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The influence of bandgap narrowing on the I–V characteristics of a MOSFET
So/M-Stale Eleclronics Vol. 36, No. 8, pp. 1129-I 134, 1993
0038-I lOI/ $6.00 + 0.00 Copyright 0 1993 Pergamon Press Ltd
Printed in Great Britain. All rights reserved
THE INFLUENCE OF BANDGAP Z-V CHARACTERISTICS
NARROWING ON THE OF A MOSFET
A. JAKUBOWSKI and L. LUKASIAK Institute
of Microelectronics and Optoelectronics, Warsaw University ul.Koszykowa 75, 00-062, Warsaw, Poland (Received
20 August
1992; in revised form
14 January
of Technology,
1993)
Abstract-Modifications of Pierret-Shields model which take into account either apparent or physical bandgap narrowing are presented. The influence of bandgap narrowing on the I-V characteristics of a MOSFET is theoretically examined through comparison of the modified models with the original PierretShields model for various device parameters. It is proved that bandgap narrowing affects significantly performance of a MOSFET in the subthreshold and the near-threshold regions.
drain-to-source voltage normalized to kT/q gate-to-source voltage normalized to kT/q surface potential normalized to kT/q surface potential at the source end of the channel normalized to kT/q source-to-bulk voltage normalized to kT/q surface potential at the drain end of the channel normalized to kT/q channel width brn] physical bandgap narrowing [eV] apparent bandgap narrowing [eV] semiconductor permittivity [F/cm] distance between the edge of the conduction band and the Fermi level normalized to kT distance between the Fermi level and the edge of the valence band normalized to kT effective mobility in the channel [cm’/Vs] quasi-Fermi level splitting normalized to kT/q quasi-Fermi level splitting at the source end of the channel normalized to kT/q quasi-Fermi level splitting at the drain end of the channel normalized to kT/q
NOTATION a c, -% E,, Ec E Cr EF
EGO E, E II
the distance between the impurity level and the edge of the valence band normalized to kT/q gate oxide capacitance per unit area [F/cm21 impurity level [eV] impurity level in the bulk of semiconductor [eV] the edge of the conduction band [eV] the edge of the conduction band in the bulk of semiconductor [eV] the Fermi level [eV] the width of the forbidden gap [eV] intrinsic Fermi level [eV] intrinsic Fermi level in the bulk of semiconductor
[evl the edge of the valence band [eV] the edge of the valence band in the bulk of semiconduction [eV] surface electric field [V/cm] E, F(u, uFe. 5) Seiwatz-Green function based on aooarent 1. bandgap narrowing Seiwatz-Green function based on physical F,(u, 5) bandgap narrowing Fermi-birac integral of the order l/2 Fl!2 F 3.2 Fermi-Dirac integral of the order 3/2 ratio of the effective capacitance of intrinsic G, semiconductor in flat-band condition to the gate oxide capacitance drain current [A] ID k the Boltzmann constant [J/K] L channel length km] effective Debye length in intrinsic semiconductor L,,
E”
E “Z
[cm1 n n, “,e N.4 N, NC NV P 4 ? U UFB UFE
electron concentration [cm-‘] intrinsic concentration [cm-j] effective intrinsic concentration [cm-‘] concentration of acceptors [cm-‘] concentration of ionized acceptors [cm-‘] effective density of states in the conduction band [cm-‘] effective density of states in the valence band [cm-‘] hole concentration [cm- ‘1 electron charge [C] total semiconductor charge per unit area [C/I&] temperature [K] potential normalized to kT/q flat band voltage normalized to kT/q effective Fermi potential normalized to kT/q
1. INTRODUCTION
The effect of bandgap narrowing in heavily doped regions of bipolar devices has been extensively studied (e.g. [l-5]) both theoretically and experimentally. Since continual miniaturization of MOSFETs is, according to scaling rules, accompanied by a corresponding increase of substrate doping the effects of degeneracy and bandgap narrowing may become of importance, especially in view of the fact that nowadays MOS devices with substrate doping on the order of IO’” cme3 and higher are reported[6,7]. This paper is aimed at theoretical investigation of the influence of bandgap narrowing on the I-V characteristics of a MOS transistor. Since in [8] it had been proved that long-channel behaviour may be retained even in the case of transistors with very short channels provided that oxide thickness is sufficiently reduced, the authors have chosen the long-channel Pierret_Shields[9] model for calculations. A modification of the Pierret-Shields model so that it takes into 1129
A.
