Metal Semiconductor Contacts

Schottky (Rectifying) Contact

Name	Formula	Related Formulae	Variables/Constants
Schottky barrier height (Schottky–Mott rule)	$ϕ_{B n} = ϕ_{M} - χ_{s} e V (n - t y p e)$ $ϕ_{B p} = E_{g} - (ϕ_{M} - χ_{s}) e V (p - t y p e)$	$ϕ_{B n} + ϕ_{B p} = E_{g} e V$	$ϕ_{B n}, ϕ_{B p} : i d e a l ba r r i er h e i g h t o n n - t y p e an d p - t y p e se mi co n d u c t or$ $ϕ_{M} : m e t a l w or k f u n c t i o n$ $χ_{s} : se mi co n d u c t or e l ec t r o n a f f ini t y$
Built-in potential (metal-semiconductor junction)	$V_{bi} = ϕ_{B n} - ϕ_{n} (n - t y p e)$ $V_{bi} = ϕ_{B p} - ϕ_{p} (p - t y p e)$	$ϕ_{n} = \frac{k _{B} T}{q} ln (\frac{N _{C}}{N _{D}})$ $ϕ_{p} = \frac{k _{B} T}{q} ln (\frac{N _{V}}{N _{A}})$	$V_{bi} : b u i l t - in p o t e n t ia l$ $ϕ_{n}, ϕ_{p} : d e pt h o f E_{F} b e l o w E_{C} (or ab o v e E_{V}) in t h e n e u t r a l se mi co n d u c t or$ $k_{B} : B o l t z mann co n s t an t$
Depletion region width (one-sided junction)	$W = \frac{2 ε _{s} ( V _{bi} - V )}{q N _{D}} (n - t y p e)$ $W = \frac{2 ε _{s} ( V _{bi} - V )}{q N _{A}} (p - t y p e)$	$W \to PN junction general form as N_{A, m e t a l - s i d e} \to \infty (see PN Junction Diode page)$	$W : d e pl e t i o n w i d t h e x t e n d in g e n t i r e l y in t o t h e se mi co n d u c t or$ $V : a ppl i e d bia s$ $N_{D}, N_{A} : se mi co n d u c t or d o p in g co n ce n t r a t i o n$
Maximum electric field	$E_{ma x} = \frac{2 q N _{D} ( V _{bi} - V )}{ε _{s}}$	$E_{ma x} = \frac{2 ( V _{bi} - V )}{W}$	$E_{ma x} : p e ak f i e l d a t t h e m e t a l - se mi co n d u c t or in t er f a ce$
Depletion (junction) capacitance	$C_{j} = \frac{ε _{s} A}{W} = A \frac{q ε _{s} N _{D}}{2 ( V _{bi} - V )}$	$C_{j 0} = C_{j} (V = 0)$	$C_{j} : d e pl e t i o n c a p a c i t an ce$ $A : co n t a c t a r e a$
Image-force (Schottky) barrier lowering	$Δ ϕ_{B} = \frac{q E _{ma x}}{4 π ε _{s}}$	$ϕ_{B, e f f} = ϕ_{B} - Δ ϕ_{B}$	$Δ ϕ_{B} : ba r r i er l o w er in g d u e t o t h e ima g e - c ha r g e f or ce$ $ϕ_{B, e f f} : e f f ec t i v e ba r r i er h e i g h t a f t er l o w er in g$
Thermionic emission current density	$J = A^{*} T^{2} exp (- \frac{q ϕ _{B}}{k _{B} T}) [exp (\frac{q V}{k _{B} T}) - 1]$	$A^{} = \frac{4 π q m ^{} k _{B}^{2}}{h ^{3}}$	$J : c u r r e n t d e n s i t y o v er t h e ba r r i er$ $A^{*} : e f f ec t i v e R i c ha r d so n co n s t an t$ $k_{B} : B o l t z mann co n s t an t$
Specific contact resistivity (thermionic-emission regime)	$ρ_{c} = (\frac{\partial J}{\partial V})_{V = 0}^{- 1} = \frac{k _{B}}{q A ^{*} T} exp (\frac{q ϕ _{B}}{k _{B} T})$		$ρ_{c} : s p ec i f i c co n t a c t r es i s t i v i t y, v a l i d f or l i g h tl y d o p e d (r ec t i f y in g) co n t a c t s$ $k_{B} : B o l t z mann co n s t an t$

