DE102013202480B4 - Method for determining the actual carrier lifetime of a semiconductor substrate from a dynamic and differential measurement of the relaxation time of free charge carriers - Google Patents

Abstract

A method for determining the actual carrier lifetime of a semiconductor substrate from a dynamic and differential measurement of free carrier relaxation time, comprising the following steps: A generating a first excess carrier density Δn and at least one second excess carrier density Δn in the semiconductor substrate, wherein the first excess carrier density Δn is different than the second excess carrier density Δn and dynamic and differentially measuring a first carrier relaxation time τ and a first generation rate Gin the semiconductor substrate from first excess carrier density Δn and at least one second carrier relaxation time τ and at least one second generation rate Gin the semiconductor substrate from the second excess carrier density Δn, B calculating the actual carrier lifetime τ ( i = 1.2) at the excess charge carrier density Δn (i = 1,2) depending on the measured variables τ, G, τ and G determined in method step A, characterized in that in method step A the carrier relaxation time τ is the characteristic relaxation time of the excess charge carrier density after a change in the generation rate G → ξ · G (i = 1,2), where 0≤ξ ≠ 1 is measured, and that in method step B the actual carrier lifetime τ (i = 1,2) is determined on the basis of an analytical relationship between τ and the measured quantities τ and G (i = 1,2 ) is determined.

Description

The invention relates to a method for determining the actual charge carrier lifetime of a semiconductor substrate from a dynamic and differential measurement of the relaxation time of free charge carriers.

When using semiconductor substrates for the production of semiconductor components, for example photovoltaic solar cells or other electronic components, an exact determination of material parameters is of great interest. In particular, carrier lifetime is an important parameter for many semiconductor substrate applications, so there is a need for robust, cost effective, and accurate carrier lifetime measurement.

It is known to measure the time decay behavior of free charge carriers in a semiconductor substrate and to determine therefrom a carrier lifetime. It is furthermore known that carrying out a dynamic measurement of a charge carrier lifetime from the time decay behavior of free charge carriers under time-constant background charge carrier injection, such as, for example, by constant illumination (bias light), does not determine the actual but a so-called differential charge carrier Lifetime leads. This is in R. Brendel, Appl. Phys. A 60, 523 (1995) and J. Schmidt, IEEE Transactions on Electron Devices 46, 2018 (1999) described.

Here and below, "injection" refers to the generation of free charge carrier pairs in the semiconductor substrate. This can be achieved, for example, by illumination and / or by electrical contacting and generating a current flow through the semiconductor.

Therefore, methods are used in which an actual charge carrier lifetime is calculated from a differential measurement by integration of the differential measurement result over the entire injection range below an excess charge carrier density of interest. This is in AG Aberle, J. Schmidt, and R. Brendel, J. Appl. Phys. 79, 1491 (1996) and J. Schmidt, IEEE Transactions on Electron Devices 46, 2018 (1999) described.

However, such a procedure for determining an actual charge carrier lifetime has several disadvantages:

First, the measurability of carrier lifetimes up to arbitrarily small injection densities is assumed; However, this is not sufficiently fulfilled under real measurement conditions due to the noise-induced sensitivity limits of a measuring apparatus.

Second, the measurement of carrier lifetime across all excess charge carrier densities below an excess charge carrier density of interest is disproportionately expensive, if only one or only a few specific excess charge carrier densities are of interest. Such measuring methods are therefore expensive and / or error-prone.

Here and below, the excess charge carrier density denotes the density of the free charge carriers in the semiconductor substrate, in the case of a doped semiconductor substrate the density of the free minority carriers.

The invention is therefore based on the object to provide a method for determining the actual charge carrier lifetime of a semiconductor substrate, which overcomes the aforementioned disadvantages.

This object is achieved by a method according to claim 1. Advantageous embodiments of the method according to the invention can be found in claims 2 to 14.

