4.5 Carrier lifetime and diffusion length

Carrier lifetime and diffusion length are crucial concepts in semiconductor physics. They describe how long carriers survive and how far they travel before recombining. These parameters significantly impact device performance, influencing everything from solar cell efficiency to transistor switching speeds.

Understanding these concepts helps engineers optimize semiconductor devices. By manipulating carrier lifetimes and diffusion lengths through material choice and device design, we can enhance the performance of various electronic and optoelectronic components.

Carrier generation and recombination

• Carrier generation and recombination are fundamental processes in semiconductor devices that involve the creation and annihilation of electron-hole pairs
• Understanding these processes is crucial for designing and optimizing semiconductor devices such as solar cells, LEDs, and transistors

Radiative recombination

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• Occurs when an electron in the conduction band directly combines with a hole in the valence band, releasing energy in the form of a photon
• The energy of the emitted photon is approximately equal to the bandgap energy of the semiconductor material
• Radiative recombination is the dominant recombination mechanism in direct bandgap semiconductors (GaAs, InP)
• The radiative recombination rate is proportional to the product of the electron and hole concentrations

Shockley-Read-Hall recombination

• Also known as trap-assisted recombination, involves the capture of an electron and a hole by an energy level within the bandgap introduced by defects or impurities
• The captured electron and hole then recombine non-radiatively, releasing energy in the form of phonons (lattice vibrations)
• SRH recombination is the dominant recombination mechanism in indirect bandgap semiconductors (Si, Ge) and at low injection levels
• The SRH recombination rate depends on the trap density, capture cross-sections, and the position of the trap level within the bandgap

Auger recombination

• A three-particle process where an electron and a hole recombine, transferring the released energy to a third carrier (electron or hole), which is then excited to a higher energy state
• The excited carrier subsequently relaxes back to its original state by releasing energy in the form of phonons
• Auger recombination becomes significant at high injection levels and in heavily doped semiconductors
• The Auger recombination rate is proportional to the product of the electron and hole concentrations and the concentration of the third carrier

Surface recombination

• Occurs at the surface of a semiconductor due to the presence of dangling bonds and surface states that act as recombination centers
• Surface recombination can be reduced by passivating the surface with dielectric layers (SiO2, Si3N4) or by introducing electric fields that repel carriers from the surface
• The surface recombination rate depends on the surface recombination velocity, which is a measure of the effectiveness of the surface in capturing and recombining carriers

Carrier lifetime

• Carrier lifetime is a critical parameter that describes the average time a carrier (electron or hole) spends in an excited state before recombining

Definition of carrier lifetime

• The carrier lifetime ($\tau$) is defined as the average time between the generation and recombination of a carrier
• Mathematically, the carrier lifetime is the ratio of the excess carrier concentration ($\Delta n$ or $\Delta p$) to the recombination rate ($R$): $\tau = \frac{\Delta n}{R}$ or $\tau = \frac{\Delta p}{R}$

Minority carrier lifetime

• In a semiconductor, the minority carrier lifetime refers to the lifetime of the less abundant carrier type (electrons in p-type, holes in n-type)
• Minority carrier lifetime is particularly important in bipolar devices (solar cells, LEDs, bipolar transistors) where the device performance depends on the minority carrier transport and recombination

Bulk vs surface lifetime

• The bulk lifetime ($\tau_{bulk}$) is the carrier lifetime in the bulk of the semiconductor, determined by the recombination processes occurring within the material (radiative, SRH, Auger)
• The surface lifetime ($\tau_{surface}$) is the carrier lifetime near the surface of the semiconductor, influenced by surface recombination
• The effective lifetime ($\tau_{eff}$) is a combination of the bulk and surface lifetimes, given by $\frac{1}{\tau_{eff}} = \frac{1}{\tau_{bulk}} + \frac{1}{\tau_{surface}}$

Effective lifetime

• The effective lifetime ($\tau_{eff}$) is the overall carrier lifetime in a semiconductor device, considering both bulk and surface recombination
• The effective lifetime is always shorter than the bulk and surface lifetimes, as it is limited by the fastest recombination process
• Improving the effective lifetime requires optimizing both the bulk and surface properties of the semiconductor

Lifetime in low vs high injection

• Low injection conditions occur when the excess carrier concentration is much lower than the majority carrier concentration ($\Delta n \ll N_D$ or $\Delta p \ll N_A$)
• High injection conditions occur when the excess carrier concentration is comparable to or higher than the majority carrier concentration ($\Delta n \geq N_D$ or $\Delta p \geq N_A$)
• The carrier lifetime can vary significantly between low and high injection conditions due to the different recombination mechanisms dominating in each regime

Radiative lifetime

• The radiative lifetime ($\tau_{rad}$) is the carrier lifetime determined by radiative recombination
• In direct bandgap semiconductors, the radiative lifetime is inversely proportional to the carrier concentration: $\tau_{rad} \propto \frac{1}{n}$ or $\tau_{rad} \propto \frac{1}{p}$
• Radiative lifetime is important in optoelectronic devices (LEDs, lasers) where radiative recombination is the desired process

