Characterization of ingots and wafers for photovoltaic applications

(1)

Characterization of ingots and wafers for photovoltaic applications

Bachelorarbeit

vorgelegt von Markus Gruber

am Fachbereich Physik der Universit¨ at Konstanz

August 2012

1. Gutachter: Prof. Dr. G. Hahn

2. Gutachter: Prof. Dr. T. Dekorsy

(2)

hohen Gehalt an Sauerstoff und Kohlenstoff, unten: Messplatz zur Bestimmung von Widerstand und Ladungstr¨agerlebensdauern an Czochralski-Siliziumkristallen

(3)

This project has been carried out in

Centro de tecnolog´ıa de silicio solar (CENTESIL) and

Instituto Energ´ıa Solar

Universidad Polit´echnica de Madrid (UPM) and was attended by

Prof. Carlos del Ca˜nizo

(4)

(5)

Zusammenfassung

Die Charakterisierung von Ingots¹ und Wafern² is wichtig f¨ur die Qualit¨atssicherung und Bewertung von neuen Fertigungsprozessen.

Die Methoden zur Messung von Ladungsträgerlebensdauer und Widerstand von Ingots wurden untersucht, einschließlich der Bestimmung der Eigenschaften des Messinstruments. Wir haben die Gleichung von Scheil bestätigt, die Eisenkonzentration bestimmt und Bor-Sauerstoff Defekte in der äußeren Schicht nachgewiesen.

Die Protokolle SEMI MF-1188-1107 und SEMI MF-1391-0704 zur Bestimmung des Gehalts an interstitiellem Sauerstoff und substitutionellem Kohlenstoff durch Infrarotspektroskopie erfordern größere Probendicken als die üblicher Wafer. Effekte durch dünnere Proben wurden untersucht. Wir fanden eine Beziehung zwischen Auflösung, Probendicke und dem Er- scheinen von Interferenzmustern. Versuche, diese Interferenzmuster durch mathematische und physikalische Methoden zu beseitigen, scheiterten. Für geringere Auflösungen wurden die Umrechnungskoeffizienten neu bestimmt.

Zur Auswertung von Methoden zur Verringerung des negativen Einflusses von interstitiellem Eisen wurde eine Methode zur ortsaufgel¨osten Bestimmung des Eisens mit Hilfe eines Microwave-Photoconductance-Decay (MWPCD)-Ger¨ats umgesetzt.

Abstract

Characterization of ingots and wafers is important for quality control and evaluation of new manufacturing processes.

On ingots the mechanisms of charge carrier lifetime and resistivity measurements have been investigated, involving the determination of the instrument characteristics. Scheil’s equation was confirmed, the iron concentration was determined and boron oxygen defects were detected in the outer layer.

The protocols SEMI MF-1188-1107 and SEMI MF-1391-0704 for determination of interstitial oxygen and substitutional carbon concentrations by infrared spectroscopy require sample thicknesses, which exceed common wafer thicknesses. Effects on thinner samples have been studied. We found a relation between resolution, sample thickness and the appearance of interference fringes. Physical and mathematical efforts to get rid of the interferences failed. A recalibration for the conversion coefficients is done for use with lower resolutions.

To evaluate defect engineering of interstitial iron, an iron mapping method has been implemented for a Microwave-Photoconductance-Decay device.

Resumen

La caracterizaci´on de lingotes y obleas es importante para controlar la calidad y evaluar nuevos

1Ingot: Fachbegriff f¨ur einen Block aus mono- oder multikristallinem Silizium. Die monokristallinen Ingots sind aufgrund des Herstellungsprozesses meist rund, die multikristallinen eckig.

2Wafer: Fachbegriff f¨ur Siliziumscheibe

(6)

portadores en lingotes, incluyendo la determinaci´on de las caracter´ısticas del dispositivo de medida. Hemos confirmado la ecuaci´on de Scheil, medido el contenido de hierro y determinado los defectos boro-ox´ıgeno en la capa exterior.

Los protocolos SEMI MF-1188-1107 y SEMI MF-1391-0704 para determinar el contenido de ox´ıgeno intersticial y carbono substitucional por espectroscop´ıa infrarroja necesitan espesores mayores que los de obleas corrientes. Se han investigado efectos por obleas más finas. Hemos encontrado una relación entre la resolución, el espesor y la apariencia de patrones de interfer- encia. Los intentos de quitar las interferencias por métodos matemáticas y f´ısicas han fallado.

Los coeficientes de conversi´on han sido recalibrados para resoluciones m´as bajos.

Para la evaluaci´on de los procesos para minimizar el efecto negativo del hierro intersticial se ha aplicado un m´etodo para hacer mapeados con un dispositivo tipo Microwave Photocon- ductance Decay (MWPCD).

(7)

1 Introduction

Since the invention of the steam engine scientists and engineers have developed many devices to make life more facile and comfortable. As a consequence mankind’s energy consumption has always increased. Our hunger for energy is appeased until today mainly by coal, petroleum and gas [Adm]. They are solar energy stored by plants long time ago. But these resources are finite, hence we have to look for alternative energy sources.

These could be geothermal energy, nuclear fusion generation or, most important and most simple, making use of the energy sent to earth by the sun, like the plants did millions of years ago. Plants can reach an efficiency of 3 % to 6 % for photosynthesis [Miy97, Sec. 1.2.1], whereas photovoltaic modules built with silicon solar cells can reach at the moment an efficiency of 18 % to 23 % [GEH⁺12, Table II]. Therefore, this technology has a great potential. However, the production is still expensive and energy-intensive.

To save energy and material, it is important to identify bad material as soon as possible in the process to recycle it. Also a good knowledge of the wafers’ properties is required to evaluate new processing methods, for example to improve the properties of lower grade feedstock. Hence, characterization of ingots and wafers plays an important role in the progress of photovoltaic devices.

This work focuses on three topics: mapping of interstitial iron (chapter 4), quantification of interstitial oxygen and substitutional carbon in wafers by infrared spectroscopy (5) and lifetime and resistivity measurements of ingots (chapter6). In chapter2, the physical basis is resumed. Based on this, the theory of all used measurement devices is described in chapter 3.

(10)

(11)

2 Fundamentals

In this chapter we will establish the physical basis for this work. First, important properties of semiconductors, especially silicon, are introduced, with special emphasis on recombination mechanisms, because they play a central role in this thesis. Finally an overview over silicon mining and crystallization techniques is given.

