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C-V profiling of delta layers in silicon by quantum and classical approaches
Microelectronic Engineering 15 (1991) 121-124 Elsevier
121
C-V PROFILING OF DELTA LAYERS IN SILICON BY QUANTUM AND CLASSICAL APPROACHES. A.C.G. Wood and A.G. O'Neill Department of Electrical and Electronic Engineering, University of Newcastle---upon-Tyne, Newcastle-upon-Tyne NEI 7RU, U.K. Abstract The resolution of the capacitance--voltage profiling technique is known to be limited by the Debye length for classical structures. When very narrow layers exist in the sample quantum effects become important and the spatial extent of the electron (or hole) wave function can become a limiting factor. In this paper the interpretation of C-V profiles of structures containing delta-doped layers is discussed, concentrating on n-type layers in silicon. C-V profiles are calculated for a range of structures by solving Poisson's equation in one dimension, and results obtained from this classical model are compared with the electron wave function width as calculated from SchriSdinger's equation.
1. INTRODUCTION MBE can be used to fabricate structures in which dopant atoms occupy a layer of atomic proportions. Such delta (&-) doped layers have been reported in both GaAs [ 1,2] and silicon [3,4]. Many applications of such structures have been suggested, and one which has stimulated much interest is the delta doped FET [5], in which the ~--layer acts as the conducting channel of a field effect transistor. Such layers can also be used as punchthrough stoppers in MOSFETs [6]. The width of the delta layer can play an important role in the performance of such a device [7], but the characterisation of very narrow layers tends to be difficult because of the high resolution needed. For ~--layers secondary ion mass spectroscopy (SIMS) and capacitance voltage (C-V) profiling are the most frequently employed techniques. In this paper we discuss the interpretation of C-V profiles and the origin of the inherent resolution limit of the approach. In the C-V method, the size of the depletion region under a Schottky gate is varied by application of a bias voltage. The rate of change of capacitance with voltage depends on the doping density at the depletion edge. Van Gorkum [8] has suggested that the well known classical expressions (equations 1 and 2 below) relating dC/dV to the doping density can be used to interpret C-V data obtained from samples containing &-layers, even though the assumptions used to derive such equations are not valid in such structures. The resolution of the technique is essentially governed by the Debye length in this approach. More recently Schubert [9] has asserted that a quantum calculation is needed to analyse such data because the very sharp potential around a &-layer causes electrons to occupy bound states which have a finite width. In this quantum picture the resolution is limited by the width of the wave function of an electron in the lowest bound state. We have performed quantum and classical calculations of the C-V profile width for a variety of dopant densities and layer widths and, for the first time, include the effects of higher subbands. We consider the effect of temperature on the width of the profile and compare the results with experimental work from the literature. We concentrate on Sb doped silicon layers due to the promise of this material system for use in submicron CMOS devices [7] and because of the lack of experimental data published for Si:B layers. 0167-9317/91/$3.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved.
