An Analysis Of The Factors Affecting

Solid-State Electronics 50 (2006) 345–354 www.elsevier.com/locate/sse

An analysis of the factors affecting the alpha parameter used for extracting surface recombination velocity in EBIC measurements Oka Kurniawan, Vincent K.S. Ong

*

School of Electrical and Electronics Engineering, Nanyang Technological University, Block S2, Nanyang Avenue, Singapore 639798, Singapore Received 25 May 2005; received in revised form 9 January 2006; accepted 9 January 2006 Available online 9 March 2006

The review of this paper was arranged by Prof. C. Tu

Abstract This paper gives an in-depth analysis of the factors affecting the alpha parameter which is used for extracting the surface recombination velocity in electron beam induced current (EBIC) line scan measurements. The analysis shows that the alpha versus normalized surface recombination velocity curve is a function of both the normalized beam depth as well as the normalized scanning range. Variations in the normalized beam depth affect the accuracy only when extracting high surface recombination velocity values. On the other hand, the variation in the normalized scanning range affects the accuracy in extracting the middle range values of the surface recombination velocity only slightly. Conditions for accurate extraction are given in this paper. The analysis was further verified with the use of computer simulation.  2006 Elsevier Ltd. All rights reserved. Keywords: EBIC; Surface recombination velocity; Diffusion length

1. Introduction A simultaneous extraction of both the minority carrier diffusion length and the surface recombination velocity with the use of EBIC was successfully demonstrated in [1]. It was shown that the EBIC current can be expressed as I ¼ kxa expðx=LÞ

ð1Þ

In this equation, L is the minority carrier diffusion length of the material, k is a constant, a (alpha) is a fitting parameter, and x is the beam distance from the junction as shown in Fig. 1. In this method, the fitting parameter alpha is used to straighten the ln(I) versus x curve, thereby enabling the minority carrier diffusion length to be extracted accurately for any values of surface recombination velocity. The alpha *

Corresponding author. Tel.: +65 6790 4708; fax: +65 6792 0415. E-mail address: [email protected] (V.K.S. Ong).

0038-1101/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.sse.2006.01.006

parameter obtained from the fitting process can then be used to extract the surface recombination velocity. In 1998, it was shown in [2] that alpha is related directly to the normalized surface recombination velocity. The relationship between alpha and the normalized surface recombination velocity can be modeled by using a normal distribution function. The results of the experiments seemed to suggest that the variation of alpha with the normalized surface recombination velocity is independent of both the minority carrier diffusion length and the depth of the generation volume. Thus, the surface recombination velocity can be extracted from alpha alone by using sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi A S ¼ g þ r1 2 ln ð2Þ aþB where S ¼ vs L=D and A = 5047, B = 0.6, g = 20, and r1 = 4.7.

ð3Þ

346

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Nomenclature a fitting parameter alpha in Eq. (1) A, B fitting parameters in Eq. (2) C1, C2 constants in current densities of Eqs. (27) and (28) d a function to vary I that defines I 0 in Eq. (6) D effective diffusion coefficient Dh diffusion coefficient for holes g fitting parameter in Eq. (2) ei ionization energy, i.e., the effective average energy required to generate a hole–electron pair Eb energy of electrons in the beam in keV f average fraction of the energy of the incident electrons lost by backscattering g total generation rate G generation factor (=Eb(1  f)/ei) I charge collection (EBIC) current I0 current that gives r2 = 1 for Eq. (5) with constant a

In Eq. (3), vs is the surface recombination velocity while S is its normalized value. L is the minority carrier diffusion length, and D is the diffusion coefficient. The errors reported were less than 20% for the surface recombination velocity range of 1 · 103 cm/s to 3.16 · 104 cm/s. Recently, Zhu et al. [3] gave an alternative method for extracting the surface recombination velocity. Their experiments, however, indicated that the alpha parameter also depends on the depth of the generation volume. If alpha depends on the depth of the generation volume, then the accuracy in extracting the surface recombination velocity using the normal distribution function would be in question. This is because the model used to extract the surface recombination velocity in [2] assumes that the relationship between alpha and the surface recombination velocity is independent of the depth.