2.1 Semiconductors

Semiconductors are colloquially defined as materials which have lower conductivity than conductors and higher conductivity than insulators. For a physical definition, we have to look at the band structure. What is the band structure? The electron of a hydrogen atom has only discrete energy levels in the bound state. If we bring two atoms together, their electron orbitals overlap and split into two states with different energies due to the Pauli principle for fermions.

As the resulting orbitals cannot be allocated to a single atom, these and the electrons in it are called delocalized. Continuing overlapping orbitals from further atoms leads to a broadening of every discrete energy level, which is called band. When there are many orbitals overlapping, the energy eigenvalues of every band can be considered continuous with an upper and a lower energy limit. The Pauli principle permits only a certain quantity of electrons in one band.

Therefore the bands are filled with electrons, from the ground state towards higher energies until all electrons are assigned to a band. The highest energy the electrons can have at 0 K is called Fermi energy. The band with the highest energy, which is occupied at 0 K, is called valence band and the lowest, almost unfilled band is called conduction band [IL09, p. 241].

The electrical properties are determined only by electrons in partially filled bands.

In conductors, valence and conduction band overlap, as shown in Figure 2.1, while there exists a so called band gap in semiconductors and insulators. The valence band is completely filled and the conductance band is empty at 0 K. The Fermi energy is in the middle of the band gap. The only difference between semiconductors and insulators is the size of the band gap, which is smaller for semiconductors. In silicon the band gap is 1.124 eV at 300 K [BOH74, Table 1]. At finite temperatures, carriers in the valence band can be thermally exited to the conduction band. The occupation probability of a state with energy E is given by the Fermi function

f(E, T) = 1 exp

E−µ kBT

+ 1

, (2.1)

whereµis the chemical potential,T the absolute temperature andk_Bthe Boltzmann constant.

The chemical potential µ is a function of the temperature, but can be approximated by the Fermi energy E_F and for E−E_F 2k_BT, the Fermi function can be approximated by the

(12)

Figure 2.1: Valence and conduction bands of conductors (left), semiconductors (middle) and insulators (right)

Maxwell-Boltzmann distribution

f(E, T) = exp

−E−E_F k_BT

(2.2) [IL09, p. 145]. At 300 K we have 2k_BT = 26 meV, which is much smaller than the band gap of silicon. Therefore this approximation is valid for every calculations concerning the carrier densities in the conduction band of silicon.

As electrons from the valence band are excited to the conduction band, they leave a vacancy in the valence band. This vacancy can be identified with a positive charge and is called hole. In pure silicon, the number of holes in the valence band is the same as the number of electrons in the conduction band. For silicon the so called intrinsic carrier density at 300 K is ni = 9.7·10⁹cm⁻³ [MT93, Table II]. The electrons in the conduction band as well as the holes in the valence band are referred to as free carriers. To obtain more electrons in the conduction band or more holes in the valence band, one can systematically add impurities with one electron more or less than the host material. This process is called doping. The dopant atom introduces additional electrons or holes, whose binding energies are so low, that most of them can be found in the conduction or valence band, even at room temperature. If additional electrons are introduced, the material is called n-type and if holes are introduced it is calledp-type. Electrons are called majority carriers inn-type regions and minority carries in p-type regions and vice versa for holes. p-type silicon is usually doped with boron and n-type with phosphorus, but other elements like aluminium are also possible. Typical dopant densities for solar grade silicon are 10¹⁵cm⁻³ to 10¹⁷cm⁻³. As nearly every dopant atom introduces an electron or hole, the intrinsic carrier density becomes negligible and the material has better conductivity with lower temperature dependence.

The resistivity ρ of a semiconductor depends on the number of free carriers, as well as on their mobility

ρ= 1

e(nµn+pµp), (2.3)

where e is the elementary charge, n and p the free electron and hole densities respectively and µ_n/p their mobilities [Sch06, p. 1]. In the case of solar grade silicon, resistivity is usually measured in Ωcm. In order to know the resistance R of a piece of silicon, the resistivity must be multiplied with the length and divided through the cross section area. In some cases one dimension of the cross section area, let’s call it thickness t, cannot be measured properly because it is very thin. Therefore the sheet resistance is defined as ρ/t. Its unit is ohm per square (Ω/2) to not misinterpret the data as bulk resistance. Sometimes it is easier to

(13)

2.1 Semiconductors

electron hole

carrier type majority minority majority minority

µmax m²/Vs 1.417·10⁻¹ 4.70·10⁻²

B1 −0.057 −0.57

B2 −0.233 −2.23

B3 2.4 2.4

B4 −0.146 −0.146

µmin m²/Vs 6.0·10⁻³ 1.60·10⁻⁴ 3.74·10⁻³ 1.55·10⁻²

Nref m⁻³ 9.64·10²² 5.6·10²² 2.82·10²³ 1.0·10²³

α 0.664 0.647 0.642 0.9

Table 2.1: Parameters for the mobility model of PC1D

use conductivity, conductance and sheet conductance, which are defined as the reciprocals of resistivity, resistance and sheet resistance, respectively.

The mobilityµ of free carriers links the drift velocity vD with the applied electric field of strength E as follows

v_D =µE

[IL09, p. 250]. Its value is determined by different scattering mechanisms, like scattering at acoustic lattice phonons, interactions with ionized impurities and carrier carrier scattering.

The first effect decreases mobility with increasing temperature, the second is mainly influenced by the dopant atoms and the last is important, when we have excess carriers generated by illumination. Dorkel and Leturcq give an overview over equations for these cases and provide simplified, but accurate equations for computational purposes [DL81]. In the solar cell simulation software PC1D [spc97] the following empirical model for mobilities in silicon is used

µ=µ_min T

T₀ B1

+ (µ_max−µ_min) (T /T₀)^B² 1 +

N Nref(T /T0)^B³

α(T /T0)^B⁴, (2.4)

whereT₀= 300 K andN_ref,µ_min/max,αas well asB_1,2,3,4 are constants usually obtained from a fit, in our case extracted from the source code [spc97] and given in Table2.1. This model was reimplemented in python and all resistivity to dopant concentration and vice versa conversions in this thesis are done with it.