1130
JAKUBOWSKI and
L.
LUKASIAK
u(x)
Ea.
-
________~___________-___________-..__ _____________.____.....___.._.____ E is
,_.-’
,_--’
EF
L._.-._._.,.-.-._._
______________-._--_---.--
pa ?a
E a*
I
E vo
m X
0
Fig. I. Band diagram of semiconductor in a MOS structure. &-the edge of the conduction band; E,the intrinsic Fermi level; E,-the Fermi level; E,-the impurity level; /?-the edge of the valence band.
account bandgap narrowing is described in Section 2. Section 3 presents the results of simulation and a comparison with the original Pierret-Shields model. Summary and conclusions are given in Section 4. Throughout this paper a n-channel MOSFET with boron-doped substrate is considered.
where k is the Boltzmann constant, and T is the temperature in Kelvin. If the total concentration of impurities NA is known, the concentration of ionized impurities NA may be calculated according to e.g. [13]: N,=N
2. THEORETICAL
CONSIDERATIONS
In Fig. I a band diagram of semiconductor making part of a MOS structure has been shown along with all the necessary notation used in further considerations. If gate voltage is applied the energy levels (except the Fermi level) become functions of x due to the band banding. Subscript w denotes values of the respective energy levels deep in the bulk of the semiconductor which is not disturbed electrically. There are two possible ways of including the effect of bandgap narrowing in MOSFET modelling. 2. I. Appurent bandgap narrowing The apparent bandgap narrowing AEoa takes into account both degeneracy and the physical bandgap narrowing at the same time. Analytical formulae relating AE,, to substrate doping were presented in e.g. [IO-121. In order to modify the Pierret-Shields model[9] the concentration of ionized impurities and concentrations of free carriers have to be formulated. The apparent bandgap narrowing relates the effective intrinsic concentration n, to the ordinary intrinsic concentration n, in the following way e.g. [lo]:
4, =
ni
ev
I (2)
*1+4exp(-q+u+a)’
where q = (EP - E,, )/kT is the normalized position of the Fermi level E, with respect to the edge of the valence band E,, in the bulk of the semiconductor, u is the band bending normalized to kT/q, and a = [E,(x) - E,(x)]/kT = (Ea,, - E,,)/kT is the normalized distance between the impurity level and the edge of the valence band, independent of x (see Fig. I). In semiconductors with low impurity concentration this distance is also independent of impurity concentration. In the case of silicon doped with boron the value of Es(x) - E,(x) is equal to 0.045 eV. On the other hand in highly doped semiconductors the distance between the impurity level and the edge of the majority carrier band becomes a function of impurity concentration [14]. In such a situation the distance may be expressed as: a = [E,(x) - E,(x)]/kT
= [0.045 - AE,,]/kT.