Ib IN J k k0 K1 L q R r r1 S vs W x z

electron beam current normalized EBIC current current density constant in Eq. (1) constant in Eq. (12) modified Bessel function of the second kind minority carrier diffusion length elementary charge (1.6 · 1019 C) electron penetration range lateral extension of the generation volume fitting parameter in Eq. (2) normalized surface recombination velocity surface recombination velocity (cm/s) width of scanning range in EBIC line scan distance between the junction and the generation volume depth of the generation volume from the free surface

In order to extract the surface recombination velocity accurately by using the alpha parameter, it is important to analyze the dependence of alpha on the physical parameters such as the depth of the generation volume, the minority carrier diffusion length, and the beam distance from the junction. This paper gives a complete analysis of the parameters that alpha depends on, based on an analytical model of the alpha equation. The analysis of the various parameters is then verified by using computer simulation. The impact on accuracy as well as the conditions for obtaining accurate surface recombination velocity is also given. 2. Analytical equation for alpha The analytical equation for alpha can be derived from Eq. (1). Taking the natural logarithm of both sides and differentiating with respect to distance will give an equation that consists of alpha, beam distance, and the minority carrier diffusion length. Rearranging the term and taking the natural logarithm of Eq. (1) gives lnðI=xa Þ ¼ lnðkÞ  x=L

ð4Þ

It can be seen that the right hand side of the equation is a straight line. Fitting this equation into the current values by adjusting the alpha parameter gives the minority carrier diffusion length and the fitting parameter alpha. It will be shown later that the actual EBIC current does not exactly yield a straight line. The error in this fitting process can be seen from the correlation coefficient, r2. Therefore, Eq. (4) needs to be rewritten as Fig. 1. Normal-collector configuration of EBIC measurement.

lnðI 0 =xa Þ ¼ lnðkÞ  x=L

ð5Þ

O. Kurniawan, V.K.S. Ong / Solid-State Electronics 50 (2006) 345–354

where we have changed I with I 0 to indicate that the current that satisfies Eq. (5) is different with the actual EBIC current. Let I0 ¼

I d

ð6Þ

where I is the actual EBIC current and d gives the error in the fitting process from a straight line. Substituting Eq. (6) into (5) gives lnðI=xa Þ  lnðdÞ ¼ lnðkÞ  x=L

ð7Þ

2

If the fitting process gives r  1, then the second term of the left hand side of Eq. (7) can be neglected. This is because for this particular case, the actual EBIC current yields almost a straight line when fitted into Eq. (4). When a is adjusted to yield a straight line in EBIC measurements, it was reported in [1] that the correlation coefficient is very close to one and thus the second term in the left hand side of Eq. (7) can be ignored. Fig. 2 shows the magnitude of this term for the case where the surface recombination velocity is 1 · 104 cm/s. It can be seen that within the scanning range (x = 9 to 33) where the equation is fitted into, the magnitude of the second term is close to zero. Fig. 2 also shows that this constant alpha makes the first term to be close to a straight line within the scanning range. Ignoring the second term and differentiating with respect to x gives d lnðIÞ a 1  ¼ dx x L

ð8Þ

347

term gives the analytical expression for alpha. The actual expressions for I and its derivative depend on the configuration of the collector. One of the more popular methods for the determination of the minority carrier diffusion length using the scanning electron microscopy (SEM) is the one where the collector is normal to the scanning direction of electron beam. This is known as the normal-collector configuration and is shown in Fig. 1. Theoretical expressions for current and its derivative for the normal-collector configuration is given in [4]. The expressions for a point source are I N ðx; zÞ ¼ expðx=LÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi Z 2S 1 u expð u2 þ 1 Lz Þ sinðux=LÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi  du p 0 ðu2 þ 1ÞðS þ u2 þ 1Þ dI N 1 ðx; zÞ ¼  expðx=LÞ L dx  pffiffiffiffiffiffiffiffiffiffiffiffi  Z 2S 1 u2 exp  u2 þ 1 Lz cosðux=LÞ pffiffiffiffiffiffiffiffiffiffiffiffi  du pL 0 ðu2 þ 1ÞðS þ u2 þ 1Þ

ð10Þ

ð11Þ

where IN = I/GIb, z is the depth of the generation volume, and the other symbols have their usual meanings. The corresponding expressions for a Gaussian source [5] are given in Eqs. (12) and (13):   2 2 u ur exp  2 2 ðu þ 1Þ 2L 0  2 pffiffiffiffiffiffiffiffiffiffiffiffi  r z S pffiffiffiffiffiffiffiffiffiffiffiffi erfc 0:57 exp  u2 þ 1 2 L S þ u2 þ 1 2L  pffiffiffiffiffiffiffiffiffiffiffiffi 

r z L2 u2 þ 1  sinðux=LÞdu ð12Þ  pffiffiffi L r2 2L

I N ðx; zÞ ¼

Rearranging the term and using the relationship d ln(I)/ dx = 1/I(dI/dx) gives   1 dI 1 þ a¼ x ð9Þ I dx L