The most important property for photovoltaic semiconductors is the ability to generate carriers by absorption of light. Photons with an energy exceeding the band gap can give their energy to electrons in the valence band, which are then excited to the conduction band. As silicon is an indirect semiconductor, a phonon has to provide the momentum required for the excited state. For the same reasons as above, a hole is excited in the valence band. These carriers, if separated to different contacts, provide the current extracted from a solar cell.

Since they are excited to a higher energy level, they are called excess carriers. If an excess electron and an excess hole meet, they can recombine, i.e. fall back into theirs ground state, giving their energy and momentum to photons or phonons. These recombination mechanisms are described in an extra section, because they have more emphasis in this thesis.

As we also treat ingots, it is crucial to know how deep the light can penetrate the material.

(14)

The skin depth of electromagnetic waves with frequency f in materials with finite resistivity ρ is given as

δ = r ρ

µ₀πf, (2.5)

where µ₀ is the magnetic constant (see [Sch06, p. 37]).

2.2 Recombination mechanisms

Once generated, the excess carriers should not disappear during their journey to the contacts of the solar cell. Therefore long lifetimes or diffusion lengths or low recombination rates are essential for efficient light to current conversion.

Lifetime τ is defined as the mean time for an excess carrier to recombine, i.e. to fall back to the ground state. The diffusion length l is the average distance a carrier can cover until it recombines. It is linked to the lifetime by

l=√ Dτ ,

with the charge carrier diffusion coefficientD, which is measured to be approximately 12.5 cm³s⁻¹ at 300 K and without external field [BJN⁺81, Fig. 4]. The recombination rateU_eis defined as the number of recombination processes per second and unit volume, which of course depend on the excess carrier density ∆n as

U_e = ∆n

τ (2.6)

This equation can also be considered as a definition of the lifetime.

When several recombination mechanisms take place, we have to add their recombination rates to get the total or effective recombination rate. Application of Eq. (2.6) to this sum leads to the calculation of the effective lifetime τeff

1

τ_eff =X

i

1

τ_i. (2.7)

From here, it can be seen that the effective lifetime is always lower as the lowest single lifetime.

An important tool for calculations with excess carriers is the continuity equation for electrons

∂ n

∂t =Ge−Ue+ divje, (2.8)

with Ge being the electron generation rate, Ue the electron recombination rate and je current density [Hof11, p. 12]. Of course, this can be also applied to holes.

There are three recombination mechanisms in semiconductors, radiative, Auger and Shockley- Read-Hall recombination. They are schematically shown in Figure 2.2 and described in the following sections. Without loss of generality, we assume that electrons recombine falling from the conduction band down to the valence band. The same is true for holes falling from the valence band to the conduction band.

(15)

2.2 Recombination mechanisms

Figure 2.2: Sketch of radiative (left), Shockley- Read-Hall (middle) and Auger (right) recombination. Black circles are electrons, white ones holes.

Straight arrows mark transitions, solid curled arrows denote photons, dotted curled arrows phonons.

2.2.1 Radiative recombination

This is the inverse of the photo effect. As excess carriers can be excited by absorption of a photon, they can also recombine releasing a photon corresponding to their energy. The probability of this effect is proportional to the excess electron density nas well as the excess hole density p, because both are required for a successful recombination. Hence we have the radiative recombination rate

U_rad=Cnp,

with a material specific constant C, which is in the range of 10⁻¹⁴cm³ s⁻¹ at 300 K [SMG74, Fig. 2]. As silicon is an indirect semiconductor not only the electron and hole must find together, but also a phonon for reasons of momentum conservation. Therefore this recombination mechanism is rather unlikely compared to the others in silicon.

2.2.2 Auger recombination

Auger recombination is a three particle process, where electron and hole recombine giving their excess energy to a third free hole or electron, pushing it into higher band. Given the free electron density nand the free hole densityp, the recombination rate is

R_Aug=C_eehn²p+C_ehhnp²

with the Auger coefficientsC_eeh andC_ehh. But there are some more effects, which can be taken into account, e.g. participation of phonons, trap assisted Auger recombination or Coulombic interactions. Therefore Kerr and Cuevas [KC02] proposed a general parametrization as follows R_Aug=np(1.8·10⁻²⁴n^0.65₀ + 6·10⁻²⁵p^0.65₀ + 3·10⁻²⁷∆n^0.8) (2.9) with equilibrium carrier densities n0, p0 and excess carrier density ∆n, giving a value in cm⁻³s⁻¹.

2.2.3 Shockley-Read-Hall recombination

This recombination mechanism is also called trap assisted recombination, because electrons are first captured by a trap having a Fermi energy in the band gap and subsequently recombines with a hole in the valence band. As the energy differences to overcome are smaller, the probability for this process is higher. This process is most effective, when the trap energy level is in the middle of the band gap. Traps can be impurities like iron or defects in the lattice,

(16)

like dislocations, vacancies or a surface. It was first theoretically described by Shockley, Read [SR52] and Hall [Hal52].

LetN be the density of a recombination center with capture cross section σ_n for electrons and σ_p for holes and a Fermi energy E_T. Then we can define the average lifetime τ_n₀ for electrons and τp0 for holes to be captured by the trap as

τ_n₀ = 1

N σ_nv_th τ_p₀ = 1 N σ_pv_th,

where vth is calledthermal velocity and has a value ofvth ≈1.1·10⁷cm⁻¹ [MGA04, p. 1022].

The probability of recombination depends on the availability of both, electrons and holes in the trap. This can be expressed by n₁ and p₁, which are the numbers of electrons in the conduction and holes in the valence band, in the case when the Fermi level coincides with the recombination center energy E_T. They are given as

n1=Ncexp

−Ec−ET

k_BT

and p1 =Nvexp

−ET −Ev

k_BT

where Nc and Nv are the effective densities of states in the conductance and valence band, respectively. In this thesis the valuesN_c= 2.86·10¹⁹cm⁻³ andN_v = 3.10·10¹⁹cm⁻³ at 300 K are used, according to [Gre90, Table I].