The invention is based on the one hand in the knowledge that not only measurements of the carrier lifetime under constant background carrier charge injection, but also continuously and in particular harmonically time-modulated measurement methods for determining a differential carrier lifetime, such as in WO 2011/134602 described method for determining material parameters of a semiconductor substrate by measuring luminescence radiation. Thus, for example, also known measurement methods from photovoltaics, such as time-modulated / quasi-static photoluminescence (as in US Pat JA Giesecke, MC Schubert, and W. Warta, Phys. Stat. Sol. (a) 209, 2286 (2012 ), in JA Giesecke, MC Schubert, D.Walter, and W. Warta, Appl. Phys. Lett. 97, 092109 (2010) or in Brüggemann and S. Reynolds, J. Non-Cryst. Solids 352, 1888 (2006) described) or MFCA (Modulated Free Carrier Absorption, as in F. Sanii, FP Giles, RJ Schwartz, and JL Gray, Solid State. Electronics 35, 311 (1992) , in SW Glunz, AB Sproul, W. Warta, and W. Wettling, J. Appl. Phys. 75, 1611 (1994) or in SW Glunz and W. Warta, J. Appl. Phys. 77, 3243 (1995) described) or it requires complex measurement and / or evaluation. Therefore, there is a need for a method which reliably and inexpensively enables the determination of the actual carrier lifetime. This is provided by the method according to the invention.

Furthermore, the invention is based on the recognition that there is an analytical connection, such that from the measurement of two pairs of differential charge carrier lifetime and generation rate in the semiconductor substrate, as explained in more detail below, using the aforementioned analytical context, the actual charge carrier lifetime can be determined.

It is thus possible for the first time, without complicated and with respect to the convergence error-prone numerical method (as in WO2011 / 134602 described) and without "integrating" the measurement data over a large charge carrier Injektonsbereich (as in AG Aberle, J. Schmidt, and R. Brendel, J. Appl. Phys. 79, 1491 (1996) described) to determine the actual charge carrier lifetime.

The method according to the invention for determining the actual charge carrier lifetime of a semiconductor substrate comprises the following method steps:

In a method step A, a first excess charge carrier density Δn ₁ and at least one second excess carrier density Δn _{2 are} generated in the semiconductor substrate, wherein the first excess carrier density Δn _{1 is} different from the second excess carrier density Δn ₂ . Furthermore, a dynamic and differential measurement of a first charge carrier relaxation time τ _{M, 1} and a first generation rate G ₁ in the semiconductor substrate is carried out on the basis of first excess charge carrier density Δn ₁ and at least one second charge carrier relaxation time τ _{M, 2} and at least one second generation rate G ₂ in FIG the semiconductor substrate, starting from the second excess charge carrier density Δn ₂ .

Dynamic and differential measurement here refers to the determination of the carrier relaxation time τ _{M, i} , either by measuring the relaxation time of the excess charge carrier density after an abrupt change in the generation rate of free charge carriers G _i → ξ. G _i with a bias light parameter ξ≥0 and ξ ≠ 1, or by measuring the phase shift between a quasi-static and continuous (in particular harmonic) time-modulated generation rate G (t) and the resulting excess charge carrier density Δn (t). This case of continuous time modulation corresponds to a superposition of infinitesimal changes of the generation rate with variable time density (limit value ξ → 1).

in der weiter unten aufgeführten Formel 1 größer Null ist. When measuring the decay time, the choice of a bias light parameter ξ <1 is customary, since usually the relaxation time of the excess charge carrier density is measured after a very short light pulse with very low intensity compared to the bias illumination. However, it would also be conceivable to measure the relaxation time of the excess charge carrier density after an abrupt increase in the generation rate, which could be described with a ξ> 1. Preferably, ξ is chosen to be sufficiently small, so that the denominator $\left({1-\xi G_i\frac{d{\tau }_{0i}}{d\mathrm{\Delta }n}|}_{\mathrm{\Delta }{n}_i}\right)$

in formula 1 below is greater than zero.

In both cases, the measurement is dynamic because the measurand is derived from the time-dependent response of an experiment. In both cases, the measurement is differential, since it can be shown that both o.g. Relaxation time as well o.g. Phase shift correspond exactly to the change in the excess charge carrier density per change in the generation rate of free charge carriers.

In a method step B, the actual carrier lifetime τ _{0, i} (i = 1,2) at the excess charge carrier density Δn _i (i = 1,2) is determined as a function of the measured variables τ _{M, 1} , G ₁ determined in method step A. , as well as at least τ _{M, 2} and G ₂ .