2 p

Z

1

Substituting the current I and its derivative with theoretical expressions that contain the surface recombination velocity 0

0.05 -0.1

0.04 -0.2 alpha

Ln (delta)

0.03 0.02

-0.3 -0.4

0.01 -0.5

0 -0.6

–0.01 0

10

20

30

40

50

x (μm) Fig. 2. The amount of the second term in the left hand side of Eq. (7). The alpha values was obtained by fitting process with L = 3 lm, z = 0.3, and scanning range from x = 9 to 33. The surface recombination velocity is 1 · 104 cm/s.

10-2

100

102

S = Vs*L/D

Fig. 3. Alpha curve from analytical equations (9)–(11) with the value x/L = 3 and z/L = 0.1.

O. Kurniawan, V.K.S. Ong / Solid-State Electronics 50 (2006) 345–354

dI N ðx; zÞ 2 ¼ dx pL

Z 0

1

  2 2 u2 ur exp  2 ðu2 þ 1Þ 2L 

3. Analysis using analytical equation

pffiffiffiffiffiffiffiffiffiffiffiffi z  r S pffiffiffiffiffiffiffiffiffiffiffiffi erfc 0:57 exp  u2 þ 1 L S þ u2 þ 1 2L2 2

 pffiffiffiffiffiffiffiffiffiffiffiffi 

r z L2 u2 þ 1  2 cosðux=LÞdu  pffiffiffi Lr 2L

ð13Þ

pffiffiffiffiffi where r ¼ z=ð0:3 15Þ ¼ z=1:162. Evaluating Eqs. (9)–(11) numerically for several values of surface recombination velocity will result in an alpha curve that is shown in Fig. 3. The curve in Fig. 3 has the same shape and the same range of alpha values as the one given in [1,2]. Eqs. (9)–(11) show that the alpha curve for a point source depends only on two parameters. They are the normalized scanning range, and the normalized depth of the generation volume, z/L. It is important to note that the term x/L in Eq. (9) refers to the location where alpha is constant. In the derivation, alpha was assumed to be a constant fitting parameter. However, Eqs. (9)–(13) show that the alpha equation has a term x/L. This is because the second term in Eq. (7) was ignored in the derivation. Therefore, Eq. (9) actually gives the alpha value where da/dx = 0. In other words, Eq. (9) gives the alpha value at a particular location x where the derivative of alpha is zero. In real measurements, the exact location where alpha is constant has to be within the scanning range where the equation is fitted into. This means that varying x/L in Eq. (9) would show the actual effect of varying the location of the scanning range with respect to the junction. For a Gaussian source, alpha also depends on r/L which is the normalized lateral extension of the generation volume. However, since r is strictly a function of depth, the alpha equation for a Gaussian source is also only a function of the normalized beam distance from the junction and the normalized depth of the generation volume. The analytical equation for alpha also shows that the alpha curve does not depend on the beam current Ib. This is because the term GIb cancels out in Eq. (9). This means that the only parameters that affect the alpha versus normalized surface recombination velocity curve are the normalized beam depth and the normalized beam distance from the junction for both the point source as well as the Gaussian source. The analysis of the alpha dependencies will, therefore, be done by investigating the effects of these two parameters. Eqs. (9)–(11) for the point source assumption will be used throughout this analysis. The point source assumption can be justified when the distance between the beam and the junction is greater than the electron penetration range in the material [6]. The accuracy in using the point source assumption will also be justified later in this paper.

Eqs. (9)–(11) show that the alpha versus normalized surface recombination velocity curve is affected by two parameters: the normalized scanning location from the junction and the normalized depth of the generation volume. The effect of changing the normalized depth on the alpha curve from Eq. (9) is shown in Fig. 4. The alpha curve is affected by the normalized depth only at higher values of surface recombination velocity. Larger z/L ratio will cause this portion of the curve to move upward. However, the alpha curve changes imperceptibly for z/L 6 0.1. In other words, the normalized depth has a negligible effect on the alpha curve when z 6 0:1 ð14Þ L The curves for different values of normalized depth show that changing z/L would affect alpha only at higher values of surface recombination velocity. This effect can be explained by considering the recombination of minority carriers at the surface for different beam depths. The deeper the generation volume, the less the effect of the surface recombination would be. Another consideration is that for lower values of surface recombination velocity, the change in the z/L ratio must be sufficiently large for it to affect the collected current. Only when the current is affected would the alpha change. When the surface recombination velocity is large, the effect on the current can be readily seen even for small changes in the z/L ratio. As the z/L ratio increases, more minority carriers will be collected instead of recombining at the surface. Therefore, the natural logarithm of the current will be less concave and the alpha value will be less negative [2]. This is the reason that the curve shifts upward at higher values of surface recombination velocity.

