MM 765: Semiconductor devices and their working principles

Assumed abrupt depletion approximation, Maxwell–Boltzmann statistics, low injection, and constant total current through the structure.

Under these assumptions the current is carried entirely by minority-carrier diffusion outside the space-charge region.

This ideal current varies as

Real diodes deviate from this at low forward bias, at high forward bias, and under reverse bias – we quantify each next.

Under reverse bias, mobile carriers are rapidly swept out of the space-charge region, meaning \( n \approx 0 \) and \( p \approx 0 \).

However, thermal energy continually generates electron-hole pairs via mid-gap defect states (traps).

Because the region is depleted, these newly generated pairs do not recombine. The strong electric field sweeps them apart.

This continuous generation and sweeping action creates an additional current component: the reverse-biased generation current (\(J_{gen}\)).

From the Shockley-Read-Hall (SRH) recombination theory, setting \(n=p=0\) and assuming a trap at midgap (\(n'=p'=n_i\)):

Defining an effective lifetime \(\tau_0 = (\tau_{p0} + \tau_{n0})/2\), the generation rate is \(G = \frac{n_i}{2\tau_0}\).

The generation current density is the integral of \(G\) over the space charge width \(W\):

Since \(W \propto \sqrt{V_{bi} + V_R}\), \(J_{gen}\) is not constant, but increases slightly with reverse bias.

The total reverse current combines ideal saturation (\(J_s\)) and generation (\(J_{gen}\)).

• \(J_s \propto n_i^2\) (Highly sensitive to bandgap)
• \(J_{gen} \propto n_i\) (Less sensitive to bandgap)

Under forward bias, electrons and holes are injected into the space-charge region from opposite sides.

Because the region now has a high concentration of both carriers, some fall into mid-gap traps and recombine before crossing the region completely.

This lost charge must be replenished by the external circuit, creating an extra current component: the forward-bias recombination current (\(J_{rec}\)).

It is most significant at low forward bias, where the ideal diffusion current is still small.

Maximum recombination occurs at the metallurgical junction, where the product \(np\) is maximized and \(n = p = n_i \exp(eV_a/2kT)\).

Substituting this into the SRH equation, the maximum recombination rate is:

Integrating this maximum rate over the effective width \(x'\) of the depletion region gives the current density:

The ideal diode theory assumed low-level injection: injected minority carriers are much fewer than the majority carriers (\(\delta p \ll n_0\)).

At very high forward bias, the injected minority carrier concentration approaches or even exceeds the equilibrium majority carrier concentration.

To maintain space-charge neutrality in the bulk region, the majority carrier concentration must also increase significantly to match the excess minority carriers!

Under high-level injection, the excess carriers dominate, so \(n \approx p\).

Using the fundamental relationship \(np = n_i^2 \exp(V_a/V_t)\), we get:

Since the diode current is proportional to the carrier concentration, the current becomes limited by this new relation:

Notice that the slope on a \(\ln(I)\) vs \(V_a\) plot returns to \(e/2kT\), making the ideality factor \(n \approx 2\) once again.

The reciprocal of the incremental conductance is the incremental resistance (or diffusion resistance):

• \(V_t = kT/e \approx 0.0259\text{ V}\) at Room Temp.
• As the DC bias current (\(I_{DQ}\)) increases, the diffusion resistance \(r_d\) decreases.
• This means the diode conducts small ac signals more easily at higher dc forward biases.

When an ac voltage \(v_1(t)\) is superimposed on the dc bias, the minority carrier concentration at the space-charge edge also fluctuates.

For holes injected into the n-region:

• As voltage increases on the positive half-cycle, more holes are injected.
• As voltage decreases on the negative half-cycle, fewer holes are injected.

Let's look at the hole concentration \(p_n(x,t)\) in the n-region.

The dashed line represents the steady-state DC distribution.

The oscillating solid line represents the total concentration swinging with the AC voltage.

The shaded region represents the AC charge \(\Delta Q\) that must be physically moved in and out of the neutral region during each cycle.

The hole concentration at \(x=0\) is driven by the total voltage \(V_0 + v_1(t)\):

If the AC signal is small, \(v_1(t) \ll V_t = kT/e\). We can expand the exponential using a Taylor series, retaining only linear terms:

Assuming a sinusoidal signal \(v_1(t) = \hat{V}_1 e^{j\omega t}\):

The admittance expression is complex and difficult to synthesize into simple circuit elements.

If we assume the frequency is not too high, \(\omega \tau_{p0} \ll 1\). We can expand the square root using Taylor series:

Substituting this back into \(Y\), we can separate it into real and imaginary parts:

Where \(g_d\) is the diffusion conductance and \(C_d\) is the diffusion capacitance.

To complete the small-signal model, we must add two more physical realities:

1. Junction Capacitance (\(C_j\)): Exists in parallel with the diffusion processes, arising from the separation of charge in the depletion region.
2. Series Resistance (\(r_s\)): The neutral p and n regions have finite resistance. The metallic contacts also add resistance.

The total admittance of the actual junction is \((g_d + j\omega C_d + j\omega C_j)\), placed in series with \(r_s\).

The stored minority charge that gives rise to \(C_d\) also affects how quickly a diode can be switched from forward to reverse bias.

Consider a diode carrying forward current \(I_F\), suddenly switched at \(t=0\) to a reverse bias.

The stored charge can't disappear instantly: it must first be removed (by reverse current and recombination) before the junction can support the reverse voltage.

This delay shows up directly in the diode current waveform.

Switching on is faster than switching off, but it isn't instantaneous either.

Two things must happen:

• 1. The depletion width must narrow from its reverse/zero-bias value down to its new, smaller forward-bias value.
• 2. The steady-state minority-carrier concentration profile must build up by diffusion outside the space-charge region.

Until both are complete, the terminal voltage and current are still settling toward their final dc values – this sets an upper limit on switching speed.

If both p and n regions are doped extremely heavily (degenerately), the Fermi level moves into the conduction band on the n-side and into the valence band on the p-side.

The depletion width becomes extremely narrow (tens of Å).

At equilibrium, the Fermi level is constant across the junction. Filled states in the n-conduction band face filled states in the p-valence band, and empty states face empty states, resulting in zero net tunneling current.

Applying a forward bias aligns filled n-states opposite to empty p-states, allowing quantum mechanical tunneling.