The concentration of majority expressed as follows: P = NvF,,z(-v
carriers
- ~1,
(3)
p may be (4)
where NV is the effective density of states in the valence band, and F,,? is the Fermi-Dirac integral of the order l/2. The normalized position of the Fermi level E, with respect to the edge of the valence band E,, in the
1131
Influence of bandgap narrowing on I-V of MOSFET may
per unit area and G, is the ratio of the intrinsic semiconductor capacitance to the gate oxide capacitance:
(5)
(loa)
which states that in the absence of electrostatic potential the concentration of free holes p is equal to the concentration of ionized acceptors NT. The concentration of minority carriers n may be easily obtained in an approximate way:
Further transformations used by Pierret and Shields lead to the final drain current formula:
bulk of the semiconductor q = (Er - E,,)/kT be evaluated from the following equation: 1 NvF,,,(--rl)
n:, n=N,(u=O)
= NA
1+4exp(-rl
+a)’
ID =
$c,
&B
uFB)("SL
-
&O)
Ml
eXp(u - <)=&cxP(u
-
UFe -
t),
(6)
where ure = In(N, (U = O)h,) is the normalized effective Fermi potential (which no longer corresponds to the distance between the Fermi level EF and the intrinsic Fermi level E,), and < is the normalized quasi Fermi-level splitting of minority carriers due to the current flow. If the Poisson equation is solved, the resulting Seiwatz-Green function[lS] F(u, uFe, 5) is as follows:
-5
‘(4 -u:o)+G
-
s
“LF(u,
” z;’
++,(-4
+ew(u-uF,-5)
(7)
where F3,? is the Fermi-Dirac integral of the order of 312. This function is different from the ordinary Kingston[l6] function used by Pierret and Shields in that it is based on the Fermi-Dirac statistics and that the bandgap narrowing is taken into account. The surface electric field may be expressed as: E.-~+,u,,.O. !e
(8)
where L,, is the intrinsic Debye length: L,,=
kT e -L, J--- Y 2nie
,e
(9)
the voltage balance equation becomes: u,s+usB--uu,,=
4 ----_q
Q,
kTC,
= W’(u,, UFer5) + 4,
uFe, 50) du
(11)
uFe
where L is the channel length, W is the channel width, peiris the effective mobility, to and tL are the normalized values of the quasi-Fermi level splitting at the source- and drain end of the channel, respectively: UW
tO="SBt %B
+
Ulb)
uDS,
2.2. Physical bandgap narrowing In the following approach the effects of physical bandgap narrowing and degeneracy are taken into account independently. As in the previous approach the key to modelling is the formulation of charge concentrations. The concentrations of ionized impurities and majority carriers are expressed by (2) and (4) respectively. The distance between the Fermi level EF and the valence band edge E,, in the bulk of semiconductor is calculated from (5) the only difference being a slightly different form of (3): a = [E,(x) - E,(x)]/kT
surface potential, cs is the semiconductor permittivity, and q is the electron charge. If the total charge in the semiconductor is described as: -~+F(rr,,u,,~),
F&t
= [0.045 - AE,]/kT,
(12)
@a)
u, is the normalized
-caEs=
0
where uDs is the normalized drain-to-source voltage, while us0= u,(t,) and u,,_= u,([,J are the normalized surface potentials at the source- and drain end of the channel, respectively and both are calculated from (10).
-u)-F3:2(-v)l
-exp(-uF,-t),
[S
0
F*(u, uFe, 5) = % In 4 : :I$;
Q,=
-
(10)
where uos, uSB,and urs are the normalized gate-tosource voltage, source-to-bulk voltage and flat band voltage, respectively, Ci is the gate-oxide capacitance
where AEo is the physical bandgap narrowing. The concentration of minority carriers n is evaluated with physical bandgap narrowing taken into account which permits a more rigorous approach to be applied when compared with (6): n=
NcF,:,(-i
-5
+u),
(13)
where Nc is the effective density of states in the conduction band, and [ = (E,, - E,)/kT is the normalized distance between the edge of the conduction band E,, in the bulk of semiconductor and the Fermi level EF, and may be expressed as: c =qEoo(T)lkT-qAEo(N,)lkT-~.
(14)
The symbol E,,(T) denotes the energy gap without the influence of high doping concentrations.
A. JAKUBOWSKI
1132
and L. LUKASAK
Once the charge concentrations are known, the solution of the Poisson equation results in a SeiwatzGreen function F,(u, 5) which takes the physical bandgap narrowing into account: F:(u,<)=~
In
4 + exp(rj - a + u)
-ld(Kuzmicz) - -Id(Slotboom)
4 + exp(fj - a)
Hi,
++-rl
-u)-F3,.2-?)I
Id(Klaassen)
e y
lo-5
2
loa
* Id
le
+~[F,.,(-r -5 +u) -Fw(--i
- 01.
P
(15)
10-7
1o-8 9 -0.4
-0.2
0.0
As it had been shown in [17] F, (u, 5) may be directly introduced into the Pierret-Shields formula (11). (A slightly different form of F, (u, 5) in [17] results from a different cause of degeneracy considered.)
