1. Introduction

With the decrease in feature size in semiconductor manufacturing, molecular contamination problems are increased significantly[1]. Recombination lifetime as well as diffusion length measurements have become ubiquitous in the semiconductor industry, because they are a good indicator of wafer contamination[2]. When the defect densities in semiconductors are low, lifetime is one of few parameters that gives significant information about the lower concentration. No other technique can detect defect densities as low as 10⁹–10¹¹ cm⁻³ in a simple, contactless room temperature measurement. In principle, there is no lower limit to the defect density determined by lifetime measurements[3]. It is for these reasons that the IC industry, largely concerned with unipolar MOS devices in which lifetime plays a minor role, has adopted lifetime measurements as a “process cleanliness monitor”[4]. Alongside this, recombination and generation lifetime as well as measurements of carrier diffusion lengths in semiconductors are also very crucial for photovoltaic devices[5] which have recently become very attractive topic to researchers.

The theory of electron-hole pair recombination through recombination centers (traps) was put forth in 50s of the last century in the well-known works by Shockley-Read-Hall (SRH)[6,7]. Here, we discuss lifetimes, their dependence on semiconductor material and device parameters like energy level, injection level & surfaces and how lifetimes are measured. Different measurement methods can give widely differing lifetimes for the same material or device. In most cases, the reasons for these discrepancies are fundamental and are not due to a deficiency of the measurement. The difficulty with defining a lifetime is that we are describing a property of a carrier within the semiconductor rather than the property of the semiconductor itself.

2. Theory

There are two principal categories of carrier lifetimes: recombination lifetimes and generation lifetimes. The concept of recombination lifetime τ_r holds when excess carriers decay as a result of recombination. Generation lifetime τ_g applies when there is a paucity of carriers, as in the space- charge region of a reverse-biased device and the device tries to attain equilibrium. During recombination an electron-hole pair ceases to exist on average after a time τ_r, illustrated in Fig. 1(a). The generation lifetime, by analogy, is the time that it takes on average to generate an ehp, illustrated in Fig. 1(b). Thus generation lifetime is a misnomer, since the creation of an ehp is measured and generation time would be more appropriate. Nevertheless, the term “generation lifetime” is commonly accepted. When these recombination and generation events occur in the bulk, they are characterized by τ_r and τ_g. When they occur at the surface, they are characterized by the surface recombination velocity s_r and the surface generation velocity s_g, also illustrated in Fig. 1. Both bulk and surface recombination or generation occur simultaneously and their separation is sometimes quite difficult. Consisting of bulk and surface components, the measured lifetimes are always effective lifetimes.

It is instructive to consider τ_r and τ_g in more detail before discussing lifetime measurement techniques. The excess ehps may have been generated by photons or particles of energy higher than the band gap or by forward biasing a pn junction. There are more carriers after the stimulus than before and the excess carriers return to equilibrium by recombination.

Figure 1. Illustration of various recombination and generation mechanisms for a (a) forward-biased and (b) reverse-biased junction

2.1. Recombination Lifetime and Surface Velocity

The departure of the carrier densities from their equilibrium values non-linearly controls the bulk recombination rate R. We consider a p-type semiconductor throughout this work and are chiefly concerned with the behavior of the minority electrons. Confining ourselves to linear, quadratic, and third order terms, R can be written as

(1)

where n = n_o + Δn, p = p_o + Δp, n_o, p_o are the equilibrium and Δn, Δp, the excess carrier densities. In the absence of trapping, Δn = Δp, allowing Eq. (1) to be simplified to

(2)

where some terms containing n_o have been dropped because n_o << p_o in a p-type material. The recombination lifetime is defined as

(3a)

giving

(3b)

Figure 2. Variuos mechanisms: (a) SRH (b) radiative and (c) Auger

There are three main recombination mechanisms those determine the recombination lifetime: SRH or multiphonon recombination characterized by τ_SRH , radiative recombination characterized by τ_rad and Auger recombination characterized by τ_Auger. These mechanisms are illustrated in Fig. 2.