It is essential that in step A, the carrier relaxation time z _{M, i} is measured as a characteristic relaxation time of the excess charge carrier density after a change in the generation rate G _i → ξ · G _i with a parameter ξ ≠ 1.

The possible variants of the generation rate change are specified below:

On the one hand, this may be a finite, preferably small, and abrupt change in the generation rate (ideally an instantaneous change, i.e. with an infinitely short time delay). In practice, the change must be made on a much shorter time scale than the carrier relaxation time for charge carrier relaxation to occur at any point of interest under constant free-carrier generation. This expression also includes light pulses (such as laser pulses), which superimpose a time-constant generation rate of free charge carriers, and whose switching time is much shorter than the carrier relaxation time.

On the other hand, it may be a continuous, preferably harmonic, and preferably quasi-static change of the generation rate act free carrier. Quasi-static here means a significant variation of the generation rate on a time scale much larger than the carrier relaxation time. This characteristic corresponds to the limiting case ξ → 1.

Furthermore, it is essential that in method step B, the actual carrier lifetime τ _{0, i} (i = 1,2) based on an analytical relationship between τ _{0, i} and the measured variables τ _{M, i} and G _i (i = 1,2) is determined.

The inventive method thus allows for the first time the determination of the actual carrier lifetime τ ₀ at the first excess carrier density Δn ₁ and the second excess carrier density Δn ₂ , without a measurement at excess carrier densities Δn smaller than the minimum of Δn ₁ and Δn _{2 is} necessary and in particular without that must be integrated via such measurement data.

The method according to the invention thus represents a fast method, which is robust with regard to convergence and thus cost-effective, for determining the actual charge carrier lifetime of a semiconductor substrate from a dynamic and differential measurement of the charge carrier relaxation time.

berechnet, mit der bei der Generationsrate G_i vorliegenden rekursiv zu bestimmenden Überschussladungsträgerdichte Δn_i=G_i·τ_0,i. Die Bestimmung der Überschussladungsträgerdichte erfolgt rekursiv, da bei Messung der Generationsrate G_i und der Ladungsträger-Relaxationszeit τ_M,i zunächst die tatsächliche Lebensdauer τ_0,i noch nicht bekannt ist.Preferably, in method step B, the actual carrier lifetime τ _{0, i} (i = 1,2) depends on the analytical relationship according to formula 1 $$\begin{array}{cc}{\tau }_{M.i}=\frac{{\tau }_{0i}}{{1-\xi G_i\frac{d{\tau }_{0i}}{d\mathrm{\Delta }n}|}_{\mathrm{\Delta }{n}_i}};& \left(i=1.2\right)\end{array}$$

calculated, with the present at the generation rate G _i recursively determined excess charge carrier density Δn _i = G _i · τ _{0, i} . The determination of the excess charge carrier density is recursive, since the actual lifetime τ _{0, i} is not yet known when measuring the generation rate G _i and the carrier relaxation time τ _{M, i} .

Applicant's investigations have shown that for many applications in method step B, the actual carrier lifetime for the excess carrier densities Δn ₁ and Δn _{2 of} interest can be assumed to be linearly dependent on the excess carrier density.

Therefore, in method step B, the actual carrier lifetime is preferably assumed to be linearly dependent on the excess charge carrier density Δn, at least for the previously specified operating points Δn _i (i = 1, 2). This achieves a further simplification and acceleration of the method.

anzunehmen.In particular, it is advantageous here, the carrier lifetime as linearly dependent with a constant k according to formula 2 $$\begin{array}{cc}{k=\frac{d\tau }{d\mathrm{\Delta }n}|}_{\mathrm{\Delta }{n}_1}& =\end{array}{\frac{d\tau }{d\mathrm{\Delta }n}|}_{\mathrm{\Delta }{n}_2};$$

to accept.