0 z/L = 0.033 z/L = 0.067 z/L = 0.1 z/L = 0.2 z/L = 0.3 z/L = 0.4 z/L = 0.5

-0.1 -0.2

alpha

348

-0.3 -0.4 -0.5 -0.6

-2

10

0

10 S = Vs*L/D

2

10

Fig. 4. Alpha curves from Eqs. (9)–(11) with x/L = 3 and L = 3 lm. Normalized surface recombination velocity (S) ranges from 0 to 459.64.

O. Kurniawan, V.K.S. Ong / Solid-State Electronics 50 (2006) 345–354

0 x/L = 2 x/L = 3 x/L = 4 x/L = 5 x/L = 10

-0.1

alpha

-0.2 -0.3 -0.4 -0.5 -0.6

-2

10

0

10

349

ing on whether a point source or a Gaussian source is used. Eq. (1) can then be fitted to these theoretical current values which will give the extracted minority carrier diffusion length as well as the alpha value. Thus, the effect of varying the width of the scanning range on the alpha curve can also be investigated. The effect of reducing the scanning range is similar to reducing the scanning location term for the alpha curve obtained from the analytical equation. This is shown in Fig. 6. On the other hand, increasing the starting location of the scanning range alone will have a similar effect as increasing the scanning location for the alpha curve obtained from the analytical equation. This agrees with the previous analysis and is shown in Fig. 7.

2

10

S = Vs*L/D

0

Fig. 5. Alpha curves from Eqs. (9)–(11) with z/L = 0.1 and L = 3 lm. Normalized surface recombination velocity (S) ranges from 0 to 459.64.

-0.1 -0.2 alpha

Changing the normalized depth could affect the accuracy in extracting higher values of surface recombination velocity. However, since the change in the alpha curve is negligible for z/L 6 0.1, the accuracy is unaffected under this condition. Increasing the scanning range location will shift the middle portion of the curve to the left and the lower portion of the curve upward as shown in Fig. 5. The rate of change in the alpha curve decreases as the scanning range location increases. Therefore, the change in the alpha curve due to the change in the scanning range location is only significant at small values. The results for different values of scanning location can be used to see the effect of varying the starting location of the scanning range in the EBIC line scan measurement. However, the scanning ranges in practical measurements have finite scanning width, and thus, this new parameter must comes into consideration. The effect of the width of the scanning range however cannot be analyzed by using Eq. (9). The following section shows the analysis for the effect of the width of the scanning range.

-0.3 -0.4 -0.5 -0.6

-2

10

0

10

10

2

S = Vs*L/D Fig. 6. Alpha curves for different scanning width. The simulation used z/L = 0.07 and L = 3 lm.

x/L = 2 to 14 x/L = 3 to 15

-0.1 -0.2 alpha

4. Effect of scanning range on the extracted alpha The analytical expression for alpha (Eqs. (9)–(11)) is able to show the changes in the alpha curve when either the normalized scanning location from the junction or the normalized beam depth changes. This equation, however, does not give any information on the width of the scanning range. Since the alpha value used for extracting surface recombination velocity comes from a fitting process within a finite width, it is important to see how the scanning range with a finite width could affect the alpha curve. In order to do this, the theoretical current values of EBIC measurement are obtained by using either Eqs. (10) or (12) depend-

x/L = 3 to 13 x/L = 3 to 7

-0.3 -0.4 -0.5 -0.6

-2

10

0

10

10

2

S = Vs*L/D Fig. 7. Alpha curves with the same scanning range but different starting location. The simulation used z/L = 0.07 and L = 3 lm.