The recombination lifetime τ_r is determined according to the relationship given below

(4)

1) SRH Recombination: During SRH recombination, ehps recombine through deep-level impurities or traps, characterized by the density N_T, energy level E_T and capture cross-sections σ_n and σ_p for electrons and holes, respectively. The energy liberated during the recombination event is dissipated by lattice vibrations or phonons, illustrated in Fig. 2(a). The SRH lifetime is given by[6]

(5)

where n₁, p₁, τ_n and τ_p are defined as

(6a)

(6b)

2) Radiative Recombination: During this recombination, ehps recombine directly from band to band with the energy carried away by photons as shown in Fig. 2(b). The radiative lifetime is[8]

(7)

B is the radiative recombination coefficient. The radiative lifetime is inversely proportional to the carrier density because in band-to-band recombination both electrons and holes must be present simultaneously.

3) Auger recombination: During Auger recombination, as illustrated in Fig. 2(c), the recombination energy is absorbed by a third carrier and the Auger lifetime is inversely proportional to the carrier density squared. The Auger lifetime is given by

(8)

where C_p is the Auger recombination coefficient for holes and C_n for electrons. Values for radiative and Auger coefficients are given in Table 1.

Table 1. Recombination Coefficients

2.2. Recombination Lifetime and Level of Injections

Equations (5) to (8) simplify for both low-level and high-level injection. Low-level injection holds when the excess minority carrier density is low compared to the equilibrium majority carrier density, Δn<<p_o. Similarly, high- level injection holds when Δn>>p_o. The injection level is important during lifetime measurements. The appropriate expressions for low-level (ll) and for high-level (hl) injection become

(9)

(9a)

where the second approximation in the τ_SRH(ll) expression holds when n₁<<p_o and p₁<<p_o.

(10)

(11)

The Si recombination lifetimes according to Eq. (4) are plotted in Fig. 3. At high carrier densities, the lifetime is controlled by Auger recombination and at low densities by SRH recombination. Auger recombination has the characteristic 1/n² dependence. The high carrier densities may be due to high doping densities or high excess carrier densities.

Whereas SRH recombination is controlled by the cleanliness of the material, Auger recombination is an intrinsic property of the semiconductor. Radiative recombination plays almost no role in Si except for very high lifetime substrates but is important in direct band gap semiconductors like GaAs.

The bulk SRH recombination rate is given by

(12)

The surface SRH recombination rate is

(13)

Where s_n = σ_nsv_thN_it; s_p = σ_psv_thN_it. The subscript “s” refers to the appropriate quantity at the surface; p_s and n_s are the hole and electron densities (cm⁻³) at the surface. The interface trap density N_it (cm⁻²) is assumed constant in Eq. (13). If not constant, the interface trap density D_it (cm⁻² eV⁻¹) must be integrated over energy with N_it in these equations given by N_it≈ kTD_it[9].

Figure 3. Recombination lifetime versus majority carrier density for n-Si with Cn = 2 × 10−31 cm6/s and B = 4.73 × 10−15 cm3/s

Figure 4. Determination of bulk lifetime, surface recombination velocity and diffusion coefficient from lifetime measurements. Data from [10]

The surface recombination velocity s_r is

(14a)

from eq. (13)

(14b)

The surface recombination velocity for low-level and high-level injection becomes

(15)

s_r depends strongly on injection level for the SiO₂/Si interface as shown in Fig. 4.

2.3. Recombination Lifetime Measurements Techniques

Recombination lifetime can be measured optically or electrically. The commonly used techniques are

1) Optical Measurements: Photoconductance Decay (PCD), Quasi-Steady-State Photoconductance (QSSPC), Short-Circuit Current/Open-Circuit Voltage Decay (SCCD/ OCVD), Photoluminescence Decay (PLD), Surface Photovoltage (SPV), Steady-State Short-Circuit Current (SSSCC), Free Carrier Absorption, Electron Beam Induced Current (EBIC) etc.

2) Electrical Measurements: Diode Current-Voltage, Reverse Recovery (RR), Open-Circuit Voltage Decay (OCVD), Pulsed MOS Capacitor, Other Techniques.

