Depletion region

The depletion region, also called the space-charge region, is the layer on both sides of a p-n junction that contains almost no free electrons or holes. Carriers diffuse across the junction when it forms, leaving fixed donor and acceptor ions whose charge sets up the field and the built-in voltage. The width ranges from tens of nanometers to several micrometers: 0.43 µm for silicon doped at $10^{16}$ cm⁻³ on both sides at zero bias, 69 nm at $5 \times 10^{17}$ cm⁻³, and typically about 1 µm in the intrinsic layer of an InGaAs PIN photodiode. As an insulating layer between two conductors, it behaves as a voltage-dependent capacitor.

Width and field

For an abrupt junction with acceptor density $N_A$ and donor density $N_D$, the depletion approximation gives

$$W = \sqrt{\frac{2\varepsilon\,(V_\text{bi} + V_R)}{q} \left(\frac{1}{N_A} + \frac{1}{N_D}\right)}$$

with $V_\text{bi}$ the built-in voltage and $V_R$ the reverse bias. Charge neutrality requires $N_A x_p = N_D x_n$, so the region extends further into the more lightly doped side; in a one-sided junction it lies almost entirely there, and $N$ in the bracket is the lighter doping.

The field is triangular for an abrupt junction, peaking at the metallurgical interface at $E_\text{max} = 2(V_\text{bi}+V_R)/W$. For the symmetric silicon junction at $10^{16}$ cm⁻³ ($V_\text{bi} = 0.714$ V, $\varepsilon_r = 11.7$), the width is 0.43 µm at zero bias and 1.2 µm at 5 V reverse bias, and the peak field at zero bias is 33 kV/cm. The reverse bias entry works a one-sided case: 0.95 µm at zero bias and 2.7 µm at 5 V for $N = 10^{15}$ cm⁻³. In a PIN structure the intrinsic layer is fully depleted at a small bias, after which $W$ stays at the layer thickness and the field becomes nearly uniform.

Junction capacitance

The depletion capacitance per unit area is that of a parallel-plate capacitor,

$$\frac{C}{A} = \frac{\varepsilon}{W}$$

which for the symmetric silicon junction is 24 nF/cm², or 241 pF per mm², at zero bias, falling to 85 pF per mm² at 5 V. A 30 µm diameter InGaAs PIN with a 1 µm depleted layer ($\varepsilon_r \approx 13.9$) has 87 fF, an RC limit of 37 GHz into 50 Ω. A plot of $1/C^2$ against reverse voltage is a straight line for an abrupt junction, whose slope gives the lighter doping and whose intercept gives $V_\text{bi}$.

Role in photodiodes

In a photodiode, an electron-hole pair generated inside the depletion region is separated by the field and collected by drift within the transit time, tens of picoseconds. A pair generated in the neutral region outside must first diffuse to the edge, or recombine on the way, which leaves a slow tail on the impulse response. Silicon at 850 nm is the standard example: the absorption coefficient is about 535 cm⁻¹, so light penetrates about 19 µm, well beyond a typical depletion width. An electron generated 10 µm from the edge of the depletion region, with a diffusion coefficient of 36 cm²/s, takes of order $d^2/2D \approx 14$ ns to reach it.

The PIN photodiode addresses this by making the absorbing layer itself the depletion region. In InGaAs, with an absorption coefficient of roughly $0.7 \times 10^4$ cm⁻¹ at 1550 nm, a 1 µm depleted absorber takes in about half the light in a single pass and a 2 µm layer about three quarters, at the cost of a longer crossing. Thermal generation inside the depletion region is also the main bulk source of dark current, which grows with the depleted volume.

Role in silicon depletion modulators

A silicon depletion modulator places a reverse-biased p-n junction across the waveguide core. The edges of the depletion region move through the optical mode as the voltage changes, removing holes on one side and electrons on the other, and the plasma dispersion effect turns the change in carrier density into a change in refractive index. For a symmetric junction at $5 \times 10^{17}$ cm⁻³ ($V_\text{bi} = 0.92$ V), $W$ grows from 69 nm at zero bias to 123 nm at 2 V, so each edge moves by 27 nm. Only that thin shell of a mode several hundred nanometers wide is modulated, so phase shifters are millimeters long, with $V_\pi L$ of 1 to 3 V·cm; the response follows RC time, since no carriers need to recombine, and bandwidths of 30 to 60 GHz are typical.

Pitfalls

The depletion approximation treats the edges as sharp, whereas the carrier density actually falls off over a few Debye lengths, which matters for very thin regions. Graded junctions from diffusion or implantation follow a cube-root voltage dependence. At high photocurrent the charge of carriers in transit screens the field, so collection slows with optical power.

Common questions

Does the depletion region get wider with reverse bias?

Yes, as the square root of $V_\text{bi} + V_R$ for an abrupt junction, until a PIN diode's intrinsic layer is fully depleted or the junction reaches breakdown.

Is the depletion region the same as the intrinsic region of a PIN diode?

Under normal operating bias they nearly coincide: the intrinsic layer is fully depleted, and the depletion region extends only slightly into the heavily doped p and n contacts on either side.

References: S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (Wiley, 2007); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); R. A. Soref and B. R. Bennett, IEEE J. Quantum Electron. 23, 123 (1987).