A . C G. Wood, A.G. O'Neill / C-V profiling of delta layers in silicon
122
2. THEORY. The apparent doping profile N'(x) of a semiconducting sample can be readily obtained from the C - V profile using the following relations : - c ~ [ dc \-, (x)= --~~:)
(I)
~s
x =--
(2)
C
with q the electron charge and r, the semiconductor permittivity. The derivation of these equations assumes that the Debye length is zero, and this places a limit on the accuracy of the technique. The reliability of these equations for a given structure can be tested by calculating the capacitance--voltage curve for the device and comparing the doping profile obtained from these equations with that used in the calculation. Obtaining the theoretical C - V relation for a delta doped layer involves solving Poisson's equation in one dimension (perpendicular to the layer plane) and calculating the total space charge either by integration across the sample, or more simply from the surface electric field using Gauss' law. Our approach is essentially the same as that of van Gorkum [8]. The doping profile is assumed to be 'top--hat' shaped, and Poisson's equation is solved by a finite difference technique. The full-width at half maximum (FWHM) of the apparent doping concentration is then obtained from equations 1 and 2. For very narrow layers, or very high doping concentrations (both of these conditions are frequently satisfied for &-doped layers), quantum confinement of carriers within the potential well formed by the doping profile will become important, particularly for very high doping densities and/or low temperatures. The charge distribution will no longer follow the classical picture, and hence the apparent doping profile obtained by the C - V method will differ from the true doping profile. In order to assess the importance of this effect it is necessary to solve both the Poisson and SchriSdinger equations. Ideally these two equations should be solved self---consistently, but this is not trivial to achieve, and reasonable results can be obtained without true self--consistency. In this work we first solve Poisson's equation to obtain the potential profile of the structure, and then solve SchriSdinger's equation for this potential, using a transfer matrix method to obtain the eigenenergies of the structure. Sample results are shown in figure 1, which illustrates the dependence o f the subband edge energies on the sheet doping concentration of the ~--layer. The notation E0 and Ko refer to the ground states corresponding to the two X valley effective masses of electrons in silicon. The electron Fermi level EF Can then be calculated from the sheet carrier density (obtained from Poisson's equation) by assuming parabolic electron subbands : *D
exp[(E- ED/kT] + 1
n= i
(3)
Ei
where m: is the in--plane effective mass of the subband with band edge energy E,. Having located the Fermi level it is straightforward to obtain the relative occupancies of the subbands. The wave function width for a given subband is estimated from the width of the potential well at the energy at the bottom of that subband.
A.C.G. Wood, A.G. O'Neill
/ C-V profiling of delta layers in silicon
i00
123
E0 50" 0
400
O O
-
5
0
~
m 0
-I00"
30"
20
E,I/
E'
E2
1
E~
I
10"
1012
1013
1014
250 Sheet doping concentration
200
150
i00
(cm -2) Temperature
Figure 1. Energy levels of n-type silicon delta layer.
(K)
Figure 3. Effect of temperature on occupancy of subbands.
A
A
Classical
\\
14
,c: . . . . . . .
15-
.
O E1
m
O
10-
~ QI > ~
5/
Quantum
"
/ / /
5'
i'0
12
-E 1
lO Classical 8
O
~
6-
~
4-
' 15
True layer width (nm)
Figure 2. Comparison of real and apparent delta layer width.
-,-t
1012
1013
Sheet doping concentration
Figure 4. Effect of doping density on apparant width of Inm delta layer.
1014 (cm-2)
124
A.C.G. Wood, A.G. O'Neill / C-V profiling of delta layers in silicon
3. RESULTS AND DISCUSSION. Figure 2 shows the apparent width of the ~--layer as obtained from the C - V FWHM and the wave function width of each of the lowest four subbands as a function of the true ~--layer width for a sheet doping concentration of 2.1012cm-2. At this doping density the potential well is quite shallow and a significant proportion of carriers occupy states in the pair of first excited subbands. This results in the classical FWHM being larger than the ground state wave function width regardless of the delta layer width. If the effect of the carriers in the excited subbands is included however, and the weighted mean wave function width is considered, we find that for layers up to about 10nm wide there is now excellent agreement between the results obtained in the classical and quantum pictures. For layer widths greater than 10nm, still higher levels are occupied and the failure to include these states causes the two curves to diverge. It is interesting to note that for this particular structure there is an absolute resolution limit of about 7nm on the C - V technique. For layer widths below 10nmthe C - V method overestimates the true layer width; for wider (but still very narrow) layers the technique tends to underestimate the layer width slightly. Figure 3 shows the influence of temperature on the occupation of the subbands. As the temperature is reduced, less carriers occupy the higher energy subbands, and at 77K virtually all the electrons occupy the E0 and E'0 levels. This results in a reduction in of the average wave function width at low temperature, and explains the recent results of Schubert [9]. The slightly higher occupancy of the ~ subband over that of the Eo subband at room temperature is due to the higher density of states of the former