \(C'\) is the derivative of charge with respect to voltage:
Squaring the reciprocal yields a linear relationship with \(V_R\):
The slope is inversely proportional to doping \(N_d\), and the intercept on the voltage axis gives \(-V_{bi}\).
An electron in the semiconductor at distance \(x\) from the interface induces a positive image charge in the metal at \(-x\).
The attractive Coulomb force modifying the barrier is \(F = \frac{-e^2}{4\pi\epsilon_s (2x)^2} = \frac{-e^2}{16\pi\epsilon_s x^2}\).
The potential energy is found by integrating the force: \(PE = -\int F dx = \frac{-e^2}{16\pi\epsilon_s x}\).
When an external electric field \(E\) is applied, the total potential energy becomes:
Setting the derivative to zero finds the position of the maximum barrier, \(x_m\):
Substituting \(x_m\) back into the energy equation gives the barrier lowering \(\Delta\phi\):
As derived, the attractive image force modifies the potential energy barrier for electrons. When an external electric field \(E\) is present, the peak of the barrier is lowered.
The amount of image-force-induced lowering (or Schottky effect) is:
This peak occurs at a distance \(x_m\) from the metallurgical interface:
This lowering effectively increases the reverse-bias saturation current as applied reverse voltage increases.
The periodic crystal lattice is abruptly terminated at the metal-semiconductor interface, creating allowed electronic energy states within the bandgap called interface states.
A narrow interfacial layer of oxide (\(\delta \approx 10\text{-}20~\text{Å}\)) typically exists between the metal and semiconductor. Because it is a thin insulator, it can support a potential difference \(\Delta V = E_{ox} \delta\).
Fermi Level Pinning
Interface states store charge \(Q_{ss}\). If the density of interface states (\(D_{it}\)) is very high, even a tiny shift in the Fermi level at the surface creates a massive surface charge. This charge requires a huge voltage drop \(\Delta V\) across the \(\delta\) layer, effectively absorbing the entire work function difference.
As a result, the Fermi level becomes "stuck" or "pinned" at a neutral energy level (\(\phi_0\)). In this limit, the barrier height becomes independent of the metal work function: \(\Phi_{B0} \approx E_g - e\phi_0\).
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 junction.
The current from the semiconductor to the metal, \(J_{s \to m}\), depends exponentially on the barrier height from the semiconductor side (\(\Phi_{Bn} - V_a\)):
The current from the metal to the semiconductor, \(J_{m \to s}\), depends only on the barrier \(\Phi_{Bn}\) and remains constant at \(-J_{sT}\).
Summing these yields the total current density \(J\):
Where \(J_{sT}\) is the reverse-saturation current density: \( J_{sT} = A^* T^2 \exp\left(\frac{-e\Phi_{Bn}}{kT}\right) \)
And \(A^*\) is the effective Richardson constant (\(A^* = \frac{4\pi e m_n^* k^2}{h^3}\)), which inherently depends on the effective mass of the semiconductor.
Under zero bias, the barrier is exactly balanced. The flow of electrons from the semiconductor into the metal (\(J_{s \to m}\)) perfectly counteracts the flow from the metal into the semiconductor (\(J_{m \to s}\)).
Under forward bias (\(V_a > 0\)), the semiconductor barrier is lowered. Electrons exponentially flood over the barrier into the metal. \(J_{m \to s}\) remains constant.
Under reverse bias (\(V_a < 0\)), the semiconductor barrier increases. \(J_{s \to m}\) shrinks to zero, leaving only the small, constant reverse saturation current \(J_{m \to s} = -J_{sT}\).
The form of the ideal I-V equation is identical for both Schottky and pn junction diodes. However, the current mechanisms are completely different.
Key Consequence:
The reverse-saturation current \(J_{sT}\) for a Schottky diode is orders of magnitude larger than \(J_s\) for a pn junction.
Therefore, the effective turn-on voltage of a Schottky diode is significantly less (e.g., ~0.3V vs ~0.7V for Silicon).
The I-V equation has the same form for both diodes, but watch what is actually crossing the junction — the mechanisms are nothing alike.
Why this matters:
This is exactly why the Schottky diode has no stored minority charge to remove when switching off (next slide) — there was never any minority-carrier injection to begin with.
The second major difference lies in their frequency response and switching characteristics.
The Schottky diode is a majority carrier device.
Result: Schottky diodes are exceptionally fast. Typical switching times are in the picosecond range, compared to nanoseconds for pn junctions.
