**3.4.2 Solid phase epitaxy in furnace**

Technically the most simple way to achieve epitaxial growth is to deposit first an amorphous layer on top of the cleaned seed layer, and then to epitaxially crystallize the layer by furnace annealing in the solid state. The layer to be crystallized can already contain the desired doping profile which remains during the annealing step. The main critical point with this simple procedure is that not only an epitaxial crystallization front moves into a-Si, but also spontaneous nucleation will occur within a-Si followed by growth of crystallites. So there exists a competing process to the desired epitaxy. The question arises, which of the two succeeds. The speed of the epitaxial front of course depends on temperature (described by Jackson-Chalmers equation, see Sect. 5.1) and so does nucleation, described by classical nucleation theory (Sect. 5.2), and growth of nuclei, the latter phenomena described together by Avrami-Mehl equation (Sect. 5.4). An important point, which makes SPE possible, is that, if no nuclei pre-exist in the amorphous matrix, nucleation does not start immediately. Instead it needs some time, called time lag of nucleation, until a stationary population of nuclei evolves (Sect. 5.3). Only after that time lag the stationary nucleation rate applies at fixed temperature, described by classical nucleation theory, and crystal nuclei appear. So any successful epitaxy relies on the time lag of nucleation. The thickness of an epitaxially crystallized layer is just given by the time lag of nucleation times the speed of the epitaxial crystallization front. After the time lag, in the virgin amorphous silicon crystalline nuclei of random orientation appear resulting in fine grained material, such as is generated by direct furnace crystallization (see Sect. 2) without seed. For successful epitaxy one has to make sure that within the amorphous phase there are no nuclei present which could form during deposition already.

In the last few years we developed the technique of SPE on diode laser crystallized seed layers on borosilicate glass substrates (Andrä et al., 2008a; Schneider et al., 2010). The virgin a-Si layers including a doping profile were deposited at high rate (typically 300 nm/min) by electron beam evaporation at a substrate temperature in the 300°C range. At that temperature no nuclei form within a-Si. The layer system was then annealed in a furnace under ambient air. To control the progress of crystallization, an in situ measurement technique was installed. For this purpose, the beam of a low power test laser was sent through the sample. The transmitted intensity was monitored by a photocell. Since a-Si has a different optical absorption from c-Si, the progress of crystallization can be monitored easily. In particular, the crystallization process is complete when the transmission does not change any more. Fig. 5 shows a transmission electron micrograph of a cross section of an epitaxially thickened silicon film.

Fig. 5. Transmission electron microscopic cross section image of a film epitaxially thickened by furnace annealing.

beam evaporation at 550°C substrate temperature has successfully been demonstrated (Dogan et al., 2008). Solar cells prepared with this process reached 346 mV open circuit voltage and 2.3% efficiency, which is a bit low as compared to the values achieved by other methods.

Technically the most simple way to achieve epitaxial growth is to deposit first an amorphous layer on top of the cleaned seed layer, and then to epitaxially crystallize the layer by furnace annealing in the solid state. The layer to be crystallized can already contain the desired doping profile which remains during the annealing step. The main critical point with this simple procedure is that not only an epitaxial crystallization front moves into a-Si, but also spontaneous nucleation will occur within a-Si followed by growth of crystallites. So there exists a competing process to the desired epitaxy. The question arises, which of the two succeeds. The speed of the epitaxial front of course depends on temperature (described by Jackson-Chalmers equation, see Sect. 5.1) and so does nucleation, described by classical nucleation theory (Sect. 5.2), and growth of nuclei, the latter phenomena described together by Avrami-Mehl equation (Sect. 5.4). An important point, which makes SPE possible, is that, if no nuclei pre-exist in the amorphous matrix, nucleation does not start immediately. Instead it needs some time, called time lag of nucleation, until a stationary population of nuclei evolves (Sect. 5.3). Only after that time lag the stationary nucleation rate applies at fixed temperature, described by classical nucleation theory, and crystal nuclei appear. So any successful epitaxy relies on the time lag of nucleation. The thickness of an epitaxially crystallized layer is just given by the time lag of nucleation times the speed of the epitaxial crystallization front. After the time lag, in the virgin amorphous silicon crystalline nuclei of random orientation appear resulting in fine grained material, such as is generated by direct furnace crystallization (see Sect. 2) without seed. For successful epitaxy one has to make sure that within the amorphous

phase there are no nuclei present which could form during deposition already.

electron micrograph of a cross section of an epitaxially thickened silicon film.