3. SIMULATION
0.2
1.0
0.4
VGS -VT[V]
RESULTS
In order to estimate the influence of bandgap narrowing on the I-V characteristics of a MOSFET the original Pierret_Shields[9] model was compared to its both modified versions presented in 2.1 and 2.2 for various approximations of physical and apparent bandgap narrowing. The values of Nc, Nv, and the temperature dependence of EGOwere taken from [ 181. In the original Pierret-Shields model[9] incomplete ionization of impurities was not taken into account. The flat-band voltage and the bulk-to-source voltage were assumed to be zero. In the modified version described in 2.1 the formulae for the apparent bandgap narrowing presented in [lo] 1, (Slotboom) and [I l] In (Klaassen) were used. In the modified version described in 2.2 two formulae for physical bandgap narrowing were used: the one presented in [19] In (Kuzmicz) for all substrate dopings considered, and the one described in [12] In (Jain) for N, > IO’* cmm3. Analytical approximations of F,:, and F3,‘2were taken from [20]. In Fig. 2(a) normalized transfer characteristics according to both the original model In and its modified versions In (Slotboom), In (Klaassen) and In (Kuzmicz) are shown as a function of the difference between the gate-to-source voltage V,, and the threshold voltage VT for substrate doping equal to 5 . 10” cm- ‘. In all calculations the threshold voltage was evaluated using the physical bandgap narrowing approximation described in [19], with the surface potential chosen so that the concentration of minority carriers at the surface was equal to the concentration of ionized purities. In Fig. 2(b) normalized transfer characteristics are shown for substrate doping equal to 2. 10’scm-3. In this figure points corresponding to In (Jain) were added. Figure 3(a) and 3(b) present the relative error of the original model Zn with respect to its modified versions as a function of VGs - VT. Substrate dopings are the
1)l
J )
,* Id -0.4
-0.2
0
0.2
0.4
0.6
0.8
1
VGS-VT(V]
Fig. 2. (a) Normalized drain current as a function of the difference between gate voltage and threshold voltage according to the original model and modified ones (NA = 5. IO” cm-j). (b) Normalized drain current as a function of the difference between gate voltage and threshold voltage according to the original model and modified ones (NA = 2. IO’*cm-‘).
same as in Fig. 2(a) and 2(b). The error decreases with gate voltage due to a quick increase of drain current. Thus the effect of bandgap narrowing is important only in the subthreshold and the near threshold regions of MOS operation. It can be seen that for higher substrate doping the discrepancies between the modified models and the original one are much greater. Figure 4 shows the percentage error of the original model with respect to the modified ones in saturation as a function of gate-oxide. The gate voltage is equal to the threshold voltage. The error slightly increases for thinner oxides. Figure 5 presents the same error as in the previous figure as a function of substrate doping. Obviously the error increases when the impurity concentration grows. Finally, in Fig. 6 the error is displayed as a function of a scaling factor, which is constructed so that the product of oxide thickness and substrate doping of the considered devices is constant (according to constant field scaling rules reduction of oxide thickness by a given factor k must be accompanied by the
Influence of bandgap narrowing on I-V of MOSFET
(a)80
1133
loo1 _-80 -
tox = 10nm VGS = VT
- Id(Kuzmicz) - - Id(Slotboom) tox = 10 nm
Id(Klaassen) - Id(Kuzmicz) ’ Id(Jain) - Id(Slotboom
0-t -0.4
1 -0.2
0.0
0.2
0.4
0.6
0.6
d
Iq(Klaassen)
0
1.0
0.5
1.0
100
_________
601L
Fig. 5. Relative error of the original model with respect to the modified ones as a function of substrate doping (t,, = 10 nm).
Id(Kuzmicz)
. Id(Jain)
Id(Klaassen) 6Om
2 w z 4 :
4om
20
O-0.4
20
- - Id(Slotboom)
.
z 5
10
5.0
SUBSTRATE DOPING [lO’$m’3]
VGS . VT [V]
(b)
2.0
tax = 10nm
VDS = 1V
-0.2
I
0.0
0.2
0.4
0.6
0.8
1.0
than the “nominal” 10 nm. In the case of transistors with higher substrate doping and thinner oxides the error quickly increases. It is easily noticed that the apparent-bandgap-narrowing-based model using the approximation presented in the paper of Klaassen et al.[l l] yields values very close to those obtained using the model based on physical bandgap narrowing. On the other hand if the approximation of apparent bandgap narrowing presented in [lo] is used the two modified models differ from each other considerably.
