6.2 p-Type Czochralski ingots
6.2.1 Resistivity in axial direction
6.2 p-Type Czochralski ingots
Here in CENTESIL boron-doped Czochralski-ingots can be grown. A great improvement would be, if the quality of the ingots could be determined before cutting them into wafers, because the cutting process wastes money, time and a third of the ingot material [SE02, Fig.
7]. Therefore we are interested in the characterization of as grown ingots without cutting.
Those ingots and UFOs subject to our experiments are shown in Table6.2. Pieces, which are extracted from the melt before the main ingot is pulled in order to remove impurities on the surface are called UFOs. The ingot itself consists of a neck, where the seed crystal begins, the shoulder, where the ingot grows to its desired diameter, the body which should have a constant diameter but is often undulating because of varying temperature through the growth process, and finally the endcone.
6.2.1 Resistivity in axial direction
The dopant concentration follows Scheil’s law (2.12), so the resistivity is expected to decrease along the growth axis. The shoulder of the ingot is the first to leave the melt and the endcone the last. The quantitative description of this statement seems to be simple, as one has only to fill in the values in Scheil’s equation, but in real world it is more challenging. The reason is, that the shape of the Czochralski ingots does not allow simple percentage calculations. We will compare two models, one neglecting shoulder and endcone and the other considering different shapes for shoulder and endcone.
The simple model takes only into account the weights of UFO mUFO and ingot mi and the length of the ingot li. As the density of silicon is constant, we can also insert the mass
Name CR15 CR4 CR8 CR6 CR5 initial mass 40 kg 60 kg 35 kg 30 kg 30 kg dopant mass 1.9 mg 2.4 g 1.6 mg 1.5 mg 47.1 g ingot mass 25.82 kg 47.22 kg 29.01 kg - 23.5 kg diameter 205 mm 205 mm 229 mm 205 mm 150 mm
length 600 mm 890 mm 550 mm - 570 mm
UFO mass 9.32 kg - - 7.73 kg
-UFO name CR4b - - -
-Comments
Table 6.2: Properties of the sample ingots used in the experiments
percentage in Scheil’s equation, instead of the volume percentage. This leads to the formula C(l) = C0
k
1−mUFO+l/limi
minit
k−1
, (6.10)
which describes the dopant concentration at distance l from the top of the ingot. C0 is the initial dopant concentration in the liquid and k the segregation coefficient of boron. Finally this concentration is converted to resistivity using Eq. (2.3) and the mobility models of PC1D 5.0 [spc97].
Taking into account the shape of shoulder and endcone, the model for the body becomes more complicated. While the treatment of the UFO remains the same, the distance percentage for the ingot is no longer the volume percentage. Therefore we have to calculate the volume for every distance. Let r(x) be the radius at distance x from the top of the ingot. Assuming rotational symmetry, the volume V up to the distancedis given as
V(d) =π Z d
0
r(x)2dx. (6.11)
The solidified fraction fS, which must be inserted in Scheil’s equation, is then given as fs(d) =
Rd
0 r(x)2dx Rl
0r(x)2dx, (6.12)
where l is the total ingot length. This value can be obtained by direct measurement of the ingot, as well as the lengths of shoulder, endcone and body radius. The question is now how to model shoulder and endcone. To evaluate the different approaches, we can compare the calculated weight with the real weight of the ingot.
The first and most simple approach is to model the radii of shoulder and endcone linearly from zero at the ends to the ingot diameter, where the body begins. This linear model does not visually represent reality, but is useful to get an idea of the error committed. As shown in Table 6.3, this model overestimates the mass in shoulder and endcone. The error is about 10 %. The second model was inspired by the visual shape, a parabola with its minimum (set to 0) at the beginning/ending of the ingot. We will refer to it as square model. The deviation
6.2 p-Type Czochralski ingots
Ingot real weight difference ’linear’ difference ’square’
CR4 47.22 kg 4.3 kg 9 % 0.9 kg 2 %
CR8 29.01 kg 3.3 kg 11 % −0.8 kg −3 %
CR15 25.82 kg 2.6 kg 10 % −0.9 kg −4 %
Table 6.3: Difference of modeled to real ingot weight. The linear model overestimates the mass in shoulder and endcone, whereas the square model tends to underestimate the mass, but reproduces the real mass within 5 %.
from the real weight is compared to the linear model more than halved and less than 5 %. The fact that we obtain values greater and smaller than the real weight, indicates that a possible systematic error is small. We also considered using a catenary, which is obtained for a minimal surface with rotational symmetry, but the resulting conditional equation can only be solved numerically and additionally is not very stable. Therefore the square model is chosen for the resistivity calculation.