Figure 5. The surface recombination velocity sr versus injection level η as a function of σps for Nit = 1010 cm−2, pos = 1016 cm−3, ETs = 0.4 eV, σns = 5 × 10−14 cm2. Data from [11]

2.4. Generation Lifetime and Surface Velocity

Every recombination process of Fig. 2 has a generation counterpart. The inverse of multiphonon recombination is thermal ehp generation in Fig. 1(b). The inverse of radiative and Auger recombination are optical and impact ionization generation. Optical generation is negligible for a device in the dark and with negligible blackbody radiation from its surroundings. Impact ionization is usually considered to be negligible for devices biased sufficiently below breakdown voltage. However, impact ionization at low ionization rates can occur at low voltages and care must be taken to eliminate this generation mechanism during τ_g measurements.

Generation dominates for pn < n²_i. Furthermore the smaller the pn product, the higher is the generation rate. R becomes negative and is then designated as the bulk generation rate G

(16)

for pn ≈ 0 with

(17)

The condition pn→0 is approximated in the scr of a reverse-biased junction.

The quantity τ_g, defined in Eq. (17), is the generation lifetime that depends inversely on the impurity density and on the capture cross-section for electrons and holes, just as recombination does. The generation lifetime can be quite high if E_T does not coincide with E_i. Generally, τ_g is higher than τ_r, at least for Si devices, where detailed comparisons have been made and τ_g ≈ (50–100)τ_r.

When p_sn_s < n²_i at the surface, we find from Eq. (13), the surface generation rate

(18)

where s_g is the surface generation velocity, given by

(19)

From Eqs. (14b) and (19), for E_it ≠ E_i it is found that s_r > s_g.

Generation lifetime is usually measured by electrical means such as Gate-Controlled Diode and Pulsed MOS Capacitor methods.

3. Methodology

Of the various lifetime measurement methods, optical technique is stated in the following section.

3.1. Recombination Lifetime: Optical Measurements

Consider a p-type semiconductor with light incident on the sample. The light may be steady state or transient. The continuity equation for uniform ehp generation and zero surface recombination is [12,13]

(20)

where Δn(t) is the time dependent excess minority carrier density, G the ehp generation rate and τ_eff the effective lifetime. Solving for τ_eff gives

(21)

In the transient PCD method, with G(t) << dΔn(t)/dt, the effective lifetime becomes τ_eff(Δn) = -Δn(t)/. In the steady-state method, with G(t) >> dΔn(t)/dt, the effective lifetime becomes τ_eff(Δn) = Δn/G and in the QSSPC method, Eq. (21) remains valid. Both Δn and G need to be known in the steady-state and QSSPC methods to determine the effective lifetime.

The excess carrier density decay for low level injection is given by Δn(t) = Δn(0)exp(-t/τ_eff) where τ_eff is calculated as

(22)

with β found from the relationship tan(βd/2) = s_r/(βD). Here τ_B is the bulk recombination lifetime, D the minority carrier diffusion constant under low injection level and the ambipolar diffusion constant under high injection level and d the sample thickness. Equation (22) holds for any optical absorption depth provided the excess carrier density has ample time to distribute uniformly, i.e., d << (Dt)^1/2. The effective lifetime of Eq. (20) is plotted in Fig. 5 versus d as a function of s_r.

For thin samples, τ_eff no longer bears any resemblance to τ_B, the bulk lifetime and is dominated by surface recombination. The surface recombination velocity must be known to determine τ_B unambiguously unless the sample is sufficiently thick. Although the surface recombination velocity of a sample is generally not known, by providing the sample with high s_r, by sandblasting for example, it is possible to determine τ_B directly. However, the sample must be extraordinarily thick. Equation (22) can be written as

(23)

where τ_s is the surface lifetime.

Figure 6. Effective lifetime versus wafer thickness as a function of surface recombination velocity. D = 30 cm2/s

Two limiting cases are of particular interest: s_r→0 gives tan(βd/2) ≈ βd/2 and s_r→∞ gives tan(βd/2)≈∞ or βd/2 ≈ π/2, making the surface lifetime

(24)

For s_r→0, a plot of 1/τ_eff versus 1/d has a slope of 2s_r and an intercept of 1/τ_B, allowing both s_r and τ_B to be determined. For s_r→∞, a plot of 1/τ_eff versus 1/d² has a slope of π²D and an intercept of 1/τ_B. Both examples are illustrated in Fig. 6. The approximation τ_s = d/2s_r holds for s_r < D/4d.