Ohmic Contact

Name	Formula	Related Formulae	Variables/Constants
Characteristic tunneling energy	$E_{00} = \frac{q ℏ}{2} \frac{N _{D}}{m ^{*} ε _{s}}$		$E_{00} : c ha r a c t er i s t i c e n er g y se tt in g w hi c h co n d u c t i o n m ec hani s m d o mina t es t h e co n t a c t$
Conduction-regime classification	$\frac{k _{B} T}{E _{00}} ≫ 1 : thermionic emission (rectifying)$ $\frac{k _{B} T}{E _{00}} \approx 1 : thermionic field emission$ $\frac{k _{B} T}{E _{00}} ≪ 1 : field emission (tunneling, ohmic)$	$E_{00} \propto N_{D}$	$(regime set by doping N_{D} through E_{00})$ $k_{B} : B o l t z mann co n s t an t$
Specific contact resistivity (field-emission / tunneling regime)	$ρ_{c} \propto exp (\frac{2 ε _{s} m ^{*}}{ℏ} \cdot \frac{ϕ _{B}}{N _{D}})$	$ρ_{c} \propto exp (\frac{ϕ _{B}}{E _{00}})$	$ρ_{c} : s p ec i f i c co n t a c t r es i s t i v i t y, v a l i d f or h e a v i l y d o p e d (t r u e o hmi c) co n t a c t s$ $N_{D} : (v er y h e a v y) d o p in g co n ce n t r a t i o n a d j a ce n t t o t h e m e t a l$
Total contact resistance	$R_{c} = \frac{ρ _{c}}{A}$		$R_{c} : t o t a l r es i s t an ce o f t h e co n t a c t$ $A : co n t a c t a r e a$

Important Questions

Ques: What physically distinguishes an Ohmic contact from a Schottky (rectifying) contact, if both are “just” a metal touching a semiconductor?

In the idealized Schottky–Mott picture, the contact type is set purely by the relative work functions: for an n-type semiconductor, $ϕ_{M} < ϕ_{S}$ produces no barrier (ohmic), while $ϕ_{M} > ϕ_{S}$ produces a rectifying barrier (Schottky) — the reverse holds for p-type. In practice, most real “ohmic” contacts are not barrier-free at all; they are deliberately engineered Schottky barriers made electrically transparent by doping the semiconductor extremely heavily right at the interface, which thins the depletion width $W$ enough that carriers tunnel directly through the barrier (field emission) instead of needing to surmount it thermionically. This is the physical reason the two tables above are organized separately: the Schottky table describes conduction over the barrier, while the Ohmic table describes conduction through it. Reference: S.M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007.

Ques: How is the conduction mechanism at a given contact actually determined — is it always a clean thermionic-vs-tunneling split?

No; the characteristic energy $E_{00}$ , compared against $k_{B} T$ , sets where a contact falls along a continuum rather than a sharp switch. At low doping ( $k_{B} T / E_{00} ≫ 1$ ), thermionic emission over the barrier dominates, giving classic Schottky diode behavior. At very heavy doping ( $k_{B} T / E_{00} ≪ 1$ ), the depletion width becomes thin enough that field emission (direct tunneling) dominates, giving a true low-resistance ohmic contact. In between, both mechanisms contribute comparably — a regime called thermionic field emission, where carriers are thermally excited partway up the barrier and then tunnel through the remaining, now-thinner portion. This intermediate-doping regime is common in real devices and is why contact resistivity cannot always be approximated by either limiting formula alone. Reference: F.A. Padovani and R. Stratton, “Field and thermionic-field emission in Schottky barriers,” Solid-State Electronics 9, 695 (1966); S.M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007.

Ques: Why does the measured Schottky barrier height often deviate significantly from the Schottky–Mott rule’s prediction?

The ideal Schottky–Mott rule assumes an atomically clean interface where the barrier height is set purely by the vacuum work functions of the two materials. In most real metal-semiconductor interfaces — particularly on covalently bonded semiconductors like Si and GaAs — a high density of interface states (dangling bonds, defects, or metal-induced gap states) pins the surface Fermi level near a fixed position within the bandgap largely independent of which metal is used, a phenomenon known as Fermi-level pinning. As a result, measured barrier heights on a given semiconductor often cluster within a narrow range across many different metals, in contrast to the wide range the Schottky–Mott rule would predict from each metal’s work function. Reference: J. Bardeen, “Surface States and Rectification at a Metal-Semi-Conductor Contact,” Phys. Rev. 71, 717 (1947); S.M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007.

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