VGS-VT [V]
Fig. 3. (a) Relative error of the original model with respect to the modified ones as a function of the difference between gate voltage and threshold voltage (NA = 5 . IO” cmm3). (b) Relative error of the original model with respect to the modified ones as a function of the difference between gate voltage and threshold voltage (NA = 2. IO’* cm-9. increase of substrate doping by the same factor k). Values of the scaling factor k > 1 correspond to devices in which substrate doping is higher than the “nominal” 5 x 10” cmm3. and oxide thickness is less
4. SUMMARY
The Pierret-Shields[9] model was modified so that the influence of bandgap narrowing on the I-V acteristics of MOS transistor be investigated The presented may use apparent or bandgap narrowing. The results of simulation indicate that in the case of MOSFETs with high doping concentration the neglect of bandgap narrowing effect leads to considerable errors especially in the subthreshold and the
Na =
lO”*k [cni’]
tox =
VGS
[nm]
V-f
.
20 50’ 2.5
- Id(Slotboom
I 5
10
25
50
100 0.1
OXIDE THICKNESS [nm]
Fig. 4. Relative error of the original model with respect to the modified ones as a function of gate oxide thickness (NA = 5 IO” cm-‘).
0.2
0.4 SCALING
Fig.
Relative error the modified
10
1.0 k
the original with respect as a of scaling
A. JAKUBOWSKIand L. LUKASIAK
1134 near-threshold mations
regions.
semiconductors. greater
Bandgap
used for analysis due
observed is greater
Therefore to
the
fact
in the space-charge
narrowing
approxi-
were intended
for neutral
these errors
may be still
that
bandgap
region
than that in the neutral
narrowing
of MOS
regions
devices
of bipolar
devices[6]. REFERENCES 1. D. D. Tang, IEEE Trans. Electron. Devices ED-27, 563 ( 1980). 2. b. E.‘Possin, M. S. Adler and B. Jayant Baliga, IEEE Trans. Electron. Devices ED-27. 983 (1980). 3. A. Neugroschel, S. C. Pao and’F. A.‘Lindholm, IEEE Trans. %ectran. Devices ED-29, 894 (1982). 4. M. S. Mock, Solid-St. Electron. 16. 1251 (1973). 5. P. P. Mertens, J. L. van Meerberger, J. F. Nijs and R. J. van Gverstraeten, IEEE Trans. Electron. Devices ED-27, 949 (1980). 6. H. C. Chen, S. S. Li and K. W. Teng, Solid-St. Electron. 32, 339 (1989). 7. J. R. Brews, in High-Speed Semiconductor Devices (Edited by S. M. Sze), pp. 139-209. Wiley, New York (1990).
8. J. R. Brews, W. Fichtner, E. Nicollian and S. M. Sze, IEDM Tech. Din., p. 10, (1979); IEEE Trans. Electron. Devices Left. EfiG1, 2 (i980): 9. R. F. Pierret and J. A. Shields. Solid-St. Elecfron. 26. 143 (1983). 10. J. W. Slotboom and H. C. de Graaff, Solid-St. Electron. 19, 857 (1976). J. W. Slotboom and H. C. de 11. D. B. M. Klaassen, Graaff, Solid-St. Elecfron. 35, 125 (1992). Solid-St. Electron. 34. 12. S. C. Jain and J. D. Rot&ton. 453 (1991). 13. M. Shur, Physics of Semiconductor Devices. PrenticeHall. Enalewood Cliffs. N.J. (1990). 14. T. F. Lee and T. C. McGill, J. appl. Phys. 46, 373 (1975). 15. R. Seiwatz and M. Green, J. appl. Phys. 29, 1034 (1958). 16. R. H. Kingston and S. F. Neustadter, J. appl. Phys. 26, 718 (1955). A. Jakubowski and L. tukasiak, IEEE 17. B. Majkusiak, Trans. Electron. Devices ED-34, 2560 (1987). IS. M. A. Green, J. appl. Phys. 67, 2944 (1990). in Physics qt 19. W. Kuzmicz and W. Zagoidion-Wosik, Semiconductor Devices, Proc. II Int. Workshop, Delhi, India. Tata-McGraw-Hill, New York (1983). F. Serra-Mestres and J. Millan, 20. X. Aymerich-Humet, J. appl. Phys. 54, 28501 (1983).