In the last few years we developed the technique of SPE on diode laser crystallized seed layers on borosilicate glass substrates (Andrä et al., 2008a; Schneider et al., 2010). The virgin a-Si layers including a doping profile were deposited at high rate (typically 300 nm/min) by electron beam evaporation at a substrate temperature in the 300°C range. At that temperature no nuclei form within a-Si. The layer system was then annealed in a furnace under ambient air. To control the progress of crystallization, an in situ measurement technique was installed. For this purpose, the beam of a low power test laser was sent through the sample. The transmitted intensity was monitored by a photocell. Since a-Si has a different optical absorption from c-Si, the progress of crystallization can be monitored easily. In particular, the crystallization process is complete when the transmission does not change any more. Fig. 5 shows a transmission

Fig. 5. Transmission electron microscopic cross section image of a film epitaxially thickened

**3.4.2 Solid phase epitaxy in furnace** 

by furnace annealing.

In summary we could epitaxially crystallize up to 1.6 µm of a-Si at a temperature of 630°C within 3 h. The epitaxial quality as determined by EBIC was best in (100) oriented grains and worst in (111) grains. Moreover, the epitaxial crystallization speed depends on orientation and on the doping level. Higher doped layers crystallize faster. Solar cells prepared on these layers reached an efficiency of 4.9% after hydrogen passivation (Schneider et al., 2010). By TEM cross section investigations it was shown that the seed layers contain only very few extended defects such as dislocations, whereas the epitaxial layer contains much more. It seems that the cleaning procedure of the seed surface prior to a-Si deposition is crucial for good epitaxial quality. At least the dislocation density in the epitaxial layer could be reduced by an additional RCA cleaning step before removal of oxide by HF. However, this did not reflect in the achieved solar cell efficiencies.

### **3.4.3 Layered laser crystallization**

The epitaxy method of layered laser crystallization has been developed in our group years ago (Andrä et al. 2005b, Andrä et al., 2008a). The principle is simple. During deposition of a-Si on top of the seed layer excimer laser pulses are applied repeatedly, which melt the newly deposited a-Si and a bit of the crystalline silicon beneath so that after each pulse epitaxial solidification occurs. Again, the layer thickness to be crystallized by one laser shot is limited by a competing nucleation process in the undercooling melt after the laser pulse. According to our experience about 200 nm of a-Si can be epitaxially crystallized by one laser pulse. The typical laser fluence needed is 550 mJ/cm². However, when during the whole thickening process the thickness of the crystalline layer beneath the newly deposited a-Si increases from the initial seed layer (say 200 nm) to the final absorber thickness (say 2 µm) the laser parameters or the thickness of the newly deposited a-Si have to be adjusted so that the laser pulse just melts the a-Si and bit of c-Si beneath. This adjustment is necessary because the thermal properties of glass, c-Si, and a-Si differ so that the temperature profiles change during the process if the laser energy would be kept constant. In the layered laser crystallization process epitaxy works independently of the grain orientation, which is an advantage since crystal orientation in the seed is at random. For the process, the laser pulse has to be fed through a window in the deposition chamber onto the growing layer. In this way the pulses can be applied without stopping deposition. For a-Si deposition we use electron beam evaporation which has first the advantage of high deposition rate, at least an order of magnitude higher than for PECVD, and secondly the advantage that deposition is directed so that no deposition occurs at the laser window. Doping is achieved by codeposition of boron or phosphorus. In our device we can deposit and laser irradiate substrates of up to 10x10 cm². The single laser spot has a size of 6x6 mm² with top hat profile. To cover the whole substrate area the laser spot is scanned over the substrate by a scanning mirror placed outside the deposition chamber. In order to avoid cracks in the glass substrate heating to about 600°C helps. Upscaling the system to m² surely is a challenge but not outside the technical possibilities. If properly optimized, about 10 laser pulses are needed at each position during absorber deposition to prepare a 2 µm thick epitaxial film. This makes sense only if the laser is fed into the deposition chamber and is applied without braking deposition, as we do it in our lab scale equipment.

In the epitaxial layer prepared by layered laser crystallization the number of extended defects like dislocations is much lower as compared to solid state epitaxy. This is because the mobility of crystallizing atoms is much higher in the melt than in a-Si so that correct placement is easier. The highest efficiencies achieved in solar cells prepared using the

Crystalline Silicon Thin Film Solar Cells 149

To improve the solar cell performance some post-crystallization treatment is required. One

In order that dopand atoms like boron or phosphorus really lead to a free carrier concentration higher than the intrinsic one, it is necessary that the dopand atoms are included substitutionally in the lattice, i.e. that they rest on regular lattice positions replacing a silicon atom. If they are included interstitially, resting not on regular lattice positions, they are useless. If the silicon lattice forms from the melt the mobility of atoms is high enough so that the dopand atoms can occupy lattice positions. In this case no additional means are needed to make them active. This is not so in case of solid state crystallization. There most of the dopand atoms are included interstitially so that they are inactive. To let them replace silicon atoms substitutionally an additional heat treatment is needed, which is realized by a rapid thermal annealing (RTA) step. In the CSG process, for example, the whole system is heated to about 900°C for 2 min to achieve dopand activation (Keevers et al., 2007). It has been a lot of speculation if this RTA step also improves the grain structure by reducing the number of