350

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It is important to extract the surface recombination velocity accurately in the range of 0.05 < S < 5 [2]. This is because in practice for S < 0.05, vs can be approximated to zero, while for S > 5, vs can be taken to be infinitely large. In order to extract the surface recombination velocity in the range of 0.05 < S < 5, it is desirable to have a steep alpha curve. This is because a steep middle portion of the alpha curve will give a more accurate result in extracting the surface recombination velocity. This is illustrated graphically in Fig. 8. The solid lines in Fig. 8 indicate the alpha values for a certain given surface recombination velocity. If a certain amount of error is introduced into the measurement, as shown by the dotted lines, then it can be seen in the figure that this error will result in a smaller variation in S on the steeper curve than on the other curve. Figs. 6 and 7 suggest that in order to have a steep alpha curve, the starting location and the width of the scanning range must be small. A small starting location, however, should not violate the condition x > 2L [1,7]. This is because a small starting location will adversely affect the accuracy of the extracted minority carrier diffusion length. Fig. 9 shows this effect. The inaccuracy that results from extracting L with a small starting location is due to the fact that in this region the EBIC current cannot be fitted exactly into Eq. (1) with a constant alpha. This is because the current variation very near to the junction no longer follows Eq. (1). This can also be seen in Fig. 2. Fig. 9 suggests that in order for the extraction of L to have an error of less than 1.5%, the starting location must obey the following relationship: x P2 ð15Þ L start The minimum width of the scanning range is also determined by the accuracy in extracting the minority carrier diffusion length. Fig. 10 shows the errors in extracting L

Fig. 9. Error in extracting L as starting location is reduced. The value of L is extracted by curve fitting Eq. (1) with theoretical current values from Eq. (10). The value of L is 3 lm, the scanning width is W/L = 9, and the depth is z/L = 0.1.

Fig. 10. Error in extracting L as the scanning range is reduced. The value of L is extracted by curve fitting Eq. (1) with theoretical current values from Eq. (10). The value of L is 3 lm, the scanning starts from x/L = 2, and the depth is z/L = 0.1.

for different scanning range values. The results show that the error in extracting L increases slightly as the width is reduced. To keep the error below 1.5%, the normalized width must be greater than or equal to 9, i.e., W P9 ð16Þ L The value of condition (16), however, depends on the starting location of scan. The nearer the starting location is from the junction, the larger the width of the scan is required to keep the error small. 5. Verification

Fig. 8. The effect of the steepness of the alpha curve on the accuracy of the surface recombination velocity extraction.

The above results from the theoretical equations were verified by using computer simulation. MEDICI, a 2-D device simulation software, was used for this purpose. A generation radius of 0.1 lm was used to simulate the generation source. This point generation source is represented in MEDICI by using a square generation area with sides of 0.2 lm [8].

O. Kurniawan, V.K.S. Ong / Solid-State Electronics 50 (2006) 345–354

The depth of the generation volume and the total generation rate can be calculated from the beam energy and the beam current information as given in [7], and reproduced here, z ¼ 3:84  1012 E1:75 ðcmÞ b

ð17Þ

18

ð18Þ

g ¼ 1:6  10 I b Eb ðEHP/sÞ

Eq. (1) was then fitted into the current values obtained from the simulation to extract the minority carrier diffusion length and the alpha parameter using the method of [1,2]. The curves for alpha with different normalized beam depth and scanning range values are shown in Figs. 11 and 12, respectively. 0 z/L = 0.3 z/L = 0.2 z/L = 0.1 z/L = 0.067 Normal distribution

-0.1

alpha

-0.2 -0.3

351

Comparing Figs. 11 and 12 with Figs. 4–6 shows that the change in the alpha curves obtained from MEDICI simulations have the same behaviour as those obtained using the analytical equation. Thus, the analytical equation for alpha can be used to predict the parameters that affect alpha as well as to predict how alpha changes with these parameters. This shows that the previous analysis which uses the analytical equations is valid for practical EBIC measurements. All conditions stated previously to extract an accurate minority carrier diffusion length and surface recombination velocity have therefore been verified to be valid. Another observation that can be seen in Fig. 12 is that the effect of the starting location of scanning on the alpha curve is more dominant than the effect of the scanning width. This is because, increasing the starting location by L while reducing the width by the same amount is similar to increasing the scanning location in the alpha analytical equation. In other words, the width must be reduced by more than one L to compensate for the effect of increasing the starting location by one L.

-0.4

6. Impact on accuracy -0.5 -0.6

-2

10

0

10

10

2

S = Vs*L/D Fig. 11. Alpha curves from MEDICI simulations for different z/L. The scanning range used is from x/L = 3 to x/L = 14 and L = 3 lm. The dotted line shows the normal distribution function from Eq. (2).

0 x/L = 2 to 14 x/L = 3 to 14 Normal distribution

-0.1

alpha

-0.2 -0.3 -0.4 -0.5 -0.6

-2

10

0

10

10

2

S = Vs*L/D Fig. 12. Alpha curves from MEDICI simulations for different scanning ranges. The simulations used z/L = 0.067 and L = 3 lm. The dotted line shows the normal distribution function from Eq. (2).

