Carrier injection

Carrier injection is the introduction of excess electrons and holes into a region of a semiconductor, above their equilibrium densities, by passing a current across a junction. Under forward bias, electrons cross from the n side of a p-n junction and holes from the p side, and where the two populations meet they recombine, emitting light in an LED or a laser diode. In a silicon injection modulator the injected carriers change the refractive index instead. Injection is how semiconductor lasers are pumped electrically: in a telecom laser diode, a current of a few milliamperes at a junction voltage slightly above the photon energy in electronvolts is enough to give its quantum wells optical gain.

The forward-biased junction

At equilibrium a p-n junction holds a built-in voltage $V_{bi}$ across its depletion region, a barrier that keeps the electrons of the n side and the holes of the p side apart. A forward bias $V_a$ lowers the barrier by $q{V}_a$, and carriers diffuse across it: electrons into the p side and holes into the n side, where they are minority carriers. At the edge of the depletion region on the p side, the electron density rises exponentially with the bias, the law of the junction:

$$n_p=n_{p0}\exp \text{\hspace{0.17em}⁣}\left(\frac{q{V}_a}{k_{B}T}\right),$$

where $n_{p0}=n_i^2\mathrm{/}{N}_A$ is the equilibrium minority density. In silicon doped with $N_A={10}^{17}$ cm⁻³, with $n_i=9.65\times {10}^9$ cm⁻³ at 300 K, $n_{p0}$ is about 930 cm⁻³, and a forward bias of 0.6 V raises it by a factor of $1.2\times {10}^{10}$ to $1.1\times {10}^{13}$ cm⁻³, still far below the doping, which is the condition of low-level injection. The injected carriers diffuse away from the junction and recombine within a diffusion length, and the two minority-carrier currents together make up the diode current, which grows as $\exp (q{V}_a\mathrm{/}{k}_{B}T)$. Out of equilibrium the single Fermi level splits into separate quasi-Fermi levels for electrons and holes, and across the junction their separation equals the applied bias, $E_{Fn}-E_{Fp}=q{V}_a$.

Heterostructures and quantum wells

In a homojunction the injected carriers spread by diffusion over a few micrometers and the light is barely guided, so a large current is needed before there is any gain; the first diode lasers, homojunctions of GaAs and GaAsP made in 1962, operated only in short pulses and, at first, only when cooled to 77 K. Kroemer proposed in 1963 to place the active layer between layers of wider bandgap, and double heterostructures made continuous operation at room temperature possible in 1970 (Hayashi et al.). The band offsets at the two heterojunctions hold the injected electrons and holes in the active layer (carrier confinement), and the lower refractive index of the wider-gap layers guides the light along it, so the carriers and the light are concentrated in the same place. Modern lasers use a separate-confinement heterostructure: quantum wells a few nanometers thick hold the carriers inside a waveguide core of intermediate bandgap, a few hundred nanometers thick, that guides the light, between cladding layers doped p and n that carry the current.

How well the barriers hold the carriers depends on the band offsets. Leakage over them rises steeply with temperature and, together with Auger recombination, lowers the characteristic temperature of InP-based lasers. In AlGaInAs heterojunctions about 72% of the bandgap difference falls in the conduction band, against about 40% in InGaAsP, so AlGaInAs wells hold the electrons, the lighter and more mobile carriers, more firmly, which suits lasers that run without temperature control (Zah et al.). Nitride lasers and LEDs add a thin AlGaN electron-blocking layer on the p side for the same reason.

Carrier density and current

The active region of a laser or LED is usually undoped, so the injected electrons and holes are equal in number, $N=P$, and their density is set by the balance between the current that supplies them and the recombination that removes them:

$$\frac{{\eta }_{i}J}{qd}=AN+B{N}^2+C{N}^3=\frac{N}{\tau },$$

where $J$ is the current density, $d$ the total thickness of the active layers, ${\eta }_i$ the injection efficiency, and $A$, $B$ and $C$ the coefficients of defect, radiative and Auger recombination, which together set the carrier lifetime $\tau$. With the illustrative coefficients of the radiative recombination entry ($A={10}^6$ s⁻¹, $B={10}^{-10}$ cm³/s, $C={10}^{-28}$ cm⁶/s), a density of $2\times {10}^{18}$ cm⁻³, of the order at which InGaAsP quantum wells reach threshold, has a lifetime of 1.66 ns and takes 116 A/cm² to sustain in each 6 nm well. Five such wells under a 2 µm × 300 µm ridge need 3.5 mA with every electron delivered to the wells, close to the 3.6 to 7.2 mA that measured threshold current densities give for a ridge of that size (threshold current). Above threshold the density stops rising: stimulated emission removes every additional carrier, the gain stays clamped at the cavity loss, and the extra current becomes laser light (laser threshold).

Not every electron that enters the contact reaches the active region. The injection efficiency ${\eta }_i$ is the fraction of the terminal current that recombines in the active layers; the rest leaks over the heterobarriers, recombines in the confinement layers, or spreads sideways past the active stripe. Above threshold it is what the internal quantum efficiency of a laser mostly measures, 0.70 to 0.90 in InGaAsP multi-quantum-well lasers and 0.85 to 0.95 in GaAs-based ones. Lasers limit the sideways spreading by etching a narrow ridge, by burying the active stripe in current-blocking or semi-insulating material, or, in a VCSEL, by oxidizing a high-aluminum AlGaAs layer from the sides, which leaves a conducting aperture at the center that channels the current.

Junction voltage

Gain at a photon energy $h\nu$ requires the quasi-Fermi levels to be separated by more than $h\nu$, the Bernard-Duraffourg condition, and injection sets that separation through the voltage across the junction, $E_{Fn}-E_{Fp}\approx q{V}_j$. A laser diode therefore needs a junction voltage above its photon energy in electronvolts, $V_j>h\nu \mathrm{/}q$:

The voltage at the terminals is higher, since the current also crosses the series resistance of the cladding and contacts and the potential steps at the heterojunctions. The excess is dissipated as heat, and the ratio $h\nu \mathrm{/}qV$, the photon energy as a fraction of the energy each electron receives from the supply, is one of the factors of the wall-plug efficiency.

Modulation speed

In an LED the light follows the carrier density, and the density responds to a change of current only as fast as the carriers recombine: for small signals the response is that of a single-pole low-pass filter with its corner at $1\mathrm{/}(2\pi {\tau }_d)$, where ${\tau }_d=(dR\mathrm{/}dN{)}^{-1}$ is the differential lifetime. It is shorter than $N\mathrm{/}R$ because radiative and Auger recombination grow faster than the density: 0.62 ns in the example above, against 1.66 ns, which puts the corner at 255 MHz. Both lifetimes shorten as the density rises, so an LED is faster at high current density.

A laser diode above threshold is not limited in this way. Its carrier density is clamped, and a change of current is taken up by the photon density through stimulated emission, so the response is a damped resonance at the relaxation oscillation frequency, which grows as the square root of the current above threshold, $f_r=D\sqrt{I-I_{\text{th}}}$. A telecom DFB laser with $D$ = 2 GHz/√mA resonates at 10 GHz when biased 25 mA above threshold, and while the damping is light its 3 dB modulation bandwidth is about $1.55f_r$, or 15.5 GHz. The carrier lifetime enters only through the damping offset, and the K-factor sets the ceiling. The injection itself adds three limits:

• Parasitics. The contact pad and the junction form a capacitance that the driver charges through its own resistance and the series resistance of the chip, a low-pass filter with its corner at $1\mathrm{/}(2\pi RC)$: 3.2 GHz for 1 pF and 50 Ω. A corner above 20 GHz needs less than 0.16 pF in this simple model, and high-speed lasers have small contact pads on thick, low-permittivity insulation for that reason.
• Transport. Carriers injected from the cladding must cross the confinement layers and be captured by the wells, from which they can also escape. Nagarajan and colleagues showed that this transport adds a low-frequency roll-off and lowers the effective differential gain, and with it the relaxation frequency.
• Turn-on delay. A laser switched from a bias $I_b$ below threshold to a current $I$ must first fill its active region to the threshold density, which takes $t_d=\tau \ln [(I-I_b)\mathrm{/}(I-I_{\text{th}})]$ if the lifetime is taken as constant. With the lifetime falling as the density rises, integrating the rate equation with the coefficients of the example gives 1.0 ns from zero bias to twice threshold, and 0.06 ns from a bias of $0.9I_{\text{th}}$, where the density already stands at 96% of its threshold value. Transmitters therefore bias the laser near or above threshold.