Together with the equilibrium carrier densities n₀ and p₀ and under the assumption that any ofn₀, n₁, p₀, p₁ is large compared toN, Shockley and Read obtain the lifetime due to this recombination mechanism as

τSRH =τp0

n0+n1+ ∆n n₀+p₀+ ∆n +τn0

p0+p1+ ∆n

n₀+p₀+ ∆n, (2.10)

where ∆n is the excess carrier density, which is assumed to be the same for electrons and holes. For highly doped samples, all traps are filled, so that the lifetime is determined by the trap capture lifetime of the minority carriers τ_p₀ in n-type and τ_n₀ in p-type material. For moderately doped p-type silicon with an acceptor concentrationNAthe equation simplifies to

τ_SRH= τp0(n1+ ∆n) +τn0(NA+p1+ ∆n)

N_A+ ∆n , (2.11)

according to [MGA04, p. 1022].

2.3 From silica to solar grade crystalline silicon

Silicon is the second most abundant element in the Earth’s crust in weight after oxygen [LH10, Sec. 5.2]. It is never found in elementary state, but as silica (SiO₂), also known as sand or quartz, or silicate, which is a salt of a silicon oxygen anion and a cation, most frequently a metal. Therefore it has to be reduced, purified and crystallized before it can be used as base material for solar cells.

First, silica is reduced with carbon at 2000^◦C in an arc furnace giving carbon mono- and dioxides and metallurgical grade silicon with a purity above 98 %. Purification can be done

(17)

2.4 Crystallization techniques

by the Siemens process leading to polysilicon or metallurgical treatments, giving upgraded metallurgical grade silicon.

In the Siemens process, the metallurgical grade silicon is first decomposed with hydrochloric acid at 300^◦C into trichlorosilane (HSiCl₃) and other gases, which are separated by distillation.

In a second step, the purified trichlorosilane flows together with hydrogen along an ultra pure silicon rod at 1000^◦C to 1200^◦C reacting to elemental silicon being deposited on the rod and hydrochloric acid, which can be introduced into the first process again [Hof11, p. 6].

Upgraded metallurgical grade silicon is obtained by pre-solidification, chemical cleaning or reactive gas blowing and subsequently purified by a plasma treatment [SE02, p. 36f]. The liquid silicon is stirred by electromagnetic forces and the induction created argon plasma is enriched with a reactive gas like oxygen or hydrogen, which partly remove impurities like boron, aluminium, carbon and calcium [DPTM08, p. 1270].

2.4 Crystallization techniques

Crystallization is used for further purification as well as for doping. The second advantage is, that efficiency of a solar cell is better, if there are less grain boundaries within a wafer.

Therefore, mono- and multicrystalline silicon is mainly used for photovoltaic purposes.

When silicon solidifies, the concentrations of impurities in the solidCs and in the liquidCl

are in an equilibrium. The ratiok= ^C_C^s

l is calledsegregation coefficient orpartition coefficient, according to [NU09, p. 14]. Some segregation coefficients can be found in [Sch64, Table 3] or [LH10, Table 5.8]. As a certain amount of the impurity solidifies, the concentration of the impurity in the liquid changes. Under the assumptions of complete mixing and no back diffusion of the impurities, the impurity concentration in the solid C_s (at the solid-liquid interface) can be described byScheil’s equation

C_s= C0

k (1−f_s)^k−1, (2.12)

whereC0 is the initial impurity concentration in the liquid andfsthe fraction of the solidified ingot mass [NU09, p. 15]. If the segregation coefficient is far from unity, most of the impurity is deposited in the last solidified part. As many metals have low segregation coefficients, this is an effective method to remove many impurities.

However, doping follows the same rule. While doping with boron which has a segregation coefficient of 0.8 is quite easy, it becomes more difficult with phosphorus, which has a coefficient of 0.35 [Sch64, Table 3]. In this case most of the initial dopant is deposited at the end.

A simple crystallization technique, is the vertical gradient freeze technique. Here, all the silicon is molten in a crucible, while the heater can be moved in height. So solidification starts at the bottom and goes on to the top of the crucible. With this method multicrystalline silicon can be obtained.

For monocrystalline silicon, the Czochralski method is used most frequently. Here the silicon is also molten in a crucible in an argon atmosphere. Then a monocrystalline seed with the desired crystal orientation is dipped into the melt and slowly pulled out, while crucible and seed crystal are rotating. Then a monocrystal growths with the given orientation. As the crucible dissolves partly during the growth process, the ingot contains a lot of oxygen.

(18)

The most pure monocrystals can be obtained with the floating zone method. Only a zone of the base material is molten by induction heating and this zone is moved through the whole ingot. Due to the low segregation coefficient, most of the impurities remain in the liquid and solidify at the end. Additionally there is no contamination during the crystallization process, because the ingot is not in contact with any crucible or wall. The crystal orientation can also be determined starting with a seed as in the Czochralski process.

(19)

3 Applied methods

In this section the fundamentals of the different measurement principles for the different properties of silicon wafers and ingots are described.

3.1 Resistivity

Many properties of silicon such as minority carrier lifetimes are influenced by the majority carrier densitynmaj. This density is linked to the resistivity by (2.3). In extrinsic, i.e. doped, silicon we can neglect the influence of the minority carrier density in this formula and obtain the relation

ρ= 1

eµmajnmaj

. (3.1)

As the mobility µmaj is roughly constant, we obtain an estimate of the majority carrier density by means of resistivity measurements. In the following we will discuss a temporary contact and two contactless methods.

3.1.1 Four-point probe

The most simple way to measure resistivity is to connect two probes to the sample, measure voltage and current and calculate resistance by the known law of Ohm. The resistivity can then be obtained from the resistance and the shape of the sample.

Unfortunately, the resistance contains not only contributions of the sample’s resistivity, but also of the probe’s resistance, the contact resistance and the spreading resistance under each probe as described in [Sch06, p. 2]. To overcome this problem one can minimize the current through the voltage probes and thereby the voltage drop caused by probe’s resistances, contact resistance and spreading resistance. This is achieved by using different probes for current injection and voltage measurement. Usually they are arranged in a line where the outer probes carry the current and the inner ones are connected to a high impedance voltmeter as shown in Figure3.1. This arrangement is calledin-line orcollinear. Other arrangements such as square arrays are also possible and are discussed by Schroder [Sch06, p. 13-17].