Equations (22)–(24) hold for samples with one dimension much smaller than the other two dimensions, for example, a wafer. For samples with none of the three dimensions very large, Eq. (23 becomes for s_r→∞

(25)

where a, b and c are the sample dimensions. It is recommended that the sample surfaces have high surface recombination velocities. The recommended dimensions and the maximum bulk lifetimes that can be determined through Eq. (23) for Si samples are given in Table 2.

Table 2. Recommended Dimensions for PCD Samples and Maximum Bulk Lifetimes for Silicon

1) Photoconductance Decay: The photoconductance decay lifetime characterization technique was proposed in 1955 [15] and has become one of the most common lifetime measurement techniques. As the name implies, ehps are created by optical excitation and their decay is monitored as a function of time following the cessation of the excitation. Other excitation means such as high-energy electrons and gamma rays can also be used. The samples may either be contactless (Fig. 7) or the measurement can be contacted with the current being monitored (Fig. 8).

Figure 7. PCD measurement schematic for contactless (a) rf bridge and (b) microwave reflectance measurements

Figure 8. Schematic diagram for contact photoconductance decay measurement

2) Quasi-Steady-State Photoconductance (QSSPC): In the QSSPC method the sample is illuminated with a flash lamp with a decay time constant of several ms and an illumination area of several cm²[7,9]. Due to the slow decay time, the sample is under quasi steady-state conditions during the measurement as the light intensity varies from its maximum to zero. The steady-state condition is maintained as long as the flash lamp time constant is longer than the effective carrier lifetime. The timevarying photoconductance is detected by inductive coupling. The excess carrier density is calculated from the photoconductance signal. The generation rate, required in Eq. (7.25), is determined from the light intensity measured with a calibrated detector. Semiconductors absorb only a fraction of the incident photons, depending on the reflectivity of the front and back surfaces, possible faceting of those surfaces, and the thickness of the wafer. The value of the absorption fraction for a polished, bare silicon wafer is f ≈ 0.6.

If the wafer has an optimized antireflection coating, f ≈ 0.9, while a textured wafer with antireflection coating can approach f ≈ 1.36 The generation rate per unit volume G can then be evaluated from the incident photon flux and the wafer thickness, according to where is the photon flux density and the sample thickness. Assuming the flash lamp light decay is exponential in time, the generation rate is higher for τ_eff < τ_flash, the sample is in quasi steady-state during the measurement. Hence, the flash lamp decay time must be sufficiently long for the QSSCP measurement to be valid.

4. Conclusions

The microwave reflection or inductive coupling photoconductance decay technique is commonly used to measure carrier lifetime. Its key strength is the contactless nature and rapidness and major weakness is the unknown surface recombination velocity. If the sample thickness can be changed, then both the bulk lifetime and the surface recombination velocity can be extracted. The quasi-steady-state photoconductance method is a more recent method and has found wide acceptance in the design of photovoltaic devices. It measures the lifetime as a function of injection level in one step but requires large sample area (several cm²) precluding high density mapping. Surface photovoltage technique is used to detect iron in p-Si. The most common electrical recombination lifetime method is the open-circuit voltage decay method. Measured τ_r or L_n mean little for thin layers, e.g., epitaxial layers on highly doped substrates, denuded zones on heavily precipitated substrates, or SOI films. Such layers are best characterized through generation lifetime characterization [16] which is commonly determined with the pulsed MOS capacitor. The Zerbst plot implementation is the most common, but the current versus inverse capacitance is easier to interpret because the doping density of the sample need not be known. Since τ_g is measured in the space-charge region of a reverse-biased device, it lends itself easily for the characterization of thin layers. Since the scr width can be varied by an applied voltage, it is possible to generate a τ_g depth profile, that is difficult to do with τ_r measurements, because the measurement depth for τ_r and L_n measurements is the minority carrier diffusion length.

ACKNOWLEDGEMENTS

Authors of this paper would like to thank the Department of EEE, BUET, Dhaka-1000, Bangladesh, for its various supports during the preparation of this manuscript.