SRH lifetime

• The SRH lifetime ($\tau_{SRH}$) is the carrier lifetime determined by Shockley-Read-Hall recombination
• The SRH lifetime depends on the trap density ($N_t$), capture cross-sections for electrons ($\sigma_n$) and holes ($\sigma_p$), and the position of the trap level within the bandgap ($E_t$)
• Minimizing the SRH lifetime requires reducing the trap density and optimizing the trap properties

Auger lifetime

• The Auger lifetime ($\tau_{Auger}$) is the carrier lifetime determined by Auger recombination
• The Auger lifetime is inversely proportional to the square of the carrier concentration: $\tau_{Auger} \propto \frac{1}{n^2}$ or $\tau_{Auger} \propto \frac{1}{p^2}$
• Auger recombination becomes significant at high injection levels and in heavily doped semiconductors, limiting the performance of devices such as solar cells and LEDs

Lifetime measurement techniques

• Several techniques are used to measure carrier lifetimes in semiconductors, including:
  1. Photoconductance decay (PCD): measures the decay of the conductance of a semiconductor after a light pulse
  2. Quasi-steady-state photoconductance (QSSPC): measures the conductance of a semiconductor under constant illumination
  3. Microwave photoconductance decay (µ-PCD): uses microwave reflectance to measure the decay of the conductance
  4. Time-resolved photoluminescence (TRPL): measures the decay of the photoluminescence intensity after a light pulse

Carrier diffusion

• Carrier diffusion is the transport of carriers (electrons or holes) in a semiconductor due to a concentration gradient

Diffusion current

• The diffusion current ($J_{diff}$) is the current resulting from the diffusion of carriers in a semiconductor
• The diffusion current is proportional to the gradient of the carrier concentration: $J_{diff,n} = qD_n\frac{dn}{dx}$ for electrons and $J_{diff,p} = -qD_p\frac{dp}{dx}$ for holes
• The proportionality constants $D_n$ and $D_p$ are the diffusion coefficients for electrons and holes, respectively

Einstein relation

• The Einstein relation connects the diffusion coefficient ($D$) and the mobility ($\mu$) of a carrier in a semiconductor
• The Einstein relation is given by $\frac{D}{\mu} = \frac{k_BT}{q}$, where $k_B$ is the Boltzmann constant, $T$ is the absolute temperature, and $q$ is the elementary charge
• The Einstein relation is a consequence of the fact that both diffusion and drift (mobility) are driven by the same underlying physical process: the random thermal motion of carriers

Ambipolar diffusion

• Ambipolar diffusion occurs when the diffusion of electrons and holes is coupled due to the requirement of charge neutrality
• In ambipolar diffusion, the faster diffusing carrier (usually electrons) is slowed down by the electric field created by the slower diffusing carrier (usually holes)
• The ambipolar diffusion coefficient ($D_a$) is a weighted average of the electron and hole diffusion coefficients: $D_a = \frac{n+p}{n/D_p + p/D_n}$, where $n$ and $p$ are the electron and hole concentrations

Diffusion length

• The diffusion length is the average distance a carrier travels before recombining

Definition of diffusion length

• The diffusion length ($L$) is defined as the square root of the product of the diffusion coefficient ($D$) and the carrier lifetime ($\tau$): $L = \sqrt{D\tau}$
• The diffusion length is a measure of how far a carrier can diffuse before recombining, and it depends on both the carrier mobility (through $D$) and the recombination processes (through $\tau$)

Minority carrier diffusion length

• The minority carrier diffusion length is the diffusion length of the less abundant carrier type in a semiconductor (electrons in p-type, holes in n-type)
• The minority carrier diffusion length is particularly important in bipolar devices (solar cells, LEDs, bipolar transistors) where the device performance depends on the minority carrier transport
• In a p-type semiconductor, the minority carrier (electron) diffusion length is given by $L_n = \sqrt{D_n\tau_n}$, while in an n-type semiconductor, the minority carrier (hole) diffusion length is given by $L_p = \sqrt{D_p\tau_p}$

Diffusion length vs carrier lifetime

• The diffusion length is directly related to the carrier lifetime, as a longer lifetime allows carriers to diffuse further before recombining
• Improving the carrier lifetime, either by reducing the defect density or by optimizing the recombination processes, leads to an increase in the diffusion length
• The relationship between the diffusion length and the carrier lifetime is given by $L = \sqrt{D\tau}$, showing that the diffusion length scales with the square root of the lifetime

Measurement of diffusion length

• Several techniques are used to measure the diffusion length in semiconductors, including:
  1. Electron beam induced current (EBIC): measures the current induced by an electron beam as a function of the distance from a collecting junction
  2. Surface photovoltage (SPV): measures the change in the surface potential of a semiconductor under illumination
  3. Cathodoluminescence (CL): measures the luminescence intensity as a function of the distance from the excitation source (electron beam)
• These techniques rely on the fact that the carrier collection efficiency depends on the diffusion length, allowing the extraction of the diffusion length from the measured data