Assuming uniform resistivity, negligible influence of minority carrier injection through current- carrying electrodes due to high recombination rates, flat surface without leakage and indepen- dent and isotropic probe properties, Valdes [Val54, Appendix] obtains the following formula for equal spaced in-line probes on a semi-infinite sample (i.e. all other surfaces are far away)

ρ= 2πV

I s, (3.2)

(20)

Figure 3.1: left:

Schematic arrangement of the probes for a collinear four-point probe measurement.

right: the circuit diagram.

whereV is the voltage between contacts (2) and (3),I is the current through (1) and (4), sis the distance between two adjacent probes (Figure3.1). Valdes also provides correction factors for measurements near the border, but they can be neglected if the distance to the border is greater than four times the distance between adjoining probes.

But as most wafers are thinner than the probes’ distance s, a correction for the finite thickness has to be considered. For a wafer with thickness w and a non-conductive bottom surface one can obtain the resistivity as

ρ= ρ₀

G ^w_s with Gw s

= 1 + 4s w

∞

X

n=1





1 q s

w

2

+ (2n)²

− 1

q 2_w^s2

+ (2n)²



,

where ρ0 is the resistivity calculated by (3.2). The correction term G(w/s) simplifies to G

w s

= 2s

wln 2, which implies ρ= π ln 2

V

I w (3.3)

in the case of infinite thin sample thickness (see [Val54, p. 427]), which is applicable for w/s≤0.1. According to Table 2 in the same article, the correction is below 1% for w/s≥5.

In our experimental setup we haves= (1.0±0.1) mm so that we can use (3.3) for wafers and (3.2) for ingots.

3.1.2 Eddy-current

Miller et al. showed in 1976 [MRW76], that the conductivity of thin slices of semiconductors or metals can be measured by coupling an oscillating magnetic field to them. The oscillation of the magnetic field causes eddy currents in the material which get absorbed by means of ohmic heating. The absorption is proportional to the conductivity.

The device design is schematically shown in Figure 3.2. The magnetic field is generated by a solenoid with n windings in a rf circuit, whose root mean square (rms) voltage is V. Assuming the sample thickness t is much smaller than the skin depthδ (see (2.5)), the power P_abs absorbed by the eddy currents’ ohmic heating is

P_abs = V²

8πn²σt, (3.4)

when the sample conductivity is σ, as stated in section II of the article. If all power in the rf circuitPrf is coupled into the conductor, we have

V I =P_rf=P_abs= V²

8πn²σt. (3.5)

(21)

3.1 Resistivity

Figure 3.2: left: Schematic eddy current sensor. The oscillating magnetic field in the solenoid causes eddy currents in the sample material. This leads to power absorption in the rf circuit (right hand side).

The rms circuit voltageV is kept constant and then the current passed into the rf circuit is proportional to the losses in the rf circuit.

In the real case due to flux leakage only a fractionK of the power couples into the sample.

Using Eq. (3.5) this power loss can be considered as parallel resistance Rsmp:= V

I = K8πn²

σt (3.6)

in the rf circuit. Usually this is not the only loss in the rf circuit. Hence another parallel loss Rrf must be introduced, leading to a total resistanceRtot of

1

R_tot = 1

R_rf + 1

R_smp. (3.7)

Now a feedback loop is introduced in order to hold V constant [MRW76, p. 800-802]. As V =I·Rtot, then

I =V 1 Rtot

.

When Eq. (3.7) is inserted andI0 defined asV /R_rf, we further obtain I−I₀ = const 1

R_smp ∝σt.

The last proportionality was found using Eq. (3.6). In absence of any conductor we haveσ = 0 and thusI =I₀. Since the coupling constantKis usually not known, only the proportionality is achieved and it is therefore necessary to calibrate the current signal by another method, for example the four point probe. Summarizing, all it was needed to do for measuring the conductivity is to measure the rms currentI once in absence of any conductor and once while the sample is placed.

The rf circuit frequency can be chosen arbitrarily, but there are some boundary conditions to match. In order to achieve a high sensitivity high currents are needed. Looking at Eq. (3.6) the coil has to be designed with few turns n. For high spatial resolution the coil should cover little area, which leads together with smallnto low inductance. Low inductance implies a high rf circuit frequency. On the other hand, electromagnetic waves with higher frequencies have a smaller skin depth. So the frequency is a compromise between sensitivity and penetration depth. Our instrument [Ins] operates at 13.56 MHz and has a sensitive area of 45 mm×15 mm.

(22)

3.1.3 Microwave reflectivity

The principle above can also be applied to electromagnetic waves. At conducting layers a fraction of their power is reflected. The rest is absorbed due to eddy currents, which get stronger with greater conductivity. Thus, the reflectivity of the sample is connected to its conductivity, but the relationship is not linear [LLb⁺08, Fig. 5].

In this work, this method will only be used to measure photoconductance, hence we can follow the approach of Kunst and Beck, who develop in [KB86] and [KB88] the theory for photoconductance measurements. They showed, that the photoconductance ∆σ is linearly linked to the change in absorption power ∆P, if ∆σis small compared to the dark conductance σ₀

∆σ = ∆P

P(σ0)A, (3.8)

where P(σ0) denotes the reflected power in the case of dark conductance and A is called sensitivity factor [KB86, Eq. 3], which is also calculated in this article.

3.2 Charge carrier lifetime

3.2.1 General measurement principle

To measure the effective minority carrier lifetime in semiconductors, the sample is illuminated and the change in the conductivity is monitored. Depending on the duration of the illumination τ_i with respect to the effective lifetime τ_eff, we can distinguish two methods: transient for τiτeffand quasi-steady-state,QSS for short, forτiτeff. Their analysis is described in the following sections.

As shown in section2.2, recombination mechanisms and thus lifetimes depend on the excess carrier density. For a detailed analysis it is therefore desirable to link the lifetimes to an excess carrier density. Fortunately the change of conductivity ∆σ due to illumination follows a linear relationship with the excess carrier density, as we will derive in the following.

The conductivity σ of a semiconductor is given by Eq. (2.3) as σ=e(µnn+µpp),

wherenandpare the free electron and hole carrier densities andµ_n/p their mobilities, respectively. In the dark, there is an equilibrium value n₀, p₀ which is the intrinsic carrier density if no dopant is present and approximately the number of dopant carriers in the other case.