0038-I lOI/ $6.00 + 0.00 Copyright 0 1993 Pergamon Press Ltd
Printed in Great Britain. All rights reserved
THE INFLUENCE OF BANDGAP Z-V CHARACTERISTICS
NARROWING ON THE OF A MOSFET
A. JAKUBOWSKI and L. LUKASIAK Institute
of Microelectronics and Optoelectronics, Warsaw University ul.Koszykowa 75, 00-062, Warsaw, Poland (Received
20 August
1992; in revised form
14 January
of Technology,
1993)
Abstract-Modifications of Pierret-Shields model which take into account either apparent or physical bandgap narrowing are presented. The influence of bandgap narrowing on the I-V characteristics of a MOSFET is theoretically examined through comparison of the modified models with the original PierretShields model for various device parameters. It is proved that bandgap narrowing affects significantly performance of a MOSFET in the subthreshold and the near-threshold regions.
drain-to-source voltage normalized to kT/q gate-to-source voltage normalized to kT/q surface potential normalized to kT/q surface potential at the source end of the channel normalized to kT/q source-to-bulk voltage normalized to kT/q surface potential at the drain end of the channel normalized to kT/q channel width brn] physical bandgap narrowing [eV] apparent bandgap narrowing [eV] semiconductor permittivity [F/cm] distance between the edge of the conduction band and the Fermi level normalized to kT distance between the Fermi level and the edge of the valence band normalized to kT effective mobility in the channel [cm’/Vs] quasi-Fermi level splitting normalized to kT/q quasi-Fermi level splitting at the source end of the channel normalized to kT/q quasi-Fermi level splitting at the drain end of the channel normalized to kT/q
NOTATION a c, -% E,, Ec E Cr EF
EGO E, E II
the distance between the impurity level and the edge of the valence band normalized to kT/q gate oxide capacitance per unit area [F/cm21 impurity level [eV] impurity level in the bulk of semiconductor [eV] the edge of the conduction band [eV] the edge of the conduction band in the bulk of semiconductor [eV] the Fermi level [eV] the width of the forbidden gap [eV] intrinsic Fermi level [eV] intrinsic Fermi level in the bulk of semiconductor
[evl the edge of the valence band [eV] the edge of the valence band in the bulk of semiconduction [eV] surface electric field [V/cm] E, F(u, uFe. 5) Seiwatz-Green function based on aooarent 1. bandgap narrowing Seiwatz-Green function based on physical F,(u, 5) bandgap narrowing Fermi-birac integral of the order l/2 Fl!2 F 3.2 Fermi-Dirac integral of the order 3/2 ratio of the effective capacitance of intrinsic G, semiconductor in flat-band condition to the gate oxide capacitance drain current [A] ID k the Boltzmann constant [J/K] L channel length km] effective Debye length in intrinsic semiconductor L,,
E”
E “Z
[cm1 n n, “,e N.4 N, NC NV P 4 ? U UFB UFE
electron concentration [cm-‘] intrinsic concentration [cm-j] effective intrinsic concentration [cm-‘] concentration of acceptors [cm-‘] concentration of ionized acceptors [cm-‘] effective density of states in the conduction band [cm-‘] effective density of states in the valence band [cm-‘] hole concentration [cm- ‘1 electron charge [C] total semiconductor charge per unit area [C/I&] temperature [K] potential normalized to kT/q flat band voltage normalized to kT/q effective Fermi potential normalized to kT/q
1. INTRODUCTION
The effect of bandgap narrowing in heavily doped regions of bipolar devices has been extensively studied (e.g. [l-5]) both theoretically and experimentally. Since continual miniaturization of MOSFETs is, according to scaling rules, accompanied by a corresponding increase of substrate doping the effects of degeneracy and bandgap narrowing may become of importance, especially in view of the fact that nowadays MOS devices with substrate doping on the order of IO’” cme3 and higher are reported[6,7]. This paper is aimed at theoretical investigation of the influence of bandgap narrowing on the I-V characteristics of a MOS transistor. Since in [8] it had been proved that long-channel behaviour may be retained even in the case of transistors with very short channels provided that oxide thickness is sufficiently reduced, the authors have chosen the long-channel Pierret_Shields[9] model for calculations. A modification of the Pierret-Shields model so that it takes into 1129
A.