In any case a hydrogen passivation step has to follow, in which different types of defects e.g. dislocations and grain boundaries, are passivated. Usually, a remote hydrogen plasma is applied to the layer system for 10 to 30 min at about 500°C. Crucial is that during cooling down at the end of the process the plasma has to be applied for some time. A lot of optimization work has been devoted to this passivation step (Rau et al., 2006), which easily

In Sect. 5 the basics of phase transformation relevant for silicon thin film crystallization, both from the melt and in the solid state are summarized (Falk & Andrä, 2006). The Section divides in the propagation of already present phase boundaries and in nucleation, including non-stationary nucleation. Kinetics of aluminum induced crystallization has already been reviewed (Pihan et al., 2007) and is not treated in the following. The facts presented in this section are the background for any successful crystallization of amorphous silicon, in the furnace or by laser irradiation. Quantitative values following from the equations depend on the material parameters of the system involved. These are rather well known for crystalline and for liquid silicon, mostly in the whole range of temperature involved in the processes. This is not the case for amorphous silicon, the properties of which strongly depend on the preparation conditions. They may appreciably differ for hydrogenated a-Si prepared by PECVD and hydrogen free a-Si deposited by electron beam evaporation. Therefore, quant-

The propagation speed of already present phase boundaries into a metastable phase, i.e. the growth of a crystal into the undercooled melt or into amorphous silicon, can quantitatively

\*/ Δμ / e 1e <sup>0</sup> *g kT kT*

*v v* (1)

**4.2 RTA and hydrogen passivation** 

point is dopand activation, the other defect passivation.

extended defects. This seems not to be the case (Brazil & Green, 2010).

can improve the open circuit voltage of the cell by 200 mV.

**5. Kinetics of phase transformation** 

itative predictions have to be taken with some care.

be described by the Jackson-Chalmers-Frenkel-Wilson equation

**5.1 Propagation of phase boundaries** 

method were 4.8% at an open circuit voltage of 517 mV (Andrä et al., 2005b; Andrä et al., 2007). These values were measured on cells without any light trapping.

#### **3.4.4 Liquid or solid phase epitaxy by diode laser irradiation**

The layered laser crystallization method described in the last section has the drawback that up-scaling into the industrial scale is not so easy. This is due to the fact, that the laser beam has to be fed into the deposition chamber and several pulses have to be applied at each position. That was the motivation for us to look for a method in which the complete absorber thickness is deposited in the amorphous state on top of the seed and to apply a single laser treatment to epitaxially crystallize the whole system in one run after deposition outside the deposition chamber.

The most obvious way to achieve epitaxy is via the liquid phase similar to layered laser crystallization. The main difference is that the whole amorphous absorber precursor layer is melted in one step down to the seed, so that epitaxial solidification is to occur after irradiation. It is a challenge to melt about 1 µm of a-Si without completely melt the about 200 nm thin c-Si seed beneath which would hamper any epitaxy. To crystallize a layer system more than 1 µm thick, a short pulse laser is useless. To get the required energy into the system the pulse fluence would have to be so large that ablation would occur at the surface. Moreover, the cooling rate of the melt after a short laser pulse would be so high, that nucleation is expected to occur in a surface near region before the epitaxial solidification front reaches the surface. Therefore we decided to use a scanned cw diode laser for this purpose with irradiation times in the ms range. In this case the cooling rate is low enough so that the melt stays long enough in a slightly undercooled state with low nucleation rate until the epitaxial solidification front reaches the surface. We succeeded in epitaxially crystallizing 500 nm in one run. However, forming of cracks is an issue. Moreover, due to the strong diffusion in the melt which intermixes any pre-existing doping profile, absorber and emitter cannot be crystallized in one step.

An alternative is solid phase epitaxy in which the amorphous layer is heated by the laser to a temperature of about 1100°C, below the melting point of a-Si. At such high temperature the solid phase epitaxial speed was determined to several 100 nm/s high so that epitaxy of 1 µm should be complete within several seconds.