In the first case there is high temperature dependency, in the latter less. Introducing light generates additional electrons ∆nand holes ∆p, hence we can write

n=n0+ ∆n and p=p0+ ∆p (3.9)

As only pairs of electrons and holes are generated, we have ∆n = ∆p. Composing these equations, we get

σ =e(µ_nn₀+µ_pp₀) +e∆n(µ_n+µ_p) =σ₀+e∆n(µ_n+µ_p),

(23)

3.2 Charge carrier lifetime

so that the change of conductivity ∆σ due to illumination is proportional to the excess carrier density ∆n, in symbols

∆σ =e∆n(µ_n+µ_p). (3.10)

The attentive reader might have noticed, that the mobilities are also a function of the excess carrier density. To obtain a self consistent value of ∆n, the right hand side can be iterated, as proposed by Sinton and Cuevas [SC96]. Iteration means to start with some mobilities and calculate the respective excess carrier density. Based on this value one can calculate the new mobilities and with those recalculate the excess carrier density. This is repeated until the mobilities obtained from the excess carrier density and from the measurement with the excess carrier density coincide.

Without loss of generality, the minority carriers are assumed to be electrons. In general, the lifetime is given by inserting the definition (2.6) into the continuity equation (2.8). Rearranging leads to

τ = ∆n

G_e+ divj_e− ^{∂ n}_∂t . (3.11)

In the following only contactless conductivity measurements will be considered, thus we can assume that there are no other sources of electrons in the material, i.e. divje= 0.

3.2.2 Transient mode

In this mode, the excess carriers are generated by a short flash. The analysis of the conductivity signal starts, when the flash has turned off. This implies Ge = 0 in Eq. (3.11) and requires carrier lifetimes significantly longer than the flash decay time to get a measurable conductivity signal. Additionally, we can separate the carrier density in the temperature and illumination depending parts as shown in Eq. (3.9). Then we get

∂ n

∂t = ∂

∂t(n0+ ∆n) = ∂∆n

∂t ,

supposing the temperature does not vary during the measurement (duration ≤ 1 s, flash not too intense). Hence the lifetime is obtained as

τ =−∆n

∂∆n

∂t

=− 1

∂

∂t ln ∆n. (3.12)

The last term shows an intuitive explanation of this lifetime measurement technique: The slope in the semilogarithmic plot is the negative inverse of the lifetime τ - this is the same relation, as we would obtain assuming an exponential decay with τ as time constant.

This method has the advantage that no calibration is needed and the optical properties such as pulse shape, light intensity and absorption coefficient do not matter. On the other hand, the turn-off time of the flash must be significantly shorter than the carrier lifetime and deriving increases the noise level, according to [Sin11, p. 31]. Additionally, if applied to ingots, the lifetime converges to the bulk lifetime, even if only a natural surface passivation is given [SSFM10, p. 317].

(24)

3.2.3 Quasi-steady-state mode

In the quasi-steady-state mode, we assume the flash decay time to be much longer as the carrier lifetime. Then we can suppose steady state conditions, i.e. the same number of excess carriers is generated by the illumination as recombines, which implies∂tn= 0. The number of excess carriers generated per second is called photogeneration rate G_e. Lifetime is then given as

τ = ∆n Ge

. (3.13)

The difficulty of this method is to determine the photogeneration rate G_e. Usually the flash intensity is measured by a calibrated reference cell, which is illuminated simultaneously. Due to different reflectivities of sample and reference cell, one has to take into account the so called optical factor, which is primarily obtained from simulations or tables [SC96]. Some factors for wafers with different passivation layers are shown in the user manual [SM06, p. 22-26].

The equation above is valid for any thickness, but the conductivity measurement methods described in section 3.1only provide an average conductivity and therefore an average excess carrier density. For thin wafers with surface passivation one can assume a homogeneous carrier profile. The surface passivation makes border effects negligible and thinness implies little difference of the generation rate between front and back due to absorption as well as effective carrier diffusion.

For thick wafers and ingots the excess carrier density profile has to be taken into account.

Bowden and Sinton developed a method, which reduces this problem to the thin wafers’ one by means of an effective thickness [BS07]. Here, just the key ideas are presented. Let ∆n(x) be the excess carrier density as function of the distance x to the surface. In the case of thin wafers (thickness t), the average excess carrier density is calculated as

∆n_av= Rt

0∆n(x)dx

t .

This equation is exact if the carrier profile is homogeneous, but for thick wafers we have

∆n(x)→0 forx→ ∞. As a consequence we underestimate the average excess carrier density as can be seen in Figure 3.3. Therefore, the regions with low ∆nshould be excluded. This is achieved by weighting the average with the excess carrier density itself

∆nav= Rt

0(∆n(x))²dx R_t

0∆n(x)dx .

The effective thickness teff is then defined as the thickness, which is necessary to obtain the real number of excess carriers assuming a constant excess carrier density ∆nav, in symbols

teff= Rt

0∆n(x)dx nav

= Rt

0∆n(x)dx 2

Rt

0(∆n(x))²dx.

For a given charge carrier distribution, we can calculate the average excess carrier density and the effective thickness. The charge carrier distribution on the other hand, can be either calculated analytically for simple cases, as done in [BS07], or simulated using PC1D. Since all

(25)

3.3 Impurities

Figure 3.3: Comparison of the different aver- aging methods and the real distribution, taken from [BS07, Fig. 1]. The solid line is the real carrier density, the dotted shows the average carrier density assuming a homogeneous distribution and the dashed line shows the average weighted by the distribution itself limited to the effective thickness.

of these calculations require a bulk carrier lifetime, one can obtain a transfer function, which relates the bulk lifetime to the effective lifetime. Of course, this transfer function depends strongly on the illumination spectrum as well as on the surface recombination velocity. But Bowden and Sinton have also shown [BS07, p. 7], that the transfer function depends only weakly on the resistivity, hence one can estimate the bulk lifetime for samples with the same surface preparation with the same transfer function.

With some more effort, but similar methods, one can also take into account that the instru- ments have only a limited sense depth. These considerations have been carried out by Swirhun et al. [SSFM10] and result in a slightly different transfer function.

3.2.4 Hardware

For charge carrier lifetime measurements, we use two devices, the Sinton BLS-I [Ins] and the Semilab WT-2000 [Sem]. The BLS-I machine can measure both in QSS and in transient mode.

The manufacturer states, that lifetimes below 200µs have to be measured in QSS mode and above can be measured in transient mode.