In this preferred embodiment of the method according to the invention, assuming a linear dependence of the charge carrier lifetime on the excess charge carrier density in the previously specified operating points, it is particularly advantageous to use a system of equations based on Formula 1 and Formula 2 for determining the unknown quantities τ _{0, i} (i = 1,2) and k to solve. This makes it possible to resort to known algebraic methods for solving a linear system of equations with unknowns.

$${\tau }_{\mathrm{0,2}}={\tau }_{\mathrm{0,1}}+k\left({\tau }_{\mathrm{0,2}}{G}_2-{\tau }_{\mathrm{0,1}}{G}_1\right);$$

und basierend auf diesem Gleichungssystem die unbekannten Größen τ_0,i (i=1,2) und k zu bestimmen.In particular, it is advantageous to set up a system of equations comprising the following formulas 3 and 4: $$\begin{array}{cc}{\tau }_{M.i}=\frac{{\tau }_{0i}}{1-\xi G.k};& \left(i=1.2\right);\end{array}$$

$${\tau }_{0.2}={\tau }_{0.1}+k\left({\tau }_{0.2}{G}_2-{\tau }_{0.1}{G}_1\right);$$

and to determine the unknown quantities τ _{0, i} (i = 1,2) and k based on this system of equations.

In an alternative preferred embodiment of the method according to the invention, an iterative determination of the actual charge carrier lifetime τ _{0, i} takes place in method step B. This preferred embodiment of the method according to the invention has the advantage that exact measurement results can be achieved even in cases in which there is a considerable non-linear dependence of the actual charge carrier lifetime on the excess charge carrier density at the specified operating points.

In particular, it is advantageous to carry out an iteration comprising the following iteration steps:

In an iteration start j = 0, a first approximation of the actual lifetime τ _{0, i, j = 0 is shown} the differential measured carrier relaxation time τ _{M, i} equated, according to formula 5: $${\tau }_{0i.j=0}={\tau }_{M.i}\left(i=1.2\right);$$

Furthermore, a fictitious differential lifetime τ _{M, i, j is} determined for each iteration step j> 0, preferably according to formula 6: $$\begin{array}{cc}{\tau }_{M.i.j}=\frac{{\tau }_{0i.j}}{{1-\xi G_i\frac{d{\tau }_{0i.j}}{d\mathrm{\Delta }n}|}_{\mathrm{\Delta }{n}_i}};& \left(i=1.2\right);\end{array}$$

mit einem Faktor ε, ε>0. Dabei wird der Faktor ε derart gewählt, dass eine Konvergenz ermöglicht wird. Vorzugsweise wird für eine hohe Konvergenzwahrscheinlichkeit der Faktor ε<1 gewählt, insbesondere bevorzugt ε≤0.05.For each iteration step j> 0 an estimation of the actual lifetime τ _{0, i, j + 1} , preferably according to formula 7, continues to take place: $${\tau }_{0i.j+1}={\tau }_{0.1j}+\epsilon \left({\tau }_{M.i.j}-{\tau }_{M.i}\right);$$

with a factor ε, ε> 0. In this case, the factor ε is chosen such that a convergence is made possible. Preferably, the factor ε <1 is selected for a high probability of convergence, in particular preferably ε≤0.05.

The iteration process continues until a predetermined abort criterion is met.

Preferably, the iteration process is continued until the relative deviation of a fictitious relaxation time τ _{M, ij} determined by formula 6 in the current iteration _step from the measured relaxation time T _{M, i is} less than a predetermined value, in particular a relative deviation of less than 1%, preferably smaller 0.1%. Typically, 50-150 iteration steps suffice (eg with a factor ε = 0.05) for a sufficiently accurate determination of the actual service life.

The method according to the invention is particularly suitable for the use of harmonically time-modulated photoluminescence measurements. In particular, the inventive method provides an improvement of in WO 2011/134602 described method, since the necessary numerical solution described there the continuity equation according to equation 4 of WO 2011/134602 in the method according to the invention described here is not necessary and thus can be dispensed with such a numerical method and its possible problems in terms of speed and robustness can be avoided.

1. i) entweder aus der Messung der Abkling- oder Relaxationszeit der Überschussladungsträgerdichte nach einem Lichtpuls oder einer abrupten finiten, vorzugsweise geringfügigen Änderung der Generationsrate freier Ladungsträger,
2. ii) oder aus der Messung einer Phasenverschiebung zwischen einer stetig (vorzugsweise harmonisch) zeitmodulierten Generationsrate und der resultierenden Überschussladungsträgerdichte auf die tatsächliche Ladungsträger-Lebensdauer schließen.