### **4. Post-crystallization treatment**

#### **4.1 Emitter preparation**

The emitter of the final solar cell can be prepared in different ways. One is to include emitter doping into the deposition sequence of the layer system so that no additional emitter preparation step is needed. This way has been chosen in the CSG process and in layered laser crystallization. It cannot be applied in case of liquid phase epitaxy of the whole layer stack (Sect. 3.4.4) since during melting for several ms, diffusion in the liquid state would intermix any dopand profile introduced during deposition. In this case, phosphorus doping of a boron doped absorber as in conventional wafer cells can be performed. The only difference is that the doping profile has to be much shallower. Another variant is to use amorphous heteroemitters. IMEC has found that this is the best emitter for their thin film solar cells prepared by the high temperature route (Gordon et al., 2007).

method were 4.8% at an open circuit voltage of 517 mV (Andrä et al., 2005b; Andrä et al.,

The layered laser crystallization method described in the last section has the drawback that up-scaling into the industrial scale is not so easy. This is due to the fact, that the laser beam has to be fed into the deposition chamber and several pulses have to be applied at each position. That was the motivation for us to look for a method in which the complete absorber thickness is deposited in the amorphous state on top of the seed and to apply a single laser treatment to epitaxially crystallize the whole system in one run after deposition

The most obvious way to achieve epitaxy is via the liquid phase similar to layered laser crystallization. The main difference is that the whole amorphous absorber precursor layer is melted in one step down to the seed, so that epitaxial solidification is to occur after irradiation. It is a challenge to melt about 1 µm of a-Si without completely melt the about 200 nm thin c-Si seed beneath which would hamper any epitaxy. To crystallize a layer system more than 1 µm thick, a short pulse laser is useless. To get the required energy into the system the pulse fluence would have to be so large that ablation would occur at the surface. Moreover, the cooling rate of the melt after a short laser pulse would be so high, that nucleation is expected to occur in a surface near region before the epitaxial solidification front reaches the surface. Therefore we decided to use a scanned cw diode laser for this purpose with irradiation times in the ms range. In this case the cooling rate is low enough so that the melt stays long enough in a slightly undercooled state with low nucleation rate until the epitaxial solidification front reaches the surface. We succeeded in epitaxially crystallizing 500 nm in one run. However, forming of cracks is an issue. Moreover, due to the strong diffusion in the melt which intermixes any pre-existing doping profile, absorber

An alternative is solid phase epitaxy in which the amorphous layer is heated by the laser to a temperature of about 1100°C, below the melting point of a-Si. At such high temperature the solid phase epitaxial speed was determined to several 100 nm/s high so that epitaxy of 1

The emitter of the final solar cell can be prepared in different ways. One is to include emitter doping into the deposition sequence of the layer system so that no additional emitter preparation step is needed. This way has been chosen in the CSG process and in layered laser crystallization. It cannot be applied in case of liquid phase epitaxy of the whole layer stack (Sect. 3.4.4) since during melting for several ms, diffusion in the liquid state would intermix any dopand profile introduced during deposition. In this case, phosphorus doping of a boron doped absorber as in conventional wafer cells can be performed. The only difference is that the doping profile has to be much shallower. Another variant is to use amorphous heteroemitters. IMEC has found that this is the best emitter for their thin film

solar cells prepared by the high temperature route (Gordon et al., 2007).

2007). These values were measured on cells without any light trapping.

**3.4.4 Liquid or solid phase epitaxy by diode laser irradiation** 

outside the deposition chamber.

and emitter cannot be crystallized in one step.

µm should be complete within several seconds.

**4. Post-crystallization treatment** 

**4.1 Emitter preparation** 

#### **4.2 RTA and hydrogen passivation**

To improve the solar cell performance some post-crystallization treatment is required. One point is dopand activation, the other defect passivation.

In order that dopand atoms like boron or phosphorus really lead to a free carrier concentration higher than the intrinsic one, it is necessary that the dopand atoms are included substitutionally in the lattice, i.e. that they rest on regular lattice positions replacing a silicon atom. If they are included interstitially, resting not on regular lattice positions, they are useless. If the silicon lattice forms from the melt the mobility of atoms is high enough so that the dopand atoms can occupy lattice positions. In this case no additional means are needed to make them active. This is not so in case of solid state crystallization. There most of the dopand atoms are included interstitially so that they are inactive. To let them replace silicon atoms substitutionally an additional heat treatment is needed, which is realized by a rapid thermal annealing (RTA) step. In the CSG process, for example, the whole system is heated to about 900°C for 2 min to achieve dopand activation (Keevers et al., 2007). It has been a lot of speculation if this RTA step also improves the grain structure by reducing the number of extended defects. This seems not to be the case (Brazil & Green, 2010).

In any case a hydrogen passivation step has to follow, in which different types of defects e.g. dislocations and grain boundaries, are passivated. Usually, a remote hydrogen plasma is applied to the layer system for 10 to 30 min at about 500°C. Crucial is that during cooling down at the end of the process the plasma has to be applied for some time. A lot of optimization work has been devoted to this passivation step (Rau et al., 2006), which easily can improve the open circuit voltage of the cell by 200 mV.