It has been shown that the shape of the alpha versus normalized surface recombination velocity curve depends mainly on the normalized depth of the generation volume as well as the normalized scanning range. This means that the accuracy of extracting the surface recombination velocity with the use of Eq. (2) could be affected by these two parameters. This is because Eq. (2) assumes that the alpha curve is invariant to both the depth and the scanning range. It is therefore important to see how changes in depth and scanning range could affect the accuracy obtained by using Eq. (2). To do this, the alpha values from MEDICI simulation data were used to extract the surface recombination velocity. To see the impact of depth on accuracy, the alpha values from Fig. 11 were substituted into Eq. (2) to obtain the normalized surface recombination velocity. The surface recombination velocity can then be extracted by using Eq. (3). On the other hand, the alpha values from Fig. 12 were used to see the impact of changing the scanning range on the accuracy. The errors in extracting the surface recombination velocity are shown in Tables 1 and 2. The results from Table 1 indicate that the accuracy is not affected much as the z/L ratio change from 0.067 to 0.3. The only obvious impact in changing z/L can be seen for the surface recombination velocity 1 · 105 cm/s. In this case, the error increases as z/L increases. For the impact in changing the scanning range on alpha, Table 2 shows that the errors for both the scanning ranges are below 15% for surface recombination velocities within the range of 3.16 · 103 cm/s to 1 · 105 cm/s. Therefore, the impact on accuracy in using Eq. (2) is not significant.

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O. Kurniawan, V.K.S. Ong / Solid-State Electronics 50 (2006) 345–354

Table 1 The impact of depth on the accuracy in extracting surface recombination velocity

0 1 · 102 1 · 103 3.16 · 103 1 · 104 3.16 · 104 1 · 105 1 · 106 1 · 107

S = 0.00459 S = 0.0459 S = 0.1452 S = 0.4596 S = 1.452 S = 4.596

Error (%) z/L = 0.3

z/L = 0.2

z/L = 0.1

z/L = 0.067

1 295.34 27.38 4.50 5.65 0.07 29.95 – –

1 291.30 25.70 2.61 7.71 3.24 22.50 – –

1 291.30 25.28 1.88 9.02 5.48 15.08 – –

1 295.34 24.45 1.59 9.15 5.93 13.18 – –

15

error (%)

vs (cm/s)

20

10

5

The scanning range is from x/L = 3 to x/L = 14 and L = 3 lm.

0 1 Table 2 The impact of scanning range on the accuracy in extracting surface recombination velocity vs (cm/s) 0 1 · 102 1 · 103 3.16 · 103 1 · 104 3.16 · 104 1 · 105 1 · 106 1 · 107

Error (%) x/L = 2 to x/L = 14

x/L = 3 to x/L = 14

1 283.22 32.82 11.00 0.90 1.55 3.76 – –

1 295.34 24.45 1.59 9.15 5.93 13.18 – –

The depth of the generation volume is z/L = 0.067 and L = 3 lm.

The effect of increasing the z/L ratio is clearly shown at the surface recombination velocity 1 · 105 cm/s. This agrees with the analysis since the effect is most obvious at higher values of surface recombination velocity. However, at the surface recombination velocity values of 1 · 104 cm/s (S = 0.460) and 3.16 · 104 cm/s (S = 1.452), the error decreases as z/L increases. The reason behind it is due to the shape of Eq. (2) as shown in Fig. 11. At the surface recombination velocity value of 1 · 105 cm/s (S = 4.596), the alpha curve moves further away from the normal distribution function as z/L increases; while this is not so for the other points. As z/L increases beyond 0.3, the accuracy in using the normal distribution function will deteriorate even further as the alpha values are higher than the corresponding points in the normal distribution function. The impact of the scanning range, on the other hand, is not very significant when the starting point is increased by one L. Fig. 12, however, shows that the normal distribution function is closer to the alpha curve for the scanning range x/L = 2 to x/L = 14. As the initial location of the scan increases beyond x/L = 3, the middle portion of the alpha curve will move further to the left, and the error will become significant at the surface recombination velocity range of 3.16 · 103 6 vs 6 1 · 105 (0.145 6 S 6 4.596).

1.5

2 x/R

2.5

3

Fig. 13. Deviation in alpha values using point source from the one using Gaussian source. The deviation is calculated as follows: Error = (alpha_point  alpha_gauss)/alpha_gauss · 100%.

