Work function with trapezoidal approximation at junction Equation, Derivation & Rearrangements

Core idea

Overview

The applied voltage tilts the barrier, so the effective barrier height is the average work function minus half the bias drop.

When to use: Use this when the wavefunction must be matched across a finite barrier or finite well.

Why it matters: The tunneling picture explains why wavefunctions oscillate in allowed regions and decay exponentially in forbidden regions.

Symbols

Variables

$Φ$ = $Φ$

Φ

\Phi

Variable

Free formulas

Rearrangements

Solve for $_{s} t a t u s$

b l oc k e d_{n} o n_{i} n v er t ib l e

Solve for reason

T h e v a r iab l es Φ_{1}, Φ_{2}, an d V_{bia s} a r e in s i d e anab so l u t e v a l u e f u n c t i o n or ana dd i t i v es u m, mak in g t h e mim p oss ib l e t o i so l a t e u ni q u e l y w i t h o u t k n o w in g t h es i g n o f t h e a r g u m e n t sor t h er e l a t i v e v a l u eso f t h e p a r am e t er s .

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Why it behaves this way

Intuition

Imagine two metal surfaces with different heights (work functions) facing each other. Without a bias, the potential barrier between them looks like a plateau or a slope connecting the two levels. When a voltage is applied, the potential at one side is pushed down, tilting the top of the barrier into a trapezoid. This equation calculates the 'average height' of that tilted plateau, which represents the effective barrier an electron must tunnel through.

Φ

Effective barrier height

The mean potential energy an electron 'sees' while traversing the gap between two materials.

Φ_{1}, Φ_{2}

Intrinsic work functions

The energy required to remove an electron from the surface of material 1 and material 2 respectively.

e V_{bias}

Applied bias energy

The energy shift caused by the external voltage, which lowers one side of the barrier and creates the trapezoidal tilt.

Signs and relationships

\frac{1}{2}(\Phi_1 + \Phi_2): This averages the two intrinsic heights to find the midpoint of the barrier before considering the external field.
-\lvert eV_{\text{bias}}\rvert: The applied bias lowers the effective potential peak because it creates a downhill slope for the electrons, narrowing the energy region where the barrier is at its maximum.

Free study cues

Insight

Canonical usage

The junction work function is calculated by averaging the work functions of the two materials at the junction and then subtracting the magnitude of the applied bias, with units of energy.

Common confusion

Students may incorrectly treat the bias term as dimensionless or fail to convert it to the same energy units as Φ_1 and Φ_2.

Dimension note

The term $∣$ eV_{\text{bias}} $∣$ has units of energy, derived from the product of elementary charge (C) and voltage (V), which equals Joules (J).

Unit systems

$Φ$ Joule (J) or electronvolt (eV) · The work function represents the minimum energy required to remove an electron from the surface of a solid.

$Φ_{1}$ Joule (J) or electronvolt (eV) · Work function of the first material.

$Φ_{2}$ Joule (J) or electronvolt (eV) · Work function of the second material.

$V_{bias}$ Volt (V) · The applied bias voltage across the junction.

$e$ Coulomb (C) · The elementary charge, used to convert voltage to energy when multiplied.

One free problem

Practice Problem

How does applying a bias voltage across a junction affect the shape of the potential barrier in a trapezoidal approximation?

Solve for: $Φ$

Hint: Consider the relationship between electrostatic potential and the tilt of the potential energy landscape.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When estimating the barrier seen by an electron at a biased metal junction, Work function with trapezoidal approximation at junction is used to calculate \Phi from the measured values. The result matters because it helps size components, compare operating conditions, or check a design margin.

Study smarter

Tips

Inside the classically allowed region you get sine and cosine solutions.
In the forbidden region the physically acceptable branch is exponential decay.
Barrier width matters exponentially, not linearly.
The trapezoid picture is a quick electrostatic approximation, not a full quantum calculation.

Avoid these traps

Common Mistakes

Using an oscillatory solution where the energy is below the barrier.
Forgetting to match both the wavefunction and its derivative at the boundaries.
Underestimating how quickly the tunneling signal drops with barrier width.
Treating the barrier height as unchanged when a finite bias is present.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Use this when the wavefunction must be matched across a finite barrier or finite well.

The tunneling picture explains why wavefunctions oscillate in allowed regions and decay exponentially in forbidden regions.

Using an oscillatory solution where the energy is below the barrier. Forgetting to match both the wavefunction and its derivative at the boundaries. Underestimating how quickly the tunneling signal drops with barrier width. Treating the barrier height as unchanged when a finite bias is present.

When estimating the barrier seen by an electron at a biased metal junction, Work function with trapezoidal approximation at junction is used to calculate \Phi from the measured values. The result matters because it helps size components, compare operating conditions, or check a design margin.

Inside the classically allowed region you get sine and cosine solutions. In the forbidden region the physically acceptable branch is exponential decay. Barrier width matters exponentially, not linearly. The trapezoid picture is a quick electrostatic approximation, not a full quantum calculation.

References

Sources

Engineering LibreTexts, finite square well and tunneling-barrier notes, accessed 2026-04-09
Peverati, The Live Textbook of Physical Chemistry 2, quantum weirdness/tunneling section, accessed 2026-04-09
Engineering LibreTexts, field enhanced emission and tunnelling effects, accessed 2026-04-09
NIST CODATA
IUPAC Gold Book
Wikipedia: Work function
Wikipedia: Electronvolt
Introduction to Solid State Physics by Charles Kittel

Work function with trapezoidal approximation at junction