#### **5. Kinetics of phase transformation**

In Sect. 5 the basics of phase transformation relevant for silicon thin film crystallization, both from the melt and in the solid state are summarized (Falk & Andrä, 2006). The Section divides in the propagation of already present phase boundaries and in nucleation, including non-stationary nucleation. Kinetics of aluminum induced crystallization has already been reviewed (Pihan et al., 2007) and is not treated in the following. The facts presented in this section are the background for any successful crystallization of amorphous silicon, in the furnace or by laser irradiation. Quantitative values following from the equations depend on the material parameters of the system involved. These are rather well known for crystalline and for liquid silicon, mostly in the whole range of temperature involved in the processes. This is not the case for amorphous silicon, the properties of which strongly depend on the preparation conditions. They may appreciably differ for hydrogenated a-Si prepared by PECVD and hydrogen free a-Si deposited by electron beam evaporation. Therefore, quantitative predictions have to be taken with some care.

#### **5.1 Propagation of phase boundaries**

The propagation speed of already present phase boundaries into a metastable phase, i.e. the growth of a crystal into the undercooled melt or into amorphous silicon, can quantitatively be described by the Jackson-Chalmers-Frenkel-Wilson equation

$$v = v\_0 \mathbf{e}^{-g^\ast / kT} \left( 1 - \mathbf{e}^{\Delta \mu / kT} \right) \tag{1}$$

Crystalline Silicon Thin Film Solar Cells 151

600 800 1000 1200 1400

Classical nucleation theory gives the nucleation rate *J*, i.e. the number of nuclei appearing in a metastable phase per volume and time interval at given temperature. The value applies after some induction time (Sect. 5.3) and as long as not too much of the parent phase is

> \* 2/3 1/2 <sup>2</sup> <sup>m</sup>

*c c kT c*

2 2 m 2

3 2 m 2 16 1 9 2 *GV j c c* 

 

σ is the interface energy between both the phases, which, however, is hard to determine independently of nucleation phenomena, and, in addition, may depend on temperature. Moreover, σ strongly influences the nucleation rate since via Eqs. 5&6 it enters Eq. 4 in the third power within the exponential. For crystallization in an undercooled silicon melt the stationary nucleation rate is plotted in Fig. 8 for a temperature dependent interfacial energy according to σ = (43,4+0.249 *T*/K) mJ/m2 (Ujihara et al., 2001). Down to about 300 K below the equilibrium melting point the nucleation rate is very low to change within 100 K of further cooling by 35 orders of magnitude. Below 1200 K the nucleation rate gets rather flat at a value of 1035 m-3s-1 = 0.1 nm-3ns-1. The stationary nucleation rate of crystallization in amorphous silicon is plotted in Fig. 9. There the values increase by 16 orders of magnitude when temperature is increased from 600 K to 1200 K. The nucleation rate then flattens off at

*Gc g*

(4)

(5)

(6)

Fig. 7. Speed of a crystallization front in amorphous silicon as depending on temperature

(36 ) <sup>3</sup>

 

1017 m-3s-1 = 0.1 µm-3s-1 up to the melting point of a-Si of 1400 K.

*<sup>j</sup> <sup>G</sup> J e <sup>V</sup> j kT*

32 <sup>3</sup> *cj V*

In this formula *V*m is the atomic volume and *j*c and Δ*G*c are the number of atoms in and the free energy of a critical nucleus of the new phase in the matrix of the parent phase,

T/K

Tma

1E-10

**5.2 Stationary nucleation rate** 

respectively. These are given by

consumed.

1E-8

1E-4

1E-2 1

1E+2

1E+4

V/µms-1

The prefactor *v*0 = *a*0γ*ν* depends on the atomic vibration frequency (Debye frequency) *ν*, the jump distance *a*0 of the order of the lattice parameter of silicon and on a geometry factor γ of the order of 1. Δµ>0 is the difference in chemical potential of the phases involved. For the transition from liquid to crystalline Δµ may be approximated by

$$
\Delta\mu = \Delta l\_{\mathbb{C}} (1 - \frac{T}{T\_{\text{mc}}}) \tag{2}
$$

where Δ*h*c is the latent heat per mole for melting and *T*mc is the equilibrium melting temperature of 1685 K. For the crystallization of amorphous silicon Δµ is given in the literature (Donovan et al., 1983). g\* is an activation energy for the jump of an atom from the parent to the final phase and is related to the self-diffusion coefficient *D* according to

$$D = \frac{a\_0^2}{\chi} \nu \,\mathrm{e}^{-\mathrm{g}^\*/kT} \tag{3}$$

Results for crystallization from the melt and in the solid state are given in Figs. 6 and 7. In the melt the crystallization speed vanishes at the equilibrium melting point *T*mc to increase to a maximum of about 16 m/s at 200 K undercooling. At even lower temperature the solidification front gets slower due to the increasing influence of the activation energy. At temperatures above the melting point the phase front runs into the crystal, i.e. the crystal melts and the speed changes sign. In Fig. 6. also the melting speed of amorphous silicon is shown (with opposite sign as compared to c-Si). Melting of a-Si starts at *T*ma, which, depending on the deposition conditions of a-Si, is 200 to 300 K lower than the melting point of c-Si.