The method according to the invention is not limited to harmonic time-modulated photoluminescence measurements, as mentioned in the specification of the excitation conditions. In particular, the method according to the invention can be used in all measuring methods which

1. i) either from the measurement of the decay or relaxation time of the excess charge carrier density after a light pulse or an abrupt finite, preferably slight change in the generation rate of free charge carriers,
2. ii) or from the measurement of a phase shift between a continuous (preferably harmonic) time-modulated generation rate and the resulting excess charge carrier density on the actual carrier lifetime.

1. i) Messungen des Photoleitfähigkeitsabklingens unter Bias-Beleuchtung (etwa via Mikrowellenreflexion, oder induktiv über elektrische Suszeptibilitätsmessung unter Radiofrequenz-Anregung), oder bei
2. ii) stetig zeitmodulierten Messungen der Ladungsträger-Relaxationszeit (etwa via Absorption freier Ladungsträger (MFCA) oder via Lumineszenz)

anwendbar.In particular, the method according to the invention is included

1. i) measurements of photoconductive decay under bias illumination (such as via microwave reflection, or inductively via electrical susceptibility measurement under radio-frequency excitation), or at
2. ii) continuously time-modulated measurements of the charge carrier relaxation time (eg via absorption of free charge carriers (MFCA) or via luminescence)

applicable.

In method step A, the charge carrier injection in the semiconductor substrate is preferably carried out by applying electrical voltage and / or electromagnetic radiation to the semiconductor substrate, and the generation rate and the differential carrier relaxation time are determined by measuring the luminescence radiation emitted by the semiconductor substrate.

Preferably, in step A, a temporally continuous, preferably continuously differentiable, modulation of the charge carrier injection takes place. In particular, it is advantageous to perform a quasi-static harmonic time modulation. In this case, the differential carrier relaxation time τ _{M, i can be determined} as a phase shift between generation rate and excess carrier density, analogous to that in FIG WO 2011/134602 described method.

Further advantages and preferably embodiments are described below with reference to FIG 1 described.

1 shows the application of the method according to the invention in the case of a highly nonlinear injection-dependent course of the carrier lifetime - based on a typical injection-dependent course of the carrier lifetime in silicon.

At the actual life curve τ ₀ (Δn), in 1 shown in a thin, solid line, was according to equation 4 of the WO 2011/134602 (which adequately describes the dynamics of free charge carriers in silicon) uses a numerical finite element approach to calculate the time-dependent excess charge carrier density for two harmonic time-modulated profiles of the free charge generation rate (sine modulation frequency 3.42 Hz).

By means of an in JA Giesecke et al., Phys. Stat. Sol. A 209, 2286 (2012) (there in particular in 1 ) was determined for both harmonic time courses of the generation rate in each case injection-dependent, the time offset between excess charge carrier density and generation rate. This time offset corresponds to the injection-dependent differential charge carrier relaxation time τ _M (τ _M × G), which is also shown in FIG 1 is shown (triangular symbols). The charge carrier relaxation time deviates significantly in the example chosen from the actual charge carrier lifetime. From the carrier relaxation time, the actual carrier lifetime τ ₀ (Δn) was calculated using the method according to the invention. Here, the iteration method according to formula 7 was used, which is characterized by using the given in formula 1 analytical relationship for calculating the differential carrier relaxation time assuming an actual charge carrier lifetime.

The resulting actual life is in 1 represented by round symbols, and as can be seen, the method according to the invention is therefore suitable for correctly mapping the actual course of the charge carrier lifetime (based on the simulation) for all excess charge carrier densities of interest.

Also in 1 Shown (solid thick line) is the result of an evaluation of the simulated time course of the excess charge carrier density by the self - consistent solution of the time - dependent continuity equation (equation 4 of the WO 2011/134602 ) according to an in T. Trupke et al., Appl. Phys. Lett. 87, 184102 (2005) described method. However, this method can determine the actual carrier lifetime only from the relative time characteristic of the excess carrier density. For a determination of the actual lifetime from pure phase information, eg accessible via a lock-in amplifier, the self-consistent method is not suitable.

Identical variables designate the same physical quantities in the course of this application.

