- Email: [email protected]
Band gap narrowing in heavily doped silicon
Solid-State Electronics Vol. 25, No. 9,9~. W-91 Printed in Great Britain.
I, 1982
00384101/82/o9o9o943$0340/0 Pergamon Press Ltd.
BAND GAP NARROWING IN HEAVILY DOPED SILICON S. R. DHARIWAL Departmentof Physics, GovernmentCollege, Ajmer305001,India and
V. N. OJHA Departmentof Physics and Astrophysics,Universityof Delhi, Delhi-l 10007,India (Received 3 November
1981; in reaised form 12 Febnmy
1982)
Abstract-Slotboom and De Graaffs empiricalformula: AE, = E, [In(NINo)+ IOn(M NO))z + Cl”? for the band gap narrowingin silicon for dopant concentration,N, below 2 x 1Ol9cm-’ has been derivedfrom the known theoretical results. Calculations give I?, = 5 kT/14 and No = [O.l8(1Mfq*~“~NY5exp (-4AE& kT)]. Numericallythese results are in good agreement with the experimental values. INTRODUCTION
Though there are many theoretical models [l-5] available
for the band gap narrowing in heavily doped silicon, for doping levels ranging from 2 x 10” cme3 to 2 x 1019cm-‘, which are of great practical interest in bipolar transistors, integrated circuits and solar cells, the experimental results are best described by the empirical formula,
there is no significant deviation in using eqn (2) instead of eqn (1) to represent the experimental results of Slotboom and De Graaff [a]. TEEORY
Now, to derive eqn (2) from theoretical considerations, we note the following points (we shall assume the semiconductor to be n-type, results can be easily extended to AE, = E,[ln (MN,,) + {(In (N/NJ)* t C}"'] (1) p-type also): (1) Experiments by Slotboom and De Graaff are based by Slotboom and De Graaff[6]. Here N is the dopant on the measurement of minority carrier current in the base of a bipolar transistor. Assuming that the dopantdensity and No, E1 and C are constants. For silicon, values of these constants as obtained by these workers atoms are fully ionized, they introduce an effective denare No = 10” cmm3,El = 9 meV and C = 0.5. The aim of sity of intrinsic carriers, ni, so that thii paper is to provide a theoretical proof for eqn (1) and to calculate values of El and No in terms of various (3) &=P&’ physical parameters of the semiconductor. For this, first of all, we note that even though Slotwhere N is the dopant impurity concentration and p. is boom and De Graaff wrote eqn (1) for ali levels of carrier the minority carrier density. In practise, as has been concentrations, band gap narrowing is observable only at shown by Heasell[3], in a heavily doped semiconductor, carrier concentrations above 2 x 10” cm-“. A look at eqn impurity atoms are not fully ionized. If we denote by & (1) (or, Fig. 7 of Ref. [6]) clearly shows that the above the density of unionized impurity atoms (donor here), and by no the number of free electrons in the conduction empirical relation combines two basic facts: (1) The band gap narrowing starts showing its effect band we have for N > 2 x 10” cme3. Below this value of dopant conN=nottld centration, the change in band gap is very small. Note (4) that for N = 2X lO”cmand No= lO”~rn-~, (In N/No)* = 0.48. Thus C= 0.5 has been used only to If we can neglect the perturbation in the intrinsic band include this fact in their formulation. edges of the semiconductor (we find this assumption to (2) For N ranging between 2~ lO”cmand 2 x be self consistent for N < 2 x lOI cme3, as our result 1019cme3, the band gap narrowing follows a logarithmic show excellent agreement with the experiments. Also relation: optical absorption experiments[7] do not show a considerable shift in the intrinsic band edges upto the dopant AE, = 2.8, In (MN,,). (2) concentration of 10’9cm-3 and thus support this view point), we have The difference in the values of AE, as obtained by eqn (5) n :o = nope (1) and (2) is always less than 5 meV, whereas the expected error in the experiments may be -+10 meV[4]. Thus, where njo is the intrinsic carrier concentration for the 909
S. R. DHARIWALand V. N. OIHA
910
unperturbed bands. n, is related with 40 by[6], cc5 = nToexp (A&/H’).
where
is the average ionization energy of the impurity atoms. Also from eqns (4), (7), (9) and (11) we get
Combining eqns (3) to (6) we get AE,=kTIn
(
y
(16)
A&=&-E,j
(6)
(7)
>
(2) Impurity band broadening starts at doping levels above 10” cm-‘. For dopant concentration of interest, following Morgan[8], we can write the density of electrons in the impurity band as
(T= (4~*q6kT/e3)“‘N”’ exp (AEd4kT).
Putting this value of u in (15) and rearranging the various factors we get ~+ht(~exp(~)]=(~)“*(~)“’ N”‘exp (- 3AEJ4kT).