Silicon injection modulators

Silicon has no Pockels effect, and its fast phase shifters change the index through the plasma dispersion effect, which follows the carrier density whether carriers are added or removed. In a forward-biased p-i-n diode built across the waveguide, carriers fill the undoped core: at 1550 nm, 10¹⁸ cm⁻³ of electrons and holes lowers the index by $3.0\times {10}^{-3}$ and adds 14.5 cm⁻¹ of free-carrier absorption, 63 dB/cm (Soref and Bennett). Because the stored charge grows exponentially with the forward voltage, a small voltage gives a large phase shift: a 10 Gb/s injection Mach-Zehnder modulator reached a $V_{\pi }L$ of 0.036 V·cm (Green et al.), against 1 to 3 V·cm for depletion modulators (half-wave voltage).

The injected carriers leave only by recombining, so the phase follows the drive with the carrier lifetime, which puts the corner at 160 MHz for a lifetime of 1 ns. Data rates of gigabits per second took pre-emphasis, a brief overdrive at each transition that pushes charge in and pulls it out faster than recombination would: 10 Gb/s in a Mach-Zehnder (Green et al.) and 12.5 Gb/s in a microring (Xu et al.). High-speed silicon modulators use depletion instead, whose speed is set by the junction capacitance and the electrodes rather than by a lifetime, and injection is used where speed matters less: switches, variable optical attenuators that use the added absorption, and slow phase trimming. The plasma dispersion entry compares the injection, depletion and accumulation designs.

Optical injection

Absorbed light injects carriers too, one electron-hole pair for each photon absorbed above the bandgap. Optically pumped semiconductor lasers such as the VECSEL bring their quantum wells to gain this way without a junction, and in a photodiode or a solar cell the junction works in reverse, separating the photogenerated carriers and delivering them as current. The quasi-Fermi-level splitting that light produces in an illuminated solar cell sets its open-circuit voltage, just as the splitting produced by current sets the junction voltage of a laser.

Common questions

What is carrier injection?

The introduction of electrons and holes into a semiconductor region above their equilibrium densities, usually by forward-biasing a p-n junction so that electrons cross from the n side and holes from the p side. The injected carriers recombine to emit light in LEDs and laser diodes, and change the refractive index in injection modulators.

How does carrier injection work in a laser diode?

Current through the forward-biased junction delivers electrons and holes to quantum wells between layers of wider bandgap, which confine them. When their quasi-Fermi levels separate by more than the photon energy the wells have optical gain, and at threshold the gain equals the cavity loss; above threshold the carrier density stays clamped and the additional current becomes laser output.

What is injection efficiency?

The fraction of the current through a device that recombines in its active region. The rest leaks over the heterobarriers, recombines in neighboring layers or spreads outside the active stripe. Above threshold the injection efficiency of a good laser is about 0.7 to 0.95.

Does the carrier lifetime limit how fast a laser can be modulated?

Not above threshold. The lifetime sets the speed of an LED and of a silicon injection modulator, a corner near $1\mathrm{/}(2\pi \tau )$, but a laser above threshold responds through stimulated emission at its relaxation oscillation frequency, which rises with the drive current into the gigahertz range. The lifetime matters for a laser switched on from below threshold, where it delays the light pulse.

Why do silicon modulators use depletion rather than injection?

Injected carriers leave only by recombining, which limits an injection modulator to a corner of $1\mathrm{/}(2\pi \tau )$, 160 MHz for a 1 ns lifetime, unless the drive is pre-emphasized, and the carriers add absorption. A depletion modulator moves carriers with the junction field, so its speed is limited by capacitance and electrodes and reaches tens of gigahertz, at the cost of a $V_{\pi }L$ of 1 to 3 V·cm against a few hundredths of a V·cm for injection.

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