Claims (14)

A method for determining the actual carrier lifetime of a semiconductor substrate from a dynamic and differential measurement of the relaxation time of free charge carriers, comprising the following method steps: A generating a first excess carrier density Δn ₁ and at least one second excess carrier density Δn ₂ in the semiconductor substrate, wherein the first excess carrier carrier density Δn _{1 is} different to the second excess charge carrier density Δn ₂ and dynamically and differentially measuring a first carrier relaxation time τ _{M, 1} and a first generation rate G ₁ in the semiconductor substrate based on first excess carrier density Δn ₁ and at least one second carrier relaxation time τ _{M, 2} and at least one second Generation rate G ₂ in the semiconductor substrate from the second excess carrier density Δn ₂ , B Calculating the actual carrier lifetime τ _{0, i} ( i = 1,2) at the excess charge carrier density Δn _i (i = 1,2) depending on the measured quantities τ _{M, 1} , G ₁ , τ _{M, 2} and G _{2 determined} in method step A, characterized in that in method step A the Charge carrier relaxation time τ _{M, i} as a characteristic relaxation time of the excess charge carrier density after a change in the generation rate G _i → ξ · G _i (i = 1,2), with 0≤ξ ≠ 1 is measured, and that in step B, the actual charge carrier Lifetime τ _{0, i} (i = 1,2) is determined on the basis of an analytical relationship between τ _{0, i} and the measured quantities τ _{M, i} and G _i (i = 1,2). Method according to Claim 1 , characterized in that in process step B, the actual charge carrier lifetime τ _{0, i} (i = 1,2) depending on the analytical context according to formula 1 $$\begin{array}{cc}{\tau }_{M.i}=\frac{{\tau }_{0i}}{{1-\xi G_i\frac{d{\tau }_{0i}}{d\mathrm{\Delta }n}|}_{\mathrm{\Delta }{n}_i}};& \left(i=1.2\right)\end{array}$$

is calculated with the present at the generation rate G _i recursively determined excess charge carrier density Δn _i = G _i · τ _{0, i} . Method according to one of the preceding claims, characterized in that in process step B, the actual charge carrier lifetime at least for in Claim 1 specified operating points Δn _i as linearly dependent on the excess carrier density Δn is assumed. Method according to Claim 3 , characterized in that a system of equations based on formula 1 and formula 2 for the determination of the unknown quantities τ _{0, i} (i = 1,2) and k is solved. Method according to one of Claims 1 to 2 , characterized in that in step B, an iterative determination of the actual carrier lifetime τ _{0, i} takes place. Method according to Claim 5 , characterized in that for an iteration start j = 0 the actual lifetime τ _{0, i, j = 0 is set} equal to the differentially measured carrier relaxation time τ _{M, i} , that for each iteration step j> 0 a fictitious differential lifetime τ _{M, i, j it is} determined that for each iteration step j> 0 an estimate of the actual lifetime τ _{0, i, j + 1} is made, with a factor ε> 0, and that the iteration continues until a predetermined termination criterion is met. Method according to one of Claims 5 to 6 , characterized in that in step A at least a third excess charge carrier density, a generation rate and a differential lifetime is measured. Method according to Claim 7 , characterized in that at n different excess carrier densities Δn _i , (i = 1, ..., n) in each case a generation rate G _i and a differential carrier relaxation time τ _{M, i is} measured. Method according to one of the preceding claims, characterized in that in step A, the charge carrier injection in the semiconductor substrate by applying the semiconductor substrate with electrical voltage and / or with electromagnetic radiation and that the generation rate and the differential carrier relaxation time by measuring the emitted from the semiconductor substrate luminescence is determined. Method according to one of Claims 1 to 8th , characterized in that in step A, the differential measurement of the carrier relaxation time τ _{M, i} respectively after an abrupt and finite change of the generation rate G _i → ξ · G _i occurs. Method according to one of the preceding claims, characterized in that in step A, a temporally continuous (preferably continuously differentiable) modulation of the charge carrier injection takes place. Method according to Claim 11 , characterized in that the modulation of the charge carrier injection takes place as a harmonic and quasi-static time modulation. Method according to Claim 12 , characterized in that the differential carrier relaxation time τ _{M, i is determined} as a phase shift between generation rate and excess charge carrier density. Method according to one of the preceding claims, characterized in that in step A, the carrier relaxation time τ _{M, i} as a characteristic relaxation time of the excess charge carrier density after a change in the generation rate G _i → ξ · G _i (i = 1,2) with a bias light parameter ξ <1 is measured.

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