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= CCs
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 Cs (at the solid-liquid interface) can be described byScheil’s equation
Cs= C0
k (1−fs)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.
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.
3 Applied methods
In this section the fundamentals of the different measurement principles for the different prop-erties 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)
Figure 3.1: left:
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
where ρ0 is the resistivity calculated by (3.2). The correction term G(w/s) simplifies to G
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 Pabs absorbed by the eddy currents’ ohmic heating is
Pabs = 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 =Prf=Pabs= V2
8πn2σt. (3.5)
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
Rtot = 1
Rrf + 1
Rsmp. (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 /Rrf, we further obtain I−I0 = const 1
Rsmp ∝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.
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.