level resulting from a higher in-plane effective mass. The effect of the sheet doping concentration on the wave function widths and classical FWHM is given in figure 4. This clearly illustrates the two regimes of Debye length and wave function width resolution limiting. For low doping densities, the potential well is very shallow, and the subbands lie very close in energy. In this regime, a large number of subbands are occupied, and the classical Debye length limited picture followed by van Gorkum [8] is appropriate. At very high doping densities however, the width of the ground state wave function imposes a fundamental quantum limit on the resolution of the C - V technique. In fact it is clear from the position of the Fermi level shown in figure 1 that as the doping density is increased beyond 3.1013cm-2 the higher subbands once more become important, in this case due to the saturation of the lowest lying subbands. In conclusion, we note that the apparent conflict between the classical [8] and quantum [9] pictures of C - V profiling can be reconciled by ensuring that the effects of all significantly occupied levels are taken into account. The quantum approach if taken to completion (including all possible states) will always tend towards the classical result, but in practice the Debye length approach is more appropriate for doping concentrations below 3.1012cm-2. For higher levels of doping, quantum effects tend to dominate. [1] A. Zrenner et al, Proc. 17th Int. Conf. Phys. Semiconductors, San Francisco, 1984, p325. [2] E.F. Schubert et al, Jap. J. Appl. Phys. 24, L608 (1985). [3] A.A. van Gorkum etal, Jap. J. Appl. Phys. 26, L1933 (1987). [4] N.L. Mattey etal, Thin Solid Films, 184, 15 (1990). [5] E.F. Schubert et al, IEEE Trans. Electron Devices, ED33, 625 (1986). [6] K. Yamaguchi and Y. Shiraki, IEEE Trans. Electron Devices ED35, 1909 (1988). [7] A.C.G. Wood and A.G. O'Neill, Proc. MRS Spring 1991 meeting, sympoisumB, paper 11.7. [8] A.A. van Gorkum and K. Yamaguchi, IEEE Trans. Electron Devices, ED36, 410 (1989). [9] E.F. Schubert et al, J. Electronic Materials, 19, 521 (1990). This work has been supported by the U.K.S.E.R.C. under I.E.D. program 1517.
121
C-V PROFILING OF DELTA LAYERS IN SILICON BY QUANTUM AND CLASSICAL APPROACHES. A.C.G. Wood and A.G. O'Neill Department of Electrical and Electronic Engineering, University of Newcastle---upon-Tyne, Newcastle-upon-Tyne NEI 7RU, U.K. Abstract The resolution of the capacitance--voltage profiling technique is known to be limited by the Debye length for classical structures. When very narrow layers exist in the sample quantum effects become important and the spatial extent of the electron (or hole) wave function can become a limiting factor. In this paper the interpretation of C-V profiles of structures containing delta-doped layers is discussed, concentrating on n-type layers in silicon. C-V profiles are calculated for a range of structures by solving Poisson's equation in one dimension, and results obtained from this classical model are compared with the electron wave function width as calculated from SchriSdinger's equation.
1. INTRODUCTION MBE can be used to fabricate structures in which dopant atoms occupy a layer of atomic proportions. Such delta (&-) doped layers have been reported in both GaAs [ 1,2] and silicon [3,4]. Many applications of such structures have been suggested, and one which has stimulated much interest is the delta doped FET [5], in which the ~--layer acts as the conducting channel of a field effect transistor. Such layers can also be used as punchthrough stoppers in MOSFETs [6]. The width of the delta layer can play an important role in the performance of such a device [7], but the characterisation of very narrow layers tends to be difficult because of the high resolution needed. For ~--layers secondary ion mass spectroscopy (SIMS) and capacitance voltage (C-V) profiling are the most frequently employed techniques. In this paper we discuss the interpretation of C-V profiles and the origin of the inherent resolution limit of the approach. In the C-V method, the size of the depletion region under a Schottky gate is varied by application of a bias voltage. The rate of change of capacitance with voltage depends on the doping density at the depletion edge. Van Gorkum [8] has suggested that the well known classical expressions (equations 1 and 2 below) relating dC/dV to the doping density can be used to interpret C-V data obtained from samples containing &-layers, even though the assumptions used to derive such equations are not valid in such structures. The resolution of the technique is essentially governed by the Debye length in this approach. More recently Schubert [9] has asserted that a quantum calculation is needed to analyse such data because the very sharp potential around a &-layer causes electrons to occupy bound states which have a finite width. In this quantum picture the resolution is limited by the width of the wave function of an electron in the lowest bound state. We have performed quantum and classical calculations of the C-V profile width for a variety of dopant densities and layer widths and, for the first time, include the effects of higher subbands. We consider the effect of temperature on the width of the profile and compare the results with experimental work from the literature. We concentrate on Sb doped silicon layers due to the promise of this material system for use in submicron CMOS devices [7] and because of the lack of experimental data published for Si:B layers. 0167-9317/91/$3.50 © 1991 - Elsevier Science Publishers B.V. All rights reserved.