(8)
nd = N erfc
Here Ed is the energy of the donor impUrity energy level, E, is the Fermi energy, and (T is the standard deviation used for the normal distribution of impurity energy levels. Note that in writing eqn (8) Fermi-Dirac statistics has been used. u can be expressed in terms of the screening length due to the free carriers and the dopant density as 181 (T= (2,rq4/e2)“2~“2N1’2 (9) (3) From the calculations by Van Overstraeten et al. [l], it is clear that for N between 2 x 10” and 2 x 1OL9 cm-‘, Fermi level is much below the conduction band edge. Therefore it is a good approximation to use Maxwell-Boltmann statistics for the free carriers and to write, no = NCexp [-(EC - Ef)/kTl. (10) (4) For non-degenerate concentration of the free carriers, we have A = (q2no/ekT)-“*
(11)
(17)
(18)
Taking logarithm on both the sides and noting that In x = x - 1 we get AE, = (5/7)kT In (N/No)
(19)
3/s
(20) Here the factor 0.18 = (ne)-4’5, e being the Naperian base. Equation (19) is identical to eqn (2). Value of E, is given by E, = 5kTll4. (21) DISCUSSION
At room temperature, E, = 9.2 meV. For silicon by considering impurity level with EC- Ed = 44 meV, we get No = 2.275 x 10” cm-‘. These values show quite a good agreement with the experimental values of Slotboom and De Graaff (E, = 9 meV, No = 10” cme3).
where c is dielectric permittivity of the semiconductor. Now we shall put these facts together to get eqn (2). For this we note that eI’fC ((&
-
E&/d/(2)(r) = 1- (2/a)“*[(& - Ef)/a]
70-
(12)
E, = E, + kT In (no/NC)
60z -E
for (Ed - Et) < 0: Also we have from eqn (lo),
(13)
w” d
40JO-
whereas in accordance with eqn (4) and (7)
20
no = N exp (- AEJkT).
(14)
Therefore, by putting eqns (8), (12), (13) and (14) together we get exP(T)=
_
5o
(~)1l*~[ln{$$exp(-AEdkT))
+ER
kT I
(15)
1
IO
0110’7 ’
I
10’6 N (cr?]
IO’9 -
Pi. 1. Empiricalresult of Slothoomand De Gcaatfis compared with the theoreticalresult obtainedhere. Essentialfeatucesof the empiricalformulaaceretainedin the theoreticalformulation.The latter shows a better agreement with the recent experimental results as obtained from Table 1 of Ref. 4, shown here by dots.
Band gap narrowingin heavily doped silicon Tbe small differences between the empirical values of El and No and those given by the present work can be
understood by noticing that the inaccuracy in the experimental determination of AEB may be more than + 10 meV [4]. Also, the values given by Slotboom are somewhat higher than reported by others[9]. In Fig. 1, values of AE, calculated from eqn (19) are compared with those given by eqn (1) (Slotboom & De GraalQ. The points shown in the figure are the experimental values as compiled in Table 1 of Ref. [4]. The result of the present theory shows a better agreement with these experimental results for N between 2 x 10” to 2 x 1O’9cmm3. To conclude, the present work gives a theoretical basis to the empirical results of Slotboom and De Graaff. At the same time the values of E, and AJ0obtained here are more accurate and are related directly with the temperature and the physical properties of the semiconductor and the dopant impurity. Acknowkedgernents-Thehelp and encouragementreceivedfrom Prof. L. S. Kothari,Prof. G. P. Srivastava,Dr. P. K. Bhatnagar,
911
Mr. K. K. Sharma and Mr. N. K. Vasu are acknowledgedwith thanks. One author (VNO)acknowledgesthe CSIR,India for the awardof a fellowship.
REFJWNCES
1. R. J. Van Overstraeten,H. De Man and R. Mertens, IEEE Tram on ElectronDeu.ED_2Q, 290(1973). 2. M. S. Mock,Solid-St Electron. 16, 1251(1973). 3. E. L. Heasell, IEEE Tram on Electron Dev. ED-%, 919 (1979). 4. H. P. D. Lanyon and R. A. Tuft, IEEE Trans on Electron Lkv. ED-26,1014(1979). 5. G. D. Mahan,I. Appl. Phys. 51,2634(1980). 6. J. W. Slothoomand H. C. De Graaff, Solid-St. E/e&on. 19, 857 (1976). 7. A. A. Vol’fsonand V. K. Subashiev,Sou. Phys.-Semicanduclors. 1, 327 (1%7). 8. T. N. Morgan,Phys. Rev. 139,343(1%5). 9. J. Van Meerhernen.J. Niis. R. Mertens and R. Van Overstraeten, Proc. 13th’IEEE-ihotovolfaic Specialists Canf. 66 (1978).