The Semilab machine can measure only in transient mode. The turn off time cannot be specified, but the whole pulse length is 200 ns, according to the manual [SRB]. Lifetimes above 1µs are therefore supposed to be trustworthy. In contrast to the general transient approach described in section 3.2.2, the machine does not evaluate the lifetime as function of the minority carrier density, but tries to fit an exponential decay. Of course, the resulting measured lifetime can then not be determined in dependence of the excess carrier density. This causes deviations from lifetimes measured with other methods.

3.3 Impurities

For solar grade silicon it is important to achieve high minority carrier lifetimes to reduce the losses during diffusion to the contacts. One important factor is the structure of the silicon itself, e.g. if there are dislocations or vacancies. Another important factor are impurities, which act as recombination centers. To analyze their influence, it is desired to know how many impurities in which configuration are present in a sample. This section describes the

(26)

characterization techniques, mainly used in this work.

3.3.1 Fourier-transform infrared (FTIR) spectroscopy

Beside the spectral fingerprint of elements in the visible spectrum, which results from transitions in the electron energy level, there is also a fingerprint for bindings in the infrared (IR), which is caused by transitions between rotational and vibrational energy levels of the same electron energy state [IvdVS⁺97, p. 94]. We will not get into the details, because the absorption bands, we are interested in, are well established.

The FTIR spectrometer consists of four main parts. A Michelson interferometer, an IR source, an IR detector and a processor, which can do fast Fourier transformations. The set-up of these parts is shown in Figure3.4a. The IR source sends white light, i.e. light consisting of a continuum of wavelengths in IR, into the interferometer. In the interferometer a beamsplitter sends half of the incoming light into one path and the rest to another. The two rays are each reflected by a mirror and combined again at the beamsplitter, where interference takes place.

Then the beam crosses the sample and its intensity is detected. By means of a movable mirror, the length of one of the branches in the interferometer can be varied. Hence one obtains an interference pattern or an intensity vs. mirror displacement curve. This curve is related to the spectrum by a Fourier transformation. The processor calculates the fast Fourier transform, which is an approximation for the Fourier transformation, and one obtains the spectrum.

The IR source usually consists of ceramic electrically heated to a temperature around 1000^◦C [Hen01]. To maintain a stable spectrum, the heating is controlled by a feed-back loop. The emitted radiation is approximately that of a black body. In Figure 3.4b we can see the spectrum of our FTIR [iFT] and additionally the shape of a black body spectrum at 880 K, which is given as

2hc² ν˜³ exp

hc˜ν kBT

−1

(3.14) with the Planck constant h, the Boltzmann constantkB and the speed of light in vacuumcin the wave number domain [KM09, Eq. 39]. This spectrum has a similar shape and the maxima of both curves are in the same region. However, there seems to be some kind of absorption over the whole range of the mid IR spectrum, which may be due to the absorption of the optical components like mirrors and beamsplitter and also due to a wavelength dependant detection sensitivity. This curve indicates, that the temperature of the IR source in our case might be around 600^◦C.

In the practical realization of the Michelson-Interferometer there are many more optical elements for example to focus the beam or make it parallel. Special care must be taken for a smooth and reproducible movement of the moving mirror. This can be achieved by an air bearing or a special geometry, where turning two mirrors has the same effect as shifting one.

An important task is to measure the distance of the mirror movement accurately. This is done by coupling a HeNe-laser into the same beam and monitor its interference pattern. As this beam only consists of one wavelength, the pattern is a shifted cosine with half the wavelength of the HeNe laser as period. The number of transitions from bright to dark multiplied by the well known wavelength is an excellent measure of the mirror movement.

TheIR detector usually is either implemented as photo resistor or relying on the pyroelectric

(27)

3.3 Impurities

(a) Schematic set-up of an FTIR instrument. Non essential optical components have been removed.

500 1000 1500 2000 2500 3000 3500

4000 wavenumber in cm⁻¹ 0

5 10 15 20 25 30 35

intensity in a.u.

empty beam 880 K

(b)IR source spectrum of Nicholet iS10 and the spectrum of a black body at 880 K

Figure 3.4: FTIR setup and example spectrum

effect. Photo resistors decrease their resistivity on illumination and the most sensitive are made of Ge or InGaAs [Hen01]. The pyroelectric effect converts a thermal flux into a measurable quantity like voltage or current [LGB98]. A temperature change leads to temporal displacement of the electrons in the crystal, which changes its polarisation and therefore introduces a voltage.

In our case, a triglycine sulfate (TGS) with protons substituted by deuterium (DTGS) is used.

Relying on the pyroelectric effect has the additional advantage, that background radiation from the housing walls or any other object emitting constant radiation is ignored, because only temperature changes lead to a signal.

An interferogram obtained with our FTIR spectroscope is shown in Figure 3.5a. One can note the asymmetric shape due to unequal contributions of the different frequencies. The whole interferogram is much longer and although the tails seem to be constant, there are also oscillations but on a smaller scale.

After a fast Fourier transform, we obtain the spectrum, shown in Figure 3.5b. For an analysis, one has to compare this spectrum to the empty spectrum shown in Figure 3.4b. As this is visually difficult, thetransmittance T is defined by

T = I_sample

I_empty. (3.15)

Of course, this definition must be applied for every single data point. In spectroscopy, the standard unit is wavenumber ˜ν, which is defined as the reciprocal of the wavelength λ. The equations

˜ ν = 1

λ = ν c = E

hc (3.16)

linking wavenumber, wavelength, frequency ν and energy E can be obtained by the known relations. c denotes the speed of light andh the Planck constant.

As transmission often decays exponentially with the distance passed through, the term absorbance A is defined as

A=−log₁₀(T) =−log₁₀

I_sample Iempty

. (3.17)

(28)

1900 1950 2000 2050 2100 2150 data points

0.80.6 0.40.2 0.00.2 0.40.6 0.81.0

detector signal in Volts

(a) Interference pattern seen by the IR detector

500 1000 1500 2000 2500 3000 3500

4000 wavenumber in cm⁻¹ 0

2 4 6 8 10 12

intensity in a.u.

(b) FFT of the interference pattern Figure 3.5: FTIR signal and conversion

Hence, the relation between absorbance and distance passed through is linear under the assumption of an exponential decay. Additionally it turns out that elimination of overlapping bands becomes easier, because rationing the transmission against a reference becomes subtrac- tion in absorbance terms.