A . C G. Wood, A.G. O'Neill / C-V profiling of delta layers in silicon
122
2. THEORY. The apparent doping profile N'(x) of a semiconducting sample can be readily obtained from the C - V profile using the following relations : - c ~ [ dc \-, (x)= --~~:)
(I)
~s
x =--
(2)
C
with q the electron charge and r, the semiconductor permittivity. The derivation of these equations assumes that the Debye length is zero, and this places a limit on the accuracy of the technique. The reliability of these equations for a given structure can be tested by calculating the capacitance--voltage curve for the device and comparing the doping profile obtained from these equations with that used in the calculation. Obtaining the theoretical C - V relation for a delta doped layer involves solving Poisson's equation in one dimension (perpendicular to the layer plane) and calculating the total space charge either by integration across the sample, or more simply from the surface electric field using Gauss' law. Our approach is essentially the same as that of van Gorkum [8]. The doping profile is assumed to be 'top--hat' shaped, and Poisson's equation is solved by a finite difference technique. The full-width at half maximum (FWHM) of the apparent doping concentration is then obtained from equations 1 and 2. For very narrow layers, or very high doping concentrations (both of these conditions are frequently satisfied for &-doped layers), quantum confinement of carriers within the potential well formed by the doping profile will become important, particularly for very high doping densities and/or low temperatures. The charge distribution will no longer follow the classical picture, and hence the apparent doping profile obtained by the C - V method will differ from the true doping profile. In order to assess the importance of this effect it is necessary to solve both the Poisson and SchriSdinger equations. Ideally these two equations should be solved self---consistently, but this is not trivial to achieve, and reasonable results can be obtained without true self--consistency. In this work we first solve Poisson's equation to obtain the potential profile of the structure, and then solve SchriSdinger's equation for this potential, using a transfer matrix method to obtain the eigenenergies of the structure. Sample results are shown in figure 1, which illustrates the dependence o f the subband edge energies on the sheet doping concentration of the ~--layer. The notation E0 and Ko refer to the ground states corresponding to the two X valley effective masses of electrons in silicon. The electron Fermi level EF Can then be calculated from the sheet carrier density (obtained from Poisson's equation) by assuming parabolic electron subbands : *D
exp[(E- ED/kT] + 1
n= i
(3)
Ei
where m: is the in--plane effective mass of the subband with band edge energy E,. Having located the Fermi level it is straightforward to obtain the relative occupancies of the subbands. The wave function width for a given subband is estimated from the width of the potential well at the energy at the bottom of that subband.
A.C.G. Wood, A.G. O'Neill
/ C-V profiling of delta layers in silicon
i00
123
E0 50" 0
400
O O
-
5
0
~
m 0
-I00"
30"
20
E,I/
E'
E2
1
E~
I
10"
1012
1013
1014
250 Sheet doping concentration
200
150
i00
(cm -2) Temperature
Figure 1. Energy levels of n-type silicon delta layer.
(K)
Figure 3. Effect of temperature on occupancy of subbands.
A
A
Classical
\\
14
,c: . . . . . . .
15-
.