I, 1982
00384101/82/o9o9o943$0340/0 Pergamon Press Ltd.
BAND GAP NARROWING IN HEAVILY DOPED SILICON S. R. DHARIWAL Departmentof Physics, GovernmentCollege, Ajmer305001,India and
V. N. OJHA Departmentof Physics and Astrophysics,Universityof Delhi, Delhi-l 10007,India (Received 3 November
1981; in reaised form 12 Febnmy
1982)
Abstract-Slotboom and De Graaffs empiricalformula: AE, = E, [In(NINo)+ IOn(M NO))z + Cl”? for the band gap narrowingin silicon for dopant concentration,N, below 2 x 1Ol9cm-’ has been derivedfrom the known theoretical results. Calculations give I?, = 5 kT/14 and No = [O.l8(1Mfq*~“~NY5exp (-4AE& kT)]. Numericallythese results are in good agreement with the experimental values. INTRODUCTION
Though there are many theoretical models [l-5] available
for the band gap narrowing in heavily doped silicon, for doping levels ranging from 2 x 10” cme3 to 2 x 1019cm-‘, which are of great practical interest in bipolar transistors, integrated circuits and solar cells, the experimental results are best described by the empirical formula,
there is no significant deviation in using eqn (2) instead of eqn (1) to represent the experimental results of Slotboom and De Graaff [a]. TEEORY
Now, to derive eqn (2) from theoretical considerations, we note the following points (we shall assume the semiconductor to be n-type, results can be easily extended to AE, = E,[ln (MN,,) + {(In (N/NJ)* t C}"'] (1) p-type also): (1) Experiments by Slotboom and De Graaff are based by Slotboom and De Graaff[6]. Here N is the dopant on the measurement of minority carrier current in the base of a bipolar transistor. Assuming that the dopantdensity and No, E1 and C are constants. For silicon, values of these constants as obtained by these workers atoms are fully ionized, they introduce an effective denare No = 10” cmm3,El = 9 meV and C = 0.5. The aim of sity of intrinsic carriers, ni, so that thii paper is to provide a theoretical proof for eqn (1) and to calculate values of El and No in terms of various (3) &=P&’ physical parameters of the semiconductor. For this, first of all, we note that even though Slotwhere N is the dopant impurity concentration and p. is boom and De Graaff wrote eqn (1) for ali levels of carrier the minority carrier density. In practise, as has been concentrations, band gap narrowing is observable only at shown by Heasell[3], in a heavily doped semiconductor, carrier concentrations above 2 x 10” cm-“. A look at eqn impurity atoms are not fully ionized. If we denote by & (1) (or, Fig. 7 of Ref. [6]) clearly shows that the above the density of unionized impurity atoms (donor here), and by no the number of free electrons in the conduction empirical relation combines two basic facts: (1) The band gap narrowing starts showing its effect band we have for N > 2 x 10” cme3. Below this value of dopant conN=nottld centration, the change in band gap is very small. Note (4) that for N = 2X lO”cmand No= lO”~rn-~, (In N/No)* = 0.48. Thus C= 0.5 has been used only to If we can neglect the perturbation in the intrinsic band include this fact in their formulation. edges of the semiconductor (we find this assumption to (2) For N ranging between 2~ lO”cmand 2 x be self consistent for N < 2 x lOI cme3, as our result 1019cme3, the band gap narrowing follows a logarithmic show excellent agreement with the experiments. Also relation: optical absorption experiments[7] do not show a considerable shift in the intrinsic band edges upto the dopant AE, = 2.8, In (MN,,). (2) concentration of 10’9cm-3 and thus support this view point), we have The difference in the values of AE, as obtained by eqn (5) n :o = nope (1) and (2) is always less than 5 meV, whereas the expected error in the experiments may be -+10 meV[4]. Thus, where njo is the intrinsic carrier concentration for the 909
S. R. DHARIWALand V. N. OIHA
910
unperturbed bands. n, is related with 40 by[6], cc5 = nToexp (A&/H’).
where
is the average ionization energy of the impurity atoms. Also from eqns (4), (7), (9) and (11) we get
Combining eqns (3) to (6) we get AE,=kTIn
(
y
(16)
A&=&-E,j
(6)
(7)
>
(2) Impurity band broadening starts at doping levels above 10” cm-‘. For dopant concentration of interest, following Morgan[8], we can write the density of electrons in the impurity band as
(T= (4~*q6kT/e3)“‘N”’ exp (AEd4kT).
Putting this value of u in (15) and rearranging the various factors we get ~+ht(~exp(~)]=(~)“*(~)“’ N”‘exp (- 3AEJ4kT).
