Diffusion length

The diffusion length is the characteristic distance that excess carriers, usually minority carriers, spread by diffusion before they recombine. Where a steady source of carriers sits at one plane, their excess density falls off away from it as $\exp(-x/L)$, with

$$L = \sqrt{D\,\tau},$$

where $D$ is the diffusion coefficient and $\tau$ the carrier lifetime. The diffusion coefficient follows from the mobility $\mu$ through the Einstein relation, $D = \mu k_B T/q$, with $k_BT/q = 25.85$ mV at 300 K. Typical values span three orders of magnitude: about 1–2 µm in the heavily injected active layer of a GaAs or InP laser, where the lifetime is a nanosecond or two, and tens to hundreds of micrometers for electrons in lightly doped silicon, and millimeters in very pure material where the lifetime reaches milliseconds.

Worked values

For holes in GaAs with a mobility of 400 cm²/V·s, the Einstein relation gives $D$ = 10.3 cm²/s. Under the high injection of a laser, electrons and holes move together with the ambipolar coefficient, close to twice the hole value, about 20 cm²/s, so a lifetime of 1 ns gives $L$ = 1.4 µm and 2 ns gives 2.0 µm. In p-type silicon, minority electrons with $D$ = 36 cm²/s (a mobility of about 1390 cm²/V·s) have

$$L = \sqrt{36\ \mathrm{cm^2/s} \times 1\ \mu\mathrm{s}} = 60\ \mu\mathrm{m},$$

190 µm for 10 µs and 600 µm for 100 µs. Both $D$ and $\tau$ fall with doping and defect density.

A related quantity is the distance diffused in a time $t$, of order $\sqrt{2Dt}$. Inverted, it gives the time a carrier takes to cross a distance $d$ by diffusion, $t \approx d^2/2D$: an electron generated 10 µm from the edge of a depletion region in silicon needs about 14 ns to reach it.

Photodiodes: collection and diffusion tails

In a photodiode an electron-hole pair created inside the depletion region is swept out by the field in a transit time of tens of picoseconds. A pair created in a neutral region contributes to the photocurrent only if it diffuses to the depletion edge before recombining, which it does with a probability that falls roughly as $\exp(-x/L)$ with its starting distance $x$. These carriers add to the quantum efficiency but arrive late, producing a slow tail on the impulse response.

Silicon at 850 nm shows both effects. With an absorption coefficient of 535 cm⁻¹, light penetrates about 19 µm. For a depletion width $W$ = 5 µm, only 23% of the light entering the silicon is absorbed in the depletion region. Gärtner's model for a deep neutral region, which neglects surface reflection, gives the internal efficiency

$$\eta = 1 - \frac{e^{-\alpha W}}{1 + \alpha L},$$

which is 40% for $L$ = 5 µm and 79% for $L$ = 50 µm. The extra collection comes at the cost of a tail lasting nanoseconds, so silicon receivers built for speed at 850 nm use thin absorbers or buried barriers that discard the slow carriers, accepting the lower responsivity. The same diffusion from the neutral regions supplies the diffusion component of dark current, which is set by the minority carriers thermally generated within about one diffusion length of the junction.

Solar cells and light emitters

A crystalline silicon solar cell relies on diffusion to collect carriers generated throughout a base that is typically under 200 µm thick; its efficiency stays high only while the minority-carrier diffusion length in the base exceeds the thickness.

In lasers and LEDs the diffusion length sets how far carriers spread laterally from the region where current is injected. Carrier confinement covers the consequences, including current spreading beyond a laser ridge and surface recombination at the sidewalls of etched micro-emitters whose size approaches $L$.

Measurement

The diffusion length is measured directly by scanning a focused electron beam (electron-beam-induced current, EBIC) or laser spot (light-beam-induced current, LBIC) away from a junction and fitting the exponential decay of the collected current. The spectral response of a photodiode or solar cell at weakly absorbed wavelengths, where carriers are generated deep in the neutral region, also yields $L$ through models such as Gärtner's. Alternatively, $\tau$ is measured by time-resolved photoluminescence or photoconductance decay and combined with a mobility from Hall measurements. Surface recombination shortens the apparent value in thin samples.

Common questions

Is the diffusion length the same for electrons and holes?

No. Each has its own mobility and lifetime, and in a doped region the relevant value is that of the minority carrier: electrons in p-type material, holes in n-type. At high injection, where both densities exceed the doping, they diffuse together with the ambipolar coefficient.

Why does a silicon photodiode respond slowly at 850 nm but not at 450 nm?

Blue light is absorbed within a fraction of a micrometer, near or inside the depletion region. At 850 nm much of it is absorbed micrometers deeper, in neutral silicon, and those carriers take nanoseconds to diffuse back.

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); W. W. Gärtner, "Depletion-layer photoeffects in semiconductors," Phys. Rev. 116, 84 (1959); M. A. Green, Solar Cells: Operating Principles, Technology and System Applications (Prentice-Hall, 1982).