Validity conditions of the Mott–Gurney equation

The Mott–Gurney equation is one of the most widely used analytical expressions in semiconductor device physics. It describes the current density in a trap-free space-charge-limited current (SCLC) device:

$$ J = \frac{9}{8}\varepsilon\mu\frac{V^2}{L^3} $$

where \(J\) is the current density, \(\varepsilon\) is the dielectric permittivity, \(\mu\) is the carrier mobility, \(V\) is the applied voltage, and \(L\) is the device thickness.

Although the equation is frequently used to extract carrier mobility from experimental current–voltage measurements, the derivation depends on several important physical assumptions. Real semiconductor devices often violate one or more of these assumptions, particularly in disordered organic semiconductors, perovskites, OLEDs, and low-mobility materials.

As a result, it is important to understand not only how the Mott–Gurney equation is derived, but also the physical conditions under which it remains valid. Failure to account for these conditions can lead to significant errors when interpreting experimental SCLC measurements.

Trap-free transport

The classical Mott–Gurney equation assumes that all injected carriers remain mobile. In other words, the semiconductor is assumed to be trap free.

In many real semiconductors, however, carriers can become trapped in localized states associated with:

• structural disorder,
• impurities,
• defects,
• grain boundaries,
• or energetic disorder.

Trapped carriers still contribute to the space charge inside the device, but do not contribute efficiently to electrical conduction. As a result, the voltage dependence of the current deviates from the ideal quadratic behaviour predicted by the Mott–Gurney equation.

Instead of:

$$ J \propto V^2 $$

trap-limited SCLC often follows:

$$ J \propto V^m $$

with:

$$ m > 2 $$

The presence of traps is therefore one of the most important limitations of ideal SCLC theory.

Single-carrier transport

The derivation of the Mott–Gurney equation assumes that only one carrier species contributes significantly to transport. This means that the device must behave either as:

• a hole-only device, or
• an electron-only device.

In practical SCLC experiments this condition is achieved by carefully engineering the contact energetics so that one carrier type is injected efficiently while the opposite carrier type is blocked.

If both electrons and holes are simultaneously injected into the device, recombination processes become important and the assumptions behind the Mott–Gurney derivation break down. The resulting current can no longer be described using a simple single-carrier SCLC model.

This is one reason why standard photovoltaic devices are generally not suitable for ideal SCLC analysis. Solar cells are intentionally designed to support bipolar transport and charge separation, whereas SCLC devices are deliberately engineered to suppress one carrier type.

Ohmic contacts and injection barriers

Another critical assumption of the Mott–Gurney equation is that the contacts inject carriers efficiently into the semiconductor. This requires approximately ohmic injection conditions.

If a large injection barrier exists at the metal–semiconductor interface, the current becomes limited by carrier injection rather than by space charge inside the bulk material.

Under ideal SCLC conditions:

• the contact barrier should be small,
• carrier injection should be efficient,
• and the injected carrier density should greatly exceed the equilibrium carrier density.

If these conditions are not satisfied, the device instead enters an injection-limited regime.

This distinction is extremely important experimentally because a non-ohmic contact can produce current–voltage curves that superficially resemble SCLC behaviour while actually being dominated by contact physics.

Negligible diffusion current

The Mott–Gurney derivation assumes that carrier transport is dominated entirely by drift current. The diffusion contribution is neglected.

The full drift–diffusion equation is:

$$ J = q\mu nE + qD\frac{dn}{dx} $$

where:

• \(q\mu nE\) is the drift current,
• and \(qD\frac{dn}{dx}\) is the diffusion current.

The Mott–Gurney approximation assumes:

$$ qD\frac{dn}{dx} \ll q\mu nE $$

This approximation is usually reasonable at high electric fields and large injected carrier densities, where field-driven transport dominates. However, at low voltages or near equilibrium conditions, diffusion can become important and the ideal SCLC approximation becomes less accurate.

In low-mobility semiconductors with strong disorder, diffusion and energetic relaxation may significantly modify the internal carrier distribution.

Uniform mobility

The classical Mott–Gurney derivation assumes that the carrier mobility is spatially uniform and independent of electric field.

In many real materials this assumption is not valid. Organic semiconductors commonly exhibit field-dependent mobility:

$$ \mu(E) = \mu_0 \exp(\gamma\sqrt{E}) $$

where \(\gamma\) is the Poole–Frenkel coefficient.

This field dependence arises because energetic disorder modifies carrier hopping rates between localized states. As the electric field increases, hopping transport becomes more efficient and the apparent mobility rises.

Under these conditions, the current–voltage relation deviates from the simple quadratic behaviour predicted by the ideal Mott–Gurney equation.

Negligible recombination

The Mott–Gurney equation assumes that recombination is negligible. This assumption is usually valid in properly designed single-carrier devices because only one carrier species is intentionally injected.

However, if minority carriers are present, recombination processes such as:

• Shockley–Read–Hall recombination,
• Langevin recombination,
• or bimolecular recombination

can modify the carrier density distribution and invalidate the ideal SCLC derivation.

Identifying true SCLC experimentally

Experimentally, the SCLC regime is usually identified from the slope of the current–voltage curve on log–log axes.

In the ohmic regime:

$$ J \propto V $$

and the slope is approximately 1.

In ideal trap-free SCLC:

$$ J \propto V^2 $$

and the slope approaches 2.

Trap-limited transport frequently produces slopes larger than 2, while injection-limited transport can generate more complicated behaviour.

Careful interpretation is therefore required. A quadratic current–voltage relation alone is not sufficient proof that ideal Mott–Gurney transport is occurring.

Relation to drift–diffusion simulation

Modern drift–diffusion simulation tools such as OghmaNano solve the coupled Poisson, continuity, and transport equations numerically. This allows realistic non-ideal effects to be included explicitly, including:

• trap distributions,
• field-dependent mobility,
• carrier recombination,
• non-ohmic contacts,
• and spatially varying electric fields.

As a result, numerical simulation often provides a more accurate description of real SCLC devices than ideal analytical theory alone.