7. Accuracy in using point source assumption All the previous discussions used point source assumption in the analysis. This assumption is justified only when the beam distance to the junction is greater than the electron penetration range [6]. The previous analysis using analytical equation for alpha used Eqs. (10) and (11) for point source to be substituted into Eq. (9). Substituting Eqs. (12) and (13) into Eq. (9) will give the analytical expression for alpha when the source is a Gaussian distribution. Fig. 13 shows the deviation of alpha values using point source from alpha values using a Gaussian source as the distance, which is normalized to the electron penetration range, is varied. The result suggests that the deviation increases as the ratio of x/R decreases. This is mainly due to the lateral extension effect of the generation volume. In order for the point source assumption to be valid, the beam distance from the junction must be greater than 1.5 times the electron penetration range, i.e., x > 1:5 ð19Þ R Under this condition the error can be kept to below 1%. The electron penetration range R is related to the center of mass of the generation volume by the following relationship [9]: z = 0.41R. Substituting this into Eq. (19) gives x > 3:66 ð20Þ z Therefore, in order for the measurement to ignore the lateral extension of generation volume, either condition (19) or (20) must be satisfied. 8. Conditions for accurate extraction The conditions for extracting the surface recombination velocity accurately with the use of the normal distribution

O. Kurniawan, V.K.S. Ong / Solid-State Electronics 50 (2006) 345–354

function were discussed in the previous sections. The two main conditions were to keep the alpha curve as constant and as steep as possible. The first condition will allow the extraction of the surface recombination velocity directly by using the alpha parameter alone. The second one will allow the surface recombination velocity to be extracted more accurately in the middle portion of the curve. From the analysis in the previous sections, it can be seen that the alpha curve can be kept constant by: (1) satisfying the condition given in Eq. (14) for the normalized depth of the generation volume, and (2) using the same normalized scanning range which includes the same normalized starting location and normalized width. The steepness of the alpha curve can be increased by two methods: (1) reducing the normalized starting location of the scan, and (2) reducing the scan width. However, reducing the starting location of the scan range is more effective since the effect on the alpha curve is more significant. Therefore, to satisfy the two main conditions for direct and accurate extraction, the condition for the EBIC measurement can be summarized as follows: Depth of generation volume z 6 0:1 L

ð21Þ

Scanning range x from ¼2  x L start to ¼ 11 L end

ð22Þ

Since the fitting parameters for the normal distribution given in [2] is not based on the conditions given above, a new set of parameter values would result in a more accurate extraction. It was found by curve fitting that a more accurate set of parameters for Eq. (2) is as follows: A ¼ 499:74; B ¼ 0:63 g ¼ 17:074; and r1 ¼ 4:67

ð23Þ

353

Table 3 shows the errors in extracting the surface recombination velocity by using Eq. (2) with the parameters given in (23). The data are taken from MEDICI simulation which satisfies conditions (21) and (22). 9. Comment on alpha values for large vs Berz and Kuiken [7] shows that the alpha value for infinite surface recombination velocity is 0.5. This alpha value comes from a theoretical derivation. However, computer simulation and the analytical equation for alpha give values that are lower than 0.5 for high surface recombination velocities. The alpha values given by simulation and analytical equation are about 0.6. The reason for this discrepancy comes from the approximation used by Berz and Kuiken to derive the value 0.5. In [7], Berz and Kuiken showed that for infinite surface recombination velocity, the current density when the beam scans the n-type region can be expressed as   2 Dh z x J¼ ð24Þ qg K 1 p D L L where Dh is the diffusion constant for holes and D is the effective diffusion constant. For low injection, D = Dh. The constant g is the total generation rate and K1(n) is the modified Bessel function of the second kind. The other symbols have their usual meanings. The above equation can also be written as x J ¼ C1  K 1 ð25Þ L where C1 is the constant with respect to x. Berz and Kuiken then made an approximation for x/L  1 by expanding the Bessel function using asymptotic series. For n  1, function K1(n) can be expanded as expðnÞ K 1 ðnÞ  rffiffiffiffiffiffiffi 2 x pL

ð26Þ

Thus, for x/L  1, Eq. (25) can be expanded and written as J ¼ C1 

Table 3 Error in extracting surface recombination velocity from MEDICI simulation data using normal distribution function with parameters given in (24) vs (cm/s)

Extracted vs (cm/s)

Error (%)

