Every semiconductor integrated circuit must communicate with external circuitry through metallic contacts. The physical nature of the metal-semiconductor interface dictates whether the contact acts as a rectifying Schottky diode or as a linear, low-resistance Ohmic interconnect. The microscopic barrier height is governed by the competition between the Schottky-Mott ideal rule and Fermi-level pinning.
1. The Ideal Schottky-Mott Rule
Consider an intimate metallurgical contact between a metal with work function (Phi_m) and an n-type semiconductor with work function (Phi_s) and electron affinity (chi_s). When (Phi_m > Phi_s), electrons transfer from the semiconductor into the metal to align their Fermi levels in thermodynamic equilibrium.
This leaves behind an uncompensated space-charge region in the semiconductor, bending the conduction band upward. According to the Schottky-Mott rule, the resulting barrier height (Phi_{Bn}) experienced by electrons in the metal seeking to enter the semiconductor is:
[
Phi_{Bn} = Phi_m – chi_s
]
The built-in potential barrier (V_{ ext{bi}}) for electrons flowing from the semiconductor into the metal is:
[
V_{ ext{bi}} = Phi_{Bn} – (E_c – E_F) = Phi_m – Phi_s
]
For a p-type semiconductor, rectifying behavior occurs when (Phi_m < Phi_s), yielding barrier height (Phi_{Bp} = rac{E_g}{e} – (Phi_m – chi_s)).
2. Current Transport: Thermionic Emission Theory
Unlike p-n junctions where current is dominated by minority carrier diffusion across neutral regions, current transport across a Schottky barrier is governed predominantly by thermionic emission of majority carriers over the top of the barrier (Bethe’s theory).
Integrating the flux of electrons with kinetic energies exceeding the barrier height ((rac{1}{2} m_e^* v_x^2 ge e(V_{ ext{bi}} – V_a))) yields the thermionic emission current-voltage relation:
[
J = J_{s} left( e^{e V_a / k_B T} – 1
ight)
]
where the saturation current density (J_s) is given by:
[
J_s = A^* T^2 e^{-e Phi_{Bn} / k_B T}
]
Here, (A^* = rac{4pi e m_e^* k_B^2}{h^3}) is the effective Richardson constant ((A^* approx 112 ext{ A/cm}^2 ext{K}^2) for n-type Silicon). Because Schottky diodes operate via majority carriers, there is no minority carrier storage delay during switching, enabling sub-nanosecond response times in high-frequency RF mixers and fast-switching power converters.
3. Bardeen Model and Fermi-Level Pinning
Experimentally, the measured Schottky barrier height on semiconductors like Silicon, GaAs, and InP shows surprisingly weak dependence on the metal work function (Phi_m). This departure from the Schottky-Mott rule is explained by the Bardeen model and Metal-Induced Gap States (MIGS):
- At the metal-semiconductor interface, the evanescent tail of the metallic electron wavefunctions penetrates a few atomic layers into the forbidden bandgap of the semiconductor.
- These states create a continuous density of interface states (D_{it}) (units of ( ext{states/cm}^2/ ext{eV})) with a characteristic charge neutrality level (E_{ ext{CNL}}).
- If (D_{it}) is sufficiently large ((D_{it} ge 10^{13} ext{ cm}^{-2} ext{eV}^{-1})), any transfer of charge across the interface is accommodated by filling or emptying interface states without altering the semiconductor bulk potential.
The Fermi level becomes pinned near (E_{ ext{CNL}}), rendering the barrier height virtually independent of the chosen metal:
[
Phi_{Bn} approx rac{E_c – E_{ ext{CNL}}}{e}
]
In GaAs, the Fermi level is notoriously pinned near mid-gap ((Phi_{Bn} approx 0.7 – 0.8 ext{ eV})) for nearly all metals.
4. Ohmic Contacts and Quantum Field-Emission Tunneling
An Ohmic contact is a non-rectifying junction exhibiting linear current-voltage characteristics with negligible contact resistance (R_c = left( rac{partial J}{partial V}
ight)^{-1}_{V=0}).
Because Fermi-level pinning prevents achieving a zero or negative barrier height ((Phi_{Bn} le 0)) in many semiconductors, practical Ohmic contacts are realized by heavy degenerating doping ((N_D > 10^{19} ext{ cm}^{-3})) directly beneath the metal contact. Under degenerate doping, the depletion width shrinks to the nanometer scale:
[
W approx sqrt{rac{2psilon_s Phi_{Bn}}{e^2 N_D}} sim 1 – 2 ext{ nm}
]
Electrons no longer need to jump over the barrier via thermionic emission; instead, they tunnel directly through the ultra-thin barrier via Field Emission (FE). The specific contact resistivity scales exponentially with the inverse square root of doping:
[
ho_c propto xpleft( rac{4pi sqrt{psilon_s m^*}}{h} rac{Phi_{Bn}}{sqrt{N_D}}
ight)
]
This quantum tunneling contact enables specific contact resistivities below (10^{-8} Omegacdot ext{cm}^2), essential for nanoscale CMOS interconnects.