(8)
nd = N erfc
Here Ed is the energy of the donor impUrity energy level, E, is the Fermi energy, and (T is the standard deviation used for the normal distribution of impurity energy levels. Note that in writing eqn (8) Fermi-Dirac statistics has been used. u can be expressed in terms of the screening length due to the free carriers and the dopant density as 181 (T= (2,rq4/e2)“2~“2N1’2 (9) (3) From the calculations by Van Overstraeten et al. [l], it is clear that for N between 2 x 10” and 2 x 1OL9 cm-‘, Fermi level is much below the conduction band edge. Therefore it is a good approximation to use Maxwell-Boltmann statistics for the free carriers and to write, no = NCexp [-(EC - Ef)/kTl. (10) (4) For non-degenerate concentration of the free carriers, we have A = (q2no/ekT)-“*
(11)
(17)
(18)
Taking logarithm on both the sides and noting that In x = x - 1 we get AE, = (5/7)kT In (N/No)
(19)
3/s
(20) Here the factor 0.18 = (ne)-4’5, e being the Naperian base. Equation (19) is identical to eqn (2). Value of E, is given by E, = 5kTll4. (21) DISCUSSION
At room temperature, E, = 9.2 meV. For silicon by considering impurity level with EC- Ed = 44 meV, we get No = 2.275 x 10” cm-‘. These values show quite a good agreement with the experimental values of Slotboom and De Graaff (E, = 9 meV, No = 10” cme3).
where c is dielectric permittivity of the semiconductor. Now we shall put these facts together to get eqn (2). For this we note that eI’fC ((&
-
E&/d/(2)(r) = 1- (2/a)“*[(& - Ef)/a]
70-
(12)
E, = E, + kT In (no/NC)
60z -E
for (Ed - Et) < 0: Also we have from eqn (lo),
(13)
w” d
40JO-
whereas in accordance with eqn (4) and (7)
20
no = N exp (- AEJkT).
(14)
Therefore, by putting eqns (8), (12), (13) and (14) together we get exP(T)=
_
5o
(~)1l*~[ln{$$exp(-AEdkT))
+ER
kT I
(15)
1
IO
0110’7 ’
I
10’6 N (cr?]
IO’9 -
Pi. 1. Empiricalresult of Slothoomand De Gcaatfis compared with the theoreticalresult obtainedhere. Essentialfeatucesof the empiricalformulaaceretainedin the theoreticalformulation.The latter shows a better agreement with the recent experimental results as obtained from Table 1 of Ref. 4, shown here by dots.
Band gap narrowingin heavily doped silicon Tbe small differences between the empirical values of El and No and those given by the present work can be
understood by noticing that the inaccuracy in the experimental determination of AEB may be more than + 10 meV [4]. Also, the values given by Slotboom are somewhat higher than reported by others[9]. In Fig. 1, values of AE, calculated from eqn (19) are compared with those given by eqn (1) (Slotboom & De GraalQ. The points shown in the figure are the experimental values as compiled in Table 1 of Ref. [4]. The result of the present theory shows a better agreement with these experimental results for N between 2 x 10” to 2 x 1O’9cmm3. To conclude, the present work gives a theoretical basis to the empirical results of Slotboom and De Graaff. At the same time the values of E, and AJ0obtained here are more accurate and are related directly with the temperature and the physical properties of the semiconductor and the dopant impurity. Acknowkedgernents-Thehelp and encouragementreceivedfrom Prof. L. S. Kothari,Prof. G. P. Srivastava,Dr. P. K. Bhatnagar,
911
Mr. K. K. Sharma and Mr. N. K. Vasu are acknowledgedwith thanks. One author (VNO)acknowledgesthe CSIR,India for the awardof a fellowship.
REFJWNCES
1. R. J. Van Overstraeten,H. De Man and R. Mertens, IEEE Tram on ElectronDeu.ED_2Q, 290(1973). 2. M. S. Mock,Solid-St Electron. 16, 1251(1973). 3. E. L. Heasell, IEEE Tram on Electron Dev. ED-%, 919 (1979). 4. H. P. D. Lanyon and R. A. Tuft, IEEE Trans on Electron Lkv. ED-26,1014(1979). 5. G. D. Mahan,I. Appl. Phys. 51,2634(1980). 6. J. W. Slothoomand H. C. De Graaff, Solid-St. E/e&on. 19, 857 (1976). 7. A. A. Vol’fsonand V. K. Subashiev,Sou. Phys.-Semicanduclors. 1, 327 (1%7). 8. T. N. Morgan,Phys. Rev. 139,343(1%5). 9. J. Van Meerhernen.J. Niis. R. Mertens and R. Van Overstraeten, Proc. 13th’IEEE-ihotovolfaic Specialists Canf. 66 (1978).