From the initial masses of silicon and boron and the relation between position and volume fraction (6.12), we can now calculate the dopant concentration and the resistivity along the ingot. The results from the square model are shown in Figure 6.6.
Those calculations are verified by resistivity measurements along the growth axis with the BLS-I instrument. The ingot CR4 could not be measured, because it is so heavily doped, that the resistivity is below the detection limit. The results of the other two ingots are shown together with the theoretical results in Figure 6.7. Due to the geometry of the device, all resistivity measurements are averaged values over a range of about 4 cm. The resistitivities are assigned to the position of the middle of the coil. Both ingots have the drawback, that the body radius oscillates in the first part. This implies, that the sensor coil cannot be applied entirely on the ingot. Hence resistivity increases as described in section 6.1.7. In the other, flat, part, the measured resistivity is near the theory curve. But we also note, that they do not coincide. The visible fluctuations between different measurements indicate, that it is difficult to provide the same measurement conditions each time. Even in the flat part, there are perceptible fluctuations, maybe due to the curved surface. But this fact can only explain deviations to higher resistivities. Lower resistivities than predicted require additional positive carriers, a biased resistivity measurement or less silicon as in the model. We cannot exclude the latter, because the residues have not been balanced, but it seems rather implausible. Even if we overestimated the shoulder by 5 kg, the resistivity would change in the range of 0.01 Ωcm - too less to explain the measurements. As described at the end of section 6.1.5, we also cannot exclude a biased resistivity measurement. Additional positive carriers can be provided by impurities, which show an acceptor like behaviour. The resistivity change from 2.75 Ωcm to 2.53 Ωcm corresponds to an additional carrier density of 4.7·1014cm−3, which is of a realistic order of magnitude for impurities, but we cannot determine type and quantity of these to check this hypothesis.
Fortunately, we have another chance to test Scheil’s equation and our models. The ingot CR5 was sold and there cut into discs. The company measured the resistivities by the four point probes method and sent us a copy of these measurements. They are shown together with the different models discussed in this section in Figure6.8a. It can be noticed, that there is little difference if we consider shoulder and endcone or not. Especially at the beginning of
(a) Ingot CR8 (b) Ingot CR15 with UFO
Figure 6.6: Resistivity calculations based on the initial dopant concentration and the ingot shape and measurements. UFO and residues were normalized to a cylinder with the same radius as the body and the height was adjusted, to obtain the measured mass.
15 20 25 30
distance to top in cm 2
3 4 5 6 7
re sis tiv ity in
Ωcm
square model exp-1 exp-2 exp-3
(a) Ingot CR8
10 15 20 25 30 35
distance to top in cm 2.0
2.5 3.0 3.5 4.0
re sis tiv ity in
Ωcm
square model exp-1 exp-2
(b) Ingot CR15
Figure 6.7: Comparison of theoretical model with measured resistivities. The measurement is an average over 4 cm and the points are plotted over the middle of the averaged area. Compared with Figure6.6only the body is shown, because the resistivity could only be measured there. The increased resistivity below 24.5 cm of ingot CR8 and below 25 cm of ingot CR15 is due to the bumpy surface.
6.2 p-Type Czochralski ingots
Figure 6.8: Four point probe resistivity measurements compared with the different models. A seg-regation coefficient of 0.75 instead of 0.8 as proposed by Sim et al. [SKL06], explains the data even better.
the ingot, all models are very close. Towards the end, both models give a resistivity which is higher as the measured one. This may be due to indiffusion of impurities like oxygen, which act in some cases as an additional donor. The simple method is closer to this data, because it supposes the dopant from the shoulder and endcone to be in the body leading to a higher dopant values and lower resistivities. For practical purposes, we consider the simple method as sufficient.
Finally, another observation can be made. Despite the value of 0.8 for the segregation coefficient is widely accepted, the resistivity at the beginning is higher as expected. Simet al.
[SKL06] propose a value of 0.75 due to the rotation during the Czochralski growth process.
This value explains our data even better as can be seen in Figure 6.8.