O E1
m
O
10-
~ QI > ~
5/
Quantum
"
/ / /
5'
i'0
12
-E 1
lO Classical 8
O
~
6-
~
4-
' 15
True layer width (nm)
Figure 2. Comparison of real and apparent delta layer width.
-,-t
1012
1013
Sheet doping concentration
Figure 4. Effect of doping density on apparant width of Inm delta layer.
1014 (cm-2)
124
A.C.G. Wood, A.G. O'Neill / C-V profiling of delta layers in silicon
3. RESULTS AND DISCUSSION. Figure 2 shows the apparent width of the ~--layer as obtained from the C - V FWHM and the wave function width of each of the lowest four subbands as a function of the true ~--layer width for a sheet doping concentration of 2.1012cm-2. At this doping density the potential well is quite shallow and a significant proportion of carriers occupy states in the pair of first excited subbands. This results in the classical FWHM being larger than the ground state wave function width regardless of the delta layer width. If the effect of the carriers in the excited subbands is included however, and the weighted mean wave function width is considered, we find that for layers up to about 10nm wide there is now excellent agreement between the results obtained in the classical and quantum pictures. For layer widths greater than 10nm, still higher levels are occupied and the failure to include these states causes the two curves to diverge. It is interesting to note that for this particular structure there is an absolute resolution limit of about 7nm on the C - V technique. For layer widths below 10nmthe C - V method overestimates the true layer width; for wider (but still very narrow) layers the technique tends to underestimate the layer width slightly. Figure 3 shows the influence of temperature on the occupation of the subbands. As the temperature is reduced, less carriers occupy the higher energy subbands, and at 77K virtually all the electrons occupy the E0 and E'0 levels. This results in a reduction in of the average wave function width at low temperature, and explains the recent results of Schubert [9]. The slightly higher occupancy of the ~ subband over that of the Eo subband at room temperature is due to the higher density of states of the former level resulting from a higher in-plane effective mass. The effect of the sheet doping concentration on the wave function widths and classical FWHM is given in figure 4. This clearly illustrates the two regimes of Debye length and wave function width resolution limiting. For low doping densities, the potential well is very shallow, and the subbands lie very close in energy. In this regime, a large number of subbands are occupied, and the classical Debye length limited picture followed by van Gorkum [8] is appropriate. At very high doping densities however, the width of the ground state wave function imposes a fundamental quantum limit on the resolution of the C - V technique. In fact it is clear from the position of the Fermi level shown in figure 1 that as the doping density is increased beyond 3.1013cm-2 the higher subbands once more become important, in this case due to the saturation of the lowest lying subbands. In conclusion, we note that the apparent conflict between the classical [8] and quantum [9] pictures of C - V profiling can be reconciled by ensuring that the effects of all significantly occupied levels are taken into account. The quantum approach if taken to completion (including all possible states) will always tend towards the classical result, but in practice the Debye length approach is more appropriate for doping concentrations below 3.1012cm-2. For higher levels of doping, quantum effects tend to dominate. [1] A. Zrenner et al, Proc. 17th Int. Conf. Phys. Semiconductors, San Francisco, 1984, p325. [2] E.F. Schubert et al, Jap. J. Appl. Phys. 24, L608 (1985). [3] A.A. van Gorkum etal, Jap. J. Appl. Phys. 26, L1933 (1987). [4] N.L. Mattey etal, Thin Solid Films, 184, 15 (1990). [5] E.F. Schubert et al, IEEE Trans. Electron Devices, ED33, 625 (1986). [6] K. Yamaguchi and Y. Shiraki, IEEE Trans. Electron Devices ED35, 1909 (1988). [7] A.C.G. Wood and A.G. O'Neill, Proc. MRS Spring 1991 meeting, sympoisumB, paper 11.7. [8] A.A. van Gorkum and K. Yamaguchi, IEEE Trans. Electron Devices, ED36, 410 (1989). [9] E.F. Schubert et al, J. Electronic Materials, 19, 521 (1990). This work has been supported by the U.K.S.E.R.C. under I.E.D. program 1517.