Schottky diodes and metal-semiconductor junctions

WHAT is it? (Definition)

WHY does a barrier form? (Derivation from first principles)

Step 1 — Barrier seen by an electron in the metal (the Schottky barrier). An electron trying to go metal → semiconductor must climb from the metal Fermi level up to $E_C$ . That height is

$\boxed{\ \phi_B = \phi_M - \chi\ }$

Why this step? The metal electron sits $\phi_M$ below vacuum; the semiconductor conduction band sits $\chi$ below vacuum. The difference is exactly the wall's height. Crucially $\phi_B$ does not depend on applied voltage — this side of the barrier is fixed.

Step 2 — Barrier seen from the semiconductor (the built-in potential). Before contact, $E_C$ of the semiconductor is $(\phi_S-\chi)$ below its own $E_F$ . After contact $E_F$ drops by $(\phi_M-\phi_S)$ to match the metal, so the band bends up by

$\boxed{\ qV_{bi} = \phi_M - \phi_S\ }$

Why this step? $V_{bi}$ is the energy the bands must bend to bring the two Fermi levels together. This is the barrier a semiconductor electron sees, and it is reduced by forward bias.

Step 3 — The two barriers are unequal, and that's the whole point. Since $\phi_B = \phi_M-\chi$ and $qV_{bi}=\phi_M-\phi_S$ , and $\phi_S=\chi+(E_C-E_F)_{\text{bulk}}$ : $\phi_B = qV_{bi} + (E_C - E_F)_{\text{bulk}}$ So the metal-side barrier $\phi_B$ is a bit taller than the semiconductor-side barrier $qV_{bi}$ .

Figure — Schottky diodes and metal-semiconductor junctions

HOW does it rectify? (Bias behaviour)

Formula Thermionic-emission I–V (why it looks like a diode equation)

Current is limited by electrons with enough thermal energy to hop the barrier — a Boltzmann tail $\propto e^{-\text{barrier}/kT}$ .

Metal→semi flow: over fixed $\phi_B$ ⇒ constant term $\propto e^{-\phi_B/kT}$ .
Semi→metal flow: over $q(V_{bi}-V)$ ⇒ grows as $e^{qV/kT}$ .

Combining and defining the saturation current $I_S = A A^{*} T^2 e^{-\phi_B/kT}$ : $\boxed{\ I = I_S\left(e^{\,qV/nkT}-1\right)\ }$ where $A^{*}$ = Richardson constant, $A$ = area, $n$ ≈ ideality factor.

Because $\phi_B$ is smaller than a p–n junction's $\sim1.1$ eV barrier, $I_S$ is much larger, so the same current is reached at a smaller $V$ → the low ~0.3 V turn-on.

Worked examples

Common mistakes (Steel-man → fix)

Flashcards

What defines a Schottky diode physically?

A rectifying junction between a metal and a lightly-doped (usually n-type) semiconductor, using only majority carriers.

Rectifying condition for a metal / n-type semiconductor contact

\phi_M > \phi_S

Rectifying condition for metal / p-type contact

\phi_M < \phi_S

Formula for the Schottky barrier height

\phi_B

(n-type)

\phi_B = \phi_M - \chi

(metal work function minus electron affinity).

Formula for the built-in potential

qV_{bi} = \phi_M - \phi_S

Does applied bias change

\phi_B

No — the metal-side barrier is fixed; only the semiconductor-side barrier

q(V_{bi}-V)

moves.

Why does a Schottky diode switch faster than a p–n diode?

Majority-carrier device → no minority-charge storage → near-zero reverse-recovery time.

Typical forward turn-on voltage of a Schottky vs Si p–n diode

~0.2–0.4 V vs ~0.7 V.

Why is the turn-on voltage lower?

Lower barrier ⇒ much larger saturation current

I_S = AA^*T^2e^{-\phi_B/kT}

⇒ same current at smaller

V

Depletion width formula

W=\sqrt{2\varepsilon_s(V_{bi}-V)/(qN_D)}

What happens if you dope the semiconductor very heavily near the metal?

Barrier becomes thin, electrons tunnel → ohmic contact (no rectification).

Current-transport mechanism in a Schottky diode

Thermionic emission of majority electrons over the barrier, giving

I=I_S(e^{qV/nkT}-1)

Relation between

\phi_B

and

qV_{bi}

\phi_B = qV_{bi} + (E_C-E_F)_{bulk}

Recall Feynman: explain to a 12-year-old

Imagine a metal and a special sand (semiconductor) touching. Tiny electric marbles (electrons) in the sand sit on a higher shelf than in the metal, so they roll down into the metal — but they leave behind a "wall" of static that stops more from following. Now: pushing electricity one way lowers the wall on the sand's side so marbles flood across → the light turns on with just a tiny push. Push the other way and the metal's wall stays tall, so almost nothing passes. And because only these fast marbles move (no slow "hole" partners to clean up), the switch flicks on and off super quickly.

Connections

PN Junction Diode — contrast: minority carriers, 0.7 V, reverse recovery.
Work Function and Electron Affinity — the energies $\phi_M,\phi_S,\chi$ come from here.
Depletion Region and Poisson's Equation — source of the $W$ formula.
Thermionic Emission and Richardson's Law — origin of $I_S=AA^*T^2e^{-\phi_B/kT}$ .
Ohmic Contacts — the non-rectifying cousin made by heavy doping.
Rectifiers and Switching Power Supplies — where Schottky's speed is exploited.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Schottky diode ek aisa diode hai jo metal ko ek lightly-doped semiconductor (mostly n-type) ke saath jodne se banta hai — normal p–n junction ki tarah do semiconductors nahi. Jab metal aur semiconductor touch karte hain, dono ke Fermi levels align ho jaate hain. Agar metal ka work function bada hai ( $\phi_M>\phi_S$ ), toh semiconductor ke electrons metal mein chale jaate hain aur peeche ek depletion region aur ek barrier ban jaata hai. Yehi barrier ek taraf current jaane deta hai, doosri taraf rokta hai — matlab rectification.

Do important barriers yaad rakho: metal side ka barrier $\phi_B=\phi_M-\chi$ jo fix rehta hai (voltage se nahi badalta), aur semiconductor side ka $qV_{bi}=\phi_M-\phi_S$ jo forward bias mein ghat jaata hai. Isliye forward mein semiconductor ke electrons aasani se cross karte hain aur current badh jaata hai. Kyunki barrier chhota hai, current bahut kam voltage (~0.3 V) par hi shuru ho jaata hai — Si p–n diode ke 0.7 V ke muqable.

Sabse bada faayda: Schottky sirf majority carriers (electrons) use karta hai, minority carrier storage bilkul nahi. Isliye reverse-recovery time almost zero hota hai aur ye bahut fast switch karta hai — high-frequency power supplies aur RF circuits mein isiliye lagta hai. Ek warning: agar semiconductor bahut heavily dope kar do, toh barrier itna patla ho jaata hai ki electrons tunnel kar jaate hain — tab wo rectifier nahi, balki ohmic contact ban jaata hai. Exam ke liye teen cheezein pakki karo: rectifying condition, dono barrier formulae, aur "fast + low drop kyun" ka reason.

Schottky diodes and metal-semiconductor junctions

WHAT is it? (Definition)

WHY does a barrier form? (Derivation from first principles)

HOW does it rectify? (Bias behaviour)

Worked examples

Common mistakes (Steel-man → fix)

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Diodes & Applications

Connections