1130
JAKUBOWSKI and
L.
LUKASIAK
u(x)
Ea.
-
________~___________-___________-..__ _____________.____.....___.._.____ E is
,_.-’
,_--’
EF
L._.-._._.,.-.-._._
______________-._--_---.--
pa ?a
E a*
I
E vo
m X
0
Fig. I. Band diagram of semiconductor in a MOS structure. &-the edge of the conduction band; E,the intrinsic Fermi level; E,-the Fermi level; E,-the impurity level; /?-the edge of the valence band.
account bandgap narrowing is described in Section 2. Section 3 presents the results of simulation and a comparison with the original Pierret-Shields model. Summary and conclusions are given in Section 4. Throughout this paper a n-channel MOSFET with boron-doped substrate is considered.
where k is the Boltzmann constant, and T is the temperature in Kelvin. If the total concentration of impurities NA is known, the concentration of ionized impurities NA may be calculated according to e.g. [13]: N,=N
2. THEORETICAL
CONSIDERATIONS
In Fig. I a band diagram of semiconductor making part of a MOS structure has been shown along with all the necessary notation used in further considerations. If gate voltage is applied the energy levels (except the Fermi level) become functions of x due to the band banding. Subscript w denotes values of the respective energy levels deep in the bulk of the semiconductor which is not disturbed electrically. There are two possible ways of including the effect of bandgap narrowing in MOSFET modelling. 2. I. Appurent bandgap narrowing The apparent bandgap narrowing AEoa takes into account both degeneracy and the physical bandgap narrowing at the same time. Analytical formulae relating AE,, to substrate doping were presented in e.g. [IO-121. In order to modify the Pierret-Shields model[9] the concentration of ionized impurities and concentrations of free carriers have to be formulated. The apparent bandgap narrowing relates the effective intrinsic concentration n, to the ordinary intrinsic concentration n, in the following way e.g. [lo]:
4, =
ni
ev
I (2)
*1+4exp(-q+u+a)’
where q = (EP - E,, )/kT is the normalized position of the Fermi level E, with respect to the edge of the valence band E,, in the bulk of the semiconductor, u is the band bending normalized to kT/q, and a = [E,(x) - E,(x)]/kT = (Ea,, - E,,)/kT is the normalized distance between the impurity level and the edge of the valence band, independent of x (see Fig. I). In semiconductors with low impurity concentration this distance is also independent of impurity concentration. In the case of silicon doped with boron the value of Es(x) - E,(x) is equal to 0.045 eV. On the other hand in highly doped semiconductors the distance between the impurity level and the edge of the majority carrier band becomes a function of impurity concentration [14]. In such a situation the distance may be expressed as: a = [E,(x) - E,(x)]/kT
= [0.045 - AE,,]/kT.