The crystallization speed in amorphous silicon shown in Fig. 7 increases with temperature, and reaches about 1 mm/s near the melting point of a-Si. At 600°C the speed is only about 0.2 nm/s which well correlates with the results obtained in furnace solid phase epitaxy (Sect. 3.4.2).

Fig. 6. Speed of the phase boundaries liquid-crystalline (lc) and amorphous liquid (al) for crystalline solidification form the melt and melting of a-Si, respectively.

The prefactor *v*0 = *a*0γ*ν* depends on the atomic vibration frequency (Debye frequency) *ν*, the jump distance *a*0 of the order of the lattice parameter of silicon and on a geometry factor γ of the order of 1. Δµ>0 is the difference in chemical potential of the phases involved. For the

> 

> > 2

γ *<sup>a</sup> g kT <sup>D</sup>* 

<sup>0</sup> \*/ <sup>e</sup>

Results for crystallization from the melt and in the solid state are given in Figs. 6 and 7. In the melt the crystallization speed vanishes at the equilibrium melting point *T*mc to increase to a maximum of about 16 m/s at 200 K undercooling. At even lower temperature the solidification front gets slower due to the increasing influence of the activation energy. At temperatures above the melting point the phase front runs into the crystal, i.e. the crystal melts and the speed changes sign. In Fig. 6. also the melting speed of amorphous silicon is shown (with opposite sign as compared to c-Si). Melting of a-Si starts at *T*ma, which, depending on the deposition conditions of a-Si, is 200 to 300 K lower than the melting point

The crystallization speed in amorphous silicon shown in Fig. 7 increases with temperature, and reaches about 1 mm/s near the melting point of a-Si. At 600°C the speed is only about 0.2 nm/s which well correlates with the results obtained in furnace solid phase epitaxy

800 1000 1200 1400 1600 1800

Fig. 6. Speed of the phase boundaries liquid-crystalline (lc) and amorphous liquid (al) for

crystalline solidification form the melt and melting of a-Si, respectively.

lc

**l→c**

al

**a→l**

(1 ) <sup>c</sup> mc *<sup>T</sup> <sup>h</sup> T*

where Δ*h*c is the latent heat per mole for melting and *T*mc is the equilibrium melting temperature of 1685 K. For the crystallization of amorphous silicon Δµ is given in the literature (Donovan et al., 1983). g\* is an activation energy for the jump of an atom from the parent to the final phase and is related to the self-diffusion coefficient *D*

(2)

T/K

cl

**c→l**

Tma Tmc

(3)

transition from liquid to crystalline Δµ may be approximated by

according to

of c-Si.

(Sect. 3.4.2).

0

10

20

v/(m /s)

Fig. 7. Speed of a crystallization front in amorphous silicon as depending on temperature

#### **5.2 Stationary nucleation rate**

Classical nucleation theory gives the nucleation rate *J*, i.e. the number of nuclei appearing in a metastable phase per volume and time interval at given temperature. The value applies after some induction time (Sect. 5.3) and as long as not too much of the parent phase is consumed.

$$J = (36\pi)^{1/2} \gamma \nu \frac{\dot{j}\_c^{2/3}}{V\_{\rm m}} \sqrt{\frac{\Delta G\_c}{3j\_c^2 \pi kT}} \ e^{\frac{\Delta G\_c + g^\*}{kT}} \tag{4}$$

In this formula *V*m is the atomic volume and *j*c and Δ*G*c are the number of atoms in and the free energy of a critical nucleus of the new phase in the matrix of the parent phase, respectively. These are given by

$$j\_c = \frac{32\pi}{3} V\_{\text{m}}^2 \frac{\sigma^2}{\Delta\mu^2} \tag{5}$$

$$
\Delta G\_c = \frac{16\pi}{9} V\_{\text{m}}^2 \frac{\sigma^3}{\Delta \mu^2} = \frac{1}{2} j\_c \Delta \mu \tag{6}
$$

σ is the interface energy between both the phases, which, however, is hard to determine independently of nucleation phenomena, and, in addition, may depend on temperature. Moreover, σ strongly influences the nucleation rate since via Eqs. 5&6 it enters Eq. 4 in the third power within the exponential. For crystallization in an undercooled silicon melt the stationary nucleation rate is plotted in Fig. 8 for a temperature dependent interfacial energy according to σ = (43,4+0.249 *T*/K) mJ/m2 (Ujihara et al., 2001). Down to about 300 K below the equilibrium melting point the nucleation rate is very low to change within 100 K of further cooling by 35 orders of magnitude. Below 1200 K the nucleation rate gets rather flat at a value of 1035 m-3s-1 = 0.1 nm-3ns-1. The stationary nucleation rate of crystallization in amorphous silicon is plotted in Fig. 9. There the values increase by 16 orders of magnitude when temperature is increased from 600 K to 1200 K. The nucleation rate then flattens off at 1017 m-3s-1 = 0.1 µm-3s-1 up to the melting point of a-Si of 1400 K.