Every semiconductor device must make contact with the outside world. This requires connections that do not restrict current flow.
An Ohmic Contact is a low-resistance junction providing conduction in both directions.
Ideally, the current through an ohmic contact is a linear function of applied voltage, and the voltage drop across the contact should be negligibly small.
Two general ways to create ohmic contacts:
Drag the slider to see, for an n-type semiconductor, exactly which choice of metal work function gives which kind of contact.
For an n-type semiconductor, an ideal nonrectifying contact is formed when the metal work function (\(\Phi_m\)) is less than the semiconductor work function (\(\Phi_s\)).
Condition: \(\Phi_m < \Phi_s\)
To align the Fermi levels in thermal equilibrium, electrons flow from the metal into the lower energy states of the semiconductor.
This creates an accumulation layer of electrons at the semiconductor surface, curving the bands downwards.
There is no depletion region and no potential barrier for electrons flowing from the semiconductor to the metal. With applied bias, electrons easily flow in both directions.
For a p-type semiconductor, an ideal nonrectifying contact is formed when the metal work function (\(\Phi_m\)) is greater than the semiconductor work function (\(\Phi_s\)).
Condition: \(\Phi_m > \Phi_s\)
To align the Fermi levels, electrons flow from the semiconductor into the metal, leaving behind empty states (holes).
This creates an accumulation layer of holes at the semiconductor surface, curving the bands upwards.
Similar to the n-type case, this accumulation region presents no barrier for majority carriers (holes), yielding an Ohmic contact.
In practice, surface states pin the Fermi level, making it extremely difficult to find a metal that forms an ideal ohmic contact. A different approach is needed.
Recall that the depletion width \(W\) is inversely proportional to the square root of the doping concentration \(N_d\):
By heavily doping the semiconductor near the interface (\(N_d \approx 10^{19} - 10^{20} \text{ cm}^{-3}\)), the depletion width becomes incredibly thin (e.g., ~10-20 Å).
At these extremely narrow widths, quantum mechanical tunneling dominates. Electrons do not need thermal energy to go over the barrier; instead, they tunnel directly through the barrier horizontally.
A figure of merit for an ohmic contact is the Specific Contact Resistance, defined as the inverse of the slope of the J-V curve at zero bias.
For low doping (Thermionic Emission dominates):
For high doping (Tunneling dominates):
A homojunction uses the same semiconductor material throughout (e.g., a Silicon pn junction).
A heterojunction is formed between two different semiconductor materials (e.g., GaAs and AlGaAs).
The most common alignment is the Straddling case, where the narrow bandgap is entirely within the wide bandgap.
According to the ideal Electron Affinity Rule, the conduction band discontinuity (\(\Delta E_c\)) is the difference between the electron affinities (\(\chi\)):
The valence band discontinuity (\(\Delta E_v\)) accounts for the rest of the bandgap difference:
Heterojunctions are classified by their doping types. We use capital letters for the wide-bandgap material and lowercase for the narrow-bandgap material.
In an nP junction, the built-in potential is the sum of the potential drops on both sides. The depletion width is shared, much like a homojunction.
In an isotype nN junction (e.g., undoped GaAs and highly doped N-AlGaAs), electrons flow from the wide-gap AlGaAs into the narrow-gap GaAs to reach thermal equilibrium.
This creates a triangular potential well at the interface on the GaAs side. Crucially, the conduction band in the GaAs dips below the Fermi level, allowing it to become populated with electrons.
Electrons are confined in this narrow well (quantized in the z-direction) but are free to move parallel to the interface (x-y plane). This forms a Two-Dimensional Electron Gas (2-DEG).
High Electron Mobility Transistor (HEMT)
Because the 2-DEG electrons reside in the undoped GaAs, they suffer minimal ionized impurity scattering, leading to exceptionally high mobility.
The triangular potential well in the GaAs is incredibly narrow, often less than 10 nm wide at the Fermi level.
This strong spatial restriction perpendicular to the interface (the z-direction) leads to quantum confinement.
The electron energy is quantized into discrete levels or subbands (\(E_0, E_1, \dots\)).
Probability Density (\(|\psi|^2\))
The graphic shows the probability density for the ground state (\(|\psi_0|^2\)) and first excited state (\(|\psi_1|^2\)). Electrons in the 2-DEG predominantly reside in the ground state \(E_0\), pushed tightly against the interface.