3.3.2 Lifetime change

Some impurities, such as iron or oxygen, form a kind of molecule with silicon or dopant atoms within the lattice. In this case, their recombination properties differ from their properties as single atoms. If the impurity additionally permits to toggle between the single and the molecule state, the difference in lifetime can be used to calculate the concentration of this impurity. This method is mainly used for the interstitial iron content in boron-doped silicon, because iron boron pairs can be easily dissociated by illumination. As a second advantage, the dissociation is reversible. The method has also been used to study light degradation of boron- doped silicon by Schmidt and Bothe [SB04]. In this section, we will derive the theory for this measurement principle following the approach of Macdonald et al. [MGA04] for interstitial iron in boron-doped silicon.

In boron-doped silicon interstitial iron Feiforms FeB pairs with the boron atoms. As there is much more dopant than impurity, due to the low solubility of iron in silicon, all interstitial iron can be assumed to be paired in the dark. Illumination separates these pairs and at first we assume all pairs to be dissociated. Later the model is extended to incomplete association and dissociation. As the inverse lifetimes add according to (2.7), the effective lifetime can be split into the contributions of interstitial iron Fe_i, iron-boron pairs FeB and other recombination mechanisms, which are not influenced by illumination. In the dark the effective lifetime is given as

1 τdark

= 1

τFeB

+ 1

τother

(29)

3.3 Impurities

and after complete dissociation as 1 τillu

= 1 τFei

+ 1

τother

.

Calculation of the difference of the inverse effective lifetimes eliminates the effect of the other recombination mechanisms, provided that they are not influenced by illumination. Therefore, this method is relatively insensitive against other recombination effects.

To calculate the iron concentration, we use the Shockley-Read-Hall lifetime model (2.11) for Fe_i as well as for FeB-pairs. For reasons of clarity, we rearrange this equation

τSRH= 1 N v_th

1 σp

n1+ ∆n N_A+ ∆n + 1

σn

1 + p1

N_A+ ∆n

(3.18) and define the latter factor as excess carrier density dependant inverse capture cross section

1

σ_eff(∆n) := 1 σ_p

n1+ ∆n N_A+ ∆n+ 1

σ_n

1 + p1

N_A+ ∆n

. (3.19)

This enables us to write the lifetimes in the simple form τ = 1/(N vthσeff). When this is inserted into the difference above, we obtain

1

τ_illu − 1

τ_dark = 1

τ_Fe_i − 1

τ_FeB =N v_th(σ_eff,Fe_i(∆n)−σ_eff,FeB(∆n)), (3.20) where N is the density of interstitial iron, which must be the same in the associated as well as the dissociated case. Solving this equation forN leads to the relation

N =C 1

τillu

− 1 τdark

(3.21) with the proportionality factor

C := 1

v_th(σ_eff,Fe_i(∆n)−σ_eff,FeB(∆n)). (3.22) Now we extend the model to incomplete association and dissociation. Letαbe the fraction of the interstitial iron to be paired with boron before dissociation and β the fraction to be paired after the dissociation process. αcan be smaller than unity because of thermal excitation and also depends on the doping level, but is usually above 99 % according to Macdonaldet al.

[MGA04, p. 1023]. Then the effective lifetime in the dark has to be substituted as follows 1

τ_dark =α 1 τFeB

+ (1−α) 1 τFei

+ 1

τ_other.

For the effective lifetime after illumination we can use the same equation, but β instead ofα.

Then we obtain for the difference 1

τ_illu− 1

τ_dark = (β−α) 1

τFeB

− 1 τFei

(30)

Defect σ_p in cm² σ_n in cm² E in eV.

Fe_i 7·10⁻¹⁷ 5·10⁻¹⁴ E_V + 0.38 [MGA04]

7·10⁻¹⁷ 4·10⁻¹⁴ EV + 0.38 [IHW99]

∗ 3.6·10⁻¹⁵ E_V + 0.39 [RG05]

FeB 2·10⁻¹⁵ 3·10⁻¹⁴ E_C−0.23 [MGA04]

3·10⁻¹⁵ 5·10⁻¹⁵ E_C−0.26 [MRD⁺06]

5.5·10⁻¹⁵ 2.5·10⁻¹⁵ EC−0.26 [RG05]

3·10⁻¹⁴ 2.5·10⁻¹⁵ E_C−0.26 [IHW99]

∗ this study used the value from [IHW99] and calculated from it σ_nwith the symmetry factor.

Table 3.1: Overview over often used values for the iron defects.

and hence for the iron concentration N = 1

β−αC 1

τ_illu − 1 τ_dark

(3.23) with the same C as above.

Beside the measurement of lifetime vs. excess carrier density curves, this method requires knowledge about the SRH parameters of the recombination centers. Some recent or often used values for the capture cross sections and energy levels are shown in Table 3.1. From this table one can see, that there is little doubt about the energy levels, but the capture cross sections vary strongly. In Figure 3.6 on the left some of the resulting conversion factors C (as defined above) are shown for different combinations. As the variation is within a factor of three ([MGA04] vs. [IHW99] & [RG05]), the absolute iron concentration may match the real iron concentration only in the order of magnitude, but if all measurements are done using the same model, relative comparisons are quite reliable. Macdonald et al. [MRD⁺06] recommend the use of the values from [IHW99] for Fe_i and the values from [MRD⁺06] for FeB. If not otherwise mentioned, we will use these values.

As one can see in Figure3.6on the right, the conversion factor depends heavily on the excess carrier density. For low excess carrier densities, the lifetime decreases after illumination and the conversion factor is almost constant. Therefore this region would be ideal to measure the iron content. But this condition can only be fulfilled by surface photovoltage (carried out by Zoth and Bergholz [ZB90]) and photoluminescence measurements (carried out by Macdonald et al. [MRD⁺06]). In our laboratory we use photoconductance methods, such as QSSPC and MWPCD. Typical excess carrier densities are between 10¹³ and 10¹⁶cm⁻³. As the calibration factor has a singularity in this region, the excess carrier density where the lifetime is reported must be chosen carefully. For high excess carrier densities the conversion factor seems to become constant, but on the other hand, other recombination processes like radiative and Auger recombination become more important. The resort is to choose an excess carrier density near the local maximum, which implies little variation. It is common to use the excess carrier density of 1·10¹⁵cm⁻³with high injection measurements [Sin11, p. 14]. At this point, lifetime increases after dissociation.

Characterization of ingots and wafers for photovoltaic applications