Crystalline Silicon Thin Film Solar Cells 153

Here g is an accommodation coefficient of the order of 1. The result for nucleation of c-Si from the melt is shown in Fig. 10. The time lag diverges at the equilibrium melting point and has a minimum of 30 ps around 1350 K. At all relevant temperatures the time lag is so small

This is different for solid phase crystallization of amorphous silicon as shown in Fig. 11. The time lag goes down from 1013 s (or 300.000 years) at 600 K to 0.01 s at the melting point of a-Si (1400 K). That means that below 300°C crystallization never occurs whereas in the CSG process of furnace crystallization at 600°C the time lag is in the range of 2 h which does not play a major role when complete crystallization takes 18 h. However, it gives an upper limit

*c c* 

that it does not play any role in laser crystallization with pulses longer than 1 ns.

1000 1200 1400 1600

Fig. 10. Time lag of nucleation for crystallization from the melt for a fixed value of interfacial

600 800 1000 1200 1400

Fig. 11. Time lag of nucleation for crystallization of amorphous silicon

2/3 *g*\*/*kT*

*gj e* (8)

Tmc T/K

Tma

T/K

βc is the attachment rate of atoms to the critical nucleus given by

for epitaxial growth by furnace annealing as described in Sect. 3.4.2.

1E-11

/s

**τ/s**

1E-2

1 1E2 1E4 1E6 1E8 1E10 1E12

energy σ of 400 mJ/m²

1E-10

1E-9

1E-8

/s

**τ/s**

Fig. 8. Stationary nucleation rate for crystallization in an undercooled silicon melt

Fig. 9. Stationary nucleation rate for crystallization of amorphous silicon

#### **5.3 Non-stationary nucleation**

When the temperature of a system is changed abruptly from a value where the parent phase is absolutely stable and there are no nuclei present to another temperature where it gets metastable, then a population of nuclei evolves. Finally, a stationary distribution of nuclei emerges which leads to the stationary nucleation rate of Eq. 4. The master equation for the population of nuclei can be solved numerically. By some approximations a closed form for non-stationary nucleation rate has been derived (Kashchiev, 1969), which leads to the stationary value after some time lag of nucleation, which is given by

$$\tau = \frac{12}{\pi^2} \frac{kT}{\Delta \mathcal{G}\_c} \frac{\dot{\mathcal{J}}\_c^2}{\mathcal{J}\_c} \tag{7}$$

*J*/m-3 s-1

1E-30

1E0

**5.3 Non-stationary nucleation** 

1E10

1E20

J/m-3s-1

1E-20

1E-10

1E0

1E10

1E20

1E30

1E40

800 1000 1200 1400 1600

600 800 1000 1200 1400

When the temperature of a system is changed abruptly from a value where the parent phase is absolutely stable and there are no nuclei present to another temperature where it gets metastable, then a population of nuclei evolves. Finally, a stationary distribution of nuclei emerges which leads to the stationary nucleation rate of Eq. 4. The master equation for the population of nuclei can be solved numerically. By some approximations a closed form for non-stationary nucleation rate has been derived (Kashchiev, 1969), which leads to the

> 2 12 *<sup>c</sup>*

2

*c c kT j G*

Fig. 9. Stationary nucleation rate for crystallization of amorphous silicon

stationary value after some time lag of nucleation, which is given by

Fig. 8. Stationary nucleation rate for crystallization in an undercooled silicon melt

*T*mc

*T*/K

Tma

T/K

(7)

βc is the attachment rate of atoms to the critical nucleus given by

$$
\mathcal{B}\_c = \gamma \,\mathrm{g} \mathbf{v} \eta\_c^{2/3} e^{-\mathbf{g}^\*/kT} \tag{8}
$$

Here g is an accommodation coefficient of the order of 1. The result for nucleation of c-Si from the melt is shown in Fig. 10. The time lag diverges at the equilibrium melting point and has a minimum of 30 ps around 1350 K. At all relevant temperatures the time lag is so small that it does not play any role in laser crystallization with pulses longer than 1 ns.

This is different for solid phase crystallization of amorphous silicon as shown in Fig. 11. The time lag goes down from 1013 s (or 300.000 years) at 600 K to 0.01 s at the melting point of a-Si (1400 K). That means that below 300°C crystallization never occurs whereas in the CSG process of furnace crystallization at 600°C the time lag is in the range of 2 h which does not play a major role when complete crystallization takes 18 h. However, it gives an upper limit for epitaxial growth by furnace annealing as described in Sect. 3.4.2.