0 1 · 102 1 · 103 3.16 · 103 1 · 104 3.16 · 104 1 · 105 1 · 106 1 · 107

1.2885 9.4568 · 10 9.8567 · 102 3.1892 · 103 1.0462 · 104 3.1669 · 104 7.5770 · 105 – –

1 5.4321 1.4331 0.9247 4.6235 0.2176 24.2295 – –

The simulation used z/L = 0.067, and the scanning range from x/L = 2 to x/L = 11, with L = 3 lm.

expðx=LÞ rffiffiffiffiffiffiffi 2 x pL

ð27Þ

which leads to J ¼ C 2 ðx=LÞ

1=2

expðx=LÞ

ð28Þ

where C2 is another constant. Comparing with Eq. (1), the alpha value for this current equation is 0.5. The approximation of the Bessel function causes the discrepancy in the alpha values. Fig. 14 shows the difference between the current values using the Bessel function and the one with an asymptotic series approximation. It can be seen that the asymptotic approximation approaches the actual Bessel function for large values of x/L. However, the EBIC current is usually scanned from x/L  2. In this region, the two functions differ. Fig. 14

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O. Kurniawan, V.K.S. Ong / Solid-State Electronics 50 (2006) 345–354

0 Modified Bessel function, K1(x/L) Asymptotic Series of K1(x/L)

-1

Ln (J/C1)

-2 -3 -4 -5 -6 -7

1

2

3

4

5

6

of the scanning range affects the accuracy of extraction only slightly. The most affected region is in the middle range of alpha values. In order to extract surface recombination velocity accurately with the use of the normal distribution function, several conditions must be met. First, the normalized depth should be less than or equal to 0.1 (condition (21)), and second, the scanning range should start close to the junction but with x/L P 2, and with the smallest possible range (condition (22)). The fitting parameters given in [2] for the normal distribution function were modified to take into account these two conditions. The error in extracting L can be kept to below 1.5% for any values of surface recombination velocity; and the error for surface recombination velocity extraction can be kept to below 25% for the range of 1 · 102 cm/s to 1 · 105 cm/s.

x/L Fig. 14. The difference between the actual Bessel function and the asymptotic series approximation in logarithmic.

shows that the logarithmic current of the actual Bessel function is more concave compared to its asymptotic approximation. A more concave curve will result in a more negative alpha value [2]. This is the reason that the simulation and analytical equation for alpha give values of about 0.6 for large values of surface recombination velocity. 10. Conclusion It has been shown that the analytical expression for alpha can be used to analyze the alpha curve for the purpose of extracting the surface recombination velocity. The shape of the alpha versus the normalized surface recombination velocity curve depends on two parameters. They are the normalized depth of the generation volume and the normalized scanning range. The accuracy in extracting the surface recombination velocity using the method given in [2] is not affected very much as the normalized depth changes. The accuracy is affected only when extracting high values of surface recombination velocity. Similarly, changing the starting location

References [1] Ong VKS, Phang JCH, Chan DSH. Direct and accurate method for the extraction of diffusion length and surface recombination velocity from an EBIC line scan. Solid-State Electron 1994;37(1):1–7. [2] Ong VKS. Direct method of extracting surface recombination velocity from an electron beam induced current line scan. Rev Sci Instrum 1998;69(4):1814. [3] Zhu S-Q, Yang F-H, Rau EI. A novel method of determining semiconductor parameters in EBIC and SEBIV modes of SEM. Semiconduct Sci Technol 2003;18(4):361–6. [4] Luke KL. Unified description of two voltage-varying methods for evaluating surface recombination velocity from electron-beam-induced current: application to normal- and planar-collector configurations. J Appl Phys 1996;79(6):3058. [5] Donolato C. On the analysis of diffusion length measurements by SEM. Solid-State Electron 1982;25(11):1077–10781. [6] Luke KL, von Roos O, Cheng LJ. Quantification of the effects of generation volume, surface recombination velocity, and diffusion length on the electron-beam-induced current and its derivative: determination of diffusion lengths in the low micron and submicron ranges. J Appl Phys 1985;57(6):1978–84. [7] Berz F, Kuiken HK. Theory of life time measurements with the scanning electron microscope: steady state. Solid-State Electron 1976;19(6):437–45. [8] Ong VKS, Phua PC. Potential sources of error in electron beam induced current simulation. Rev Sci Instrum 2001;72(1):201–6. [9] Everhart TE, Hoff PH. Determination of kilovolt electron energy dissipation vs penetration distance in solid materials 1971;42(13):5837–46.