Fig. 10. Time lag of nucleation for crystallization from the melt for a fixed value of interfacial energy σ of 400 mJ/m²

Fig. 11. Time lag of nucleation for crystallization of amorphous silicon

Crystalline Silicon Thin Film Solar Cells 155

Multi- and polycrystalline silicon thin film solar cells receive growing interest worldwide. Presently, the maximum efficiency reached by these types of cells is 10.4%. Different cell concepts and preparation methods are under investigation and no clear favourite way is identified up to now. The concepts differ in the resulting grain structure, i.e. size and quality, but also in the preparation technologies used and the processing time needed. Today it is not clear which of the methods will succeed in industrial production. In all the methods, pin holes in the films are an issue since they lead to shunting of the final cells. Another issue is dopand deployment, particularly along grain boundaries. This also may lead to shunting, which today limits the open circuit voltage to slightly above 500 mV. A further point is that TCO cannot easily be used as a front contact in superstrate cells since it hardly withstands the temperatures needed for crystallization. Usually a highly doped

Very important for thin film crystalline solar cell is a perfect light management so that about 2 µm of silicon is enough to absorb the solar spectrum. This can be achieved either by structured substrates or by texturing the surface. In the first case, the irregular substrate surface should not influence the crystallization behaviour. In the second case, the rough surface should not increase surface recombination. Generally, passivation of defects and of

Concerning the theoretical description of the processes involved in crystallization, the basic equations are well understood. However, there are some issues with the material parameters involved, which, particularly for amorphous silicon, strongly depend on deposition conditions and therefore need to be determined individually. But even if numerical predictions may not completely coincide with experiments due to inadequate numerical values of the materials parameters, general trends can reliably be predicted. All the mentioned issues need further investigation. Careful study of these topics is expected to lead to full exploitation the potential of the material. Multicrystalline thin film cells with a ratio of grain size over film thickness similar to multicrystalline wafer cells should deliver, if prepared correctly, comparable efficiencies. Therefore we expect the polyand multicrystalline silicon thin film solar cells to gain increasing significance and may replace microcrystalline silicon cells. Multicrystalline silicon also can act as one partner in

This work partly was funded by the European Commission under contract 213303 (HIGH-EF), and by the German state of Thuringia via Thüringer Aufbaubank under contract 2008 FE 9160 (SolLUX). We would like to thank J. Lábár and G. Sáfrán (MFA Budapest) for TEM

Amkreutz, M.; Müller, J.; Schmidt, M.; Haschke, J.; Hänel, T. & Schulze, T.F. (2009).

Optical and electrical properties of electron beam crystallized thin film silicon

silicon layer is used instead, which, however, has somewhat low transparency.

**6. Conclusion** 

the surface is a crucial preparation step.

**7. Acknowledgment** 

investigations.

**8. References** 

tandem cells which would further increase the efficiency.

#### **5.4 Complete kinetics of transformation**

Stationary nucleation together with the growth of supercritical nuclei according to the Jackson-Chalmers equation leads to a continuous increase of the amount of the new phase on account of the parent phase. When one takes account that during the progress of phase transformation more and more parent phase is consumed and less volume is available for actual transformation, one ends up with the Avrami-Mehl equation (Avrami, 1940) for the volumetric amount of the new phase

$$\alpha = 1 - e^{t^\star / t\_c^\star} \tag{9}$$

with the characteristic time

$$t\_c = \sqrt{\frac{3}{\pi J v^3}}\tag{10}$$

*J* is the stationary nucleation rate of Eq. 4 and *v* is the speed of propagation of a phase front according to Jackson-Chalmers Eq. 1. In deriving Eq. 9 the time lag of nucleation τ has been neglected. To include this effect, one simply replaces *t* by (*t*-τ) in Eq. 9 for *t*>τ. The resulting average grain size when the parent phase has been consumed completely is given by

$$D = 1.0374 \sqrt{\frac{v}{J}} \tag{11}$$

So the grains are the larger the higher the Jackson-Chalmers speed and the lower the nucleation rate is, which sounds reasonable. To get large grains from an undercooled melt one should keep the temperature in a range of not too high undercooling, where nucleation rate is low and growth rate is high (Figs. 6 and 8). Fig. 12 shows the expected final grain size in solid phase crystallization of amorphous silicon. It shows that in the CSG process at about 600°C (see Sect. 3.) grains of several µm are to be expected, which is in accordance with experiments. By increasing the crystallization temperature one cannot change the grain size appreciably. Lowering the temperature would lead to a rather high time needed for crystallization due to higher time lag of nucleation (Fig. 11), lower nucleation rate (Fig. 9), and lower growth rate (Fig. 7).

600 800 1000 1200 1400 1 D/µm 10 100 T/K Tma

Fig. 12. Average grain size after solid phase crystallization of amorphous silicon as depending on temperature