The concentration of majority expressed as follows: P = NvF,,z(-v
carriers
- ~1,
(3)
p may be (4)
where NV is the effective density of states in the valence band, and F,,? is the Fermi-Dirac integral of the order l/2. The normalized position of the Fermi level E, with respect to the edge of the valence band E,, in the
1131
Influence of bandgap narrowing on I-V of MOSFET may
per unit area and G, is the ratio of the intrinsic semiconductor capacitance to the gate oxide capacitance:
(5)
(loa)
which states that in the absence of electrostatic potential the concentration of free holes p is equal to the concentration of ionized acceptors NT. The concentration of minority carriers n may be easily obtained in an approximate way:
Further transformations used by Pierret and Shields lead to the final drain current formula:
bulk of the semiconductor q = (Er - E,,)/kT be evaluated from the following equation: 1 NvF,,,(--rl)
n:, n=N,(u=O)
= NA
1+4exp(-rl
+a)’
ID =
$c,
&B
uFB)("SL
-
&O)
Ml
eXp(u - <)=&cxP(u
-
UFe -
t),
(6)
where ure = In(N, (U = O)h,) is the normalized effective Fermi potential (which no longer corresponds to the distance between the Fermi level EF and the intrinsic Fermi level E,), and < is the normalized quasi Fermi-level splitting of minority carriers due to the current flow. If the Poisson equation is solved, the resulting Seiwatz-Green function[lS] F(u, uFe, 5) is as follows:
-5
‘(4 -u:o)+G
-
s
“LF(u,
” z;’
++,(-4
+ew(u-uF,-5)
(7)
where F3,? is the Fermi-Dirac integral of the order of 312. This function is different from the ordinary Kingston[l6] function used by Pierret and Shields in that it is based on the Fermi-Dirac statistics and that the bandgap narrowing is taken into account. The surface electric field may be expressed as: E.-~+,u,,.O. !e
(8)
where L,, is the intrinsic Debye length: L,,=
kT e -L, J--- Y 2nie
,e
(9)
the voltage balance equation becomes: u,s+usB--uu,,=
4 ----_q
Q,
kTC,
= W’(u,, UFer5) + 4,
uFe, 50) du
(11)
uFe
where L is the channel length, W is the channel width, peiris the effective mobility, to and tL are the normalized values of the quasi-Fermi level splitting at the source- and drain end of the channel, respectively: UW
tO="SBt %B
+
Ulb)
uDS,
2.2. Physical bandgap narrowing In the following approach the effects of physical bandgap narrowing and degeneracy are taken into account independently. As in the previous approach the key to modelling is the formulation of charge concentrations. The concentrations of ionized impurities and majority carriers are expressed by (2) and (4) respectively. The distance between the Fermi level EF and the valence band edge E,, in the bulk of semiconductor is calculated from (5) the only difference being a slightly different form of (3): a = [E,(x) - E,(x)]/kT
surface potential, cs is the semiconductor permittivity, and q is the electron charge. If the total charge in the semiconductor is described as: -~+F(rr,,u,,~),
F&t
= [0.045 - AE,]/kT,
(12)
@a)
u, is the normalized
-caEs=
0
where uDs is the normalized drain-to-source voltage, while us0= u,(t,) and u,,_= u,([,J are the normalized surface potentials at the source- and drain end of the channel, respectively and both are calculated from (10).
-u)-F3:2(-v)l
-exp(-uF,-t),
[S
0
F*(u, uFe, 5) = % In 4 : :I$;
Q,=
-
(10)
where uos, uSB,and urs are the normalized gate-tosource voltage, source-to-bulk voltage and flat band voltage, respectively, Ci is the gate-oxide capacitance
where AEo is the physical bandgap narrowing. The concentration of minority carriers n is evaluated with physical bandgap narrowing taken into account which permits a more rigorous approach to be applied when compared with (6): n=
NcF,:,(-i
-5
+u),
(13)
where Nc is the effective density of states in the conduction band, and [ = (E,, - E,)/kT is the normalized distance between the edge of the conduction band E,, in the bulk of semiconductor and the Fermi level EF, and may be expressed as: c =qEoo(T)lkT-qAEo(N,)lkT-~.
(14)
The symbol E,,(T) denotes the energy gap without the influence of high doping concentrations.
A. JAKUBOWSKI
1132
and L. LUKASAK
Once the charge concentrations are known, the solution of the Poisson equation results in a SeiwatzGreen function F,(u, 5) which takes the physical bandgap narrowing into account: F:(u,<)=~
In
4 + exp(rj - a + u)
-ld(Kuzmicz) - -Id(Slotboom)
4 + exp(fj - a)
Hi,
++-rl
-u)-F3,.2-?)I
Id(Klaassen)
e y
lo-5
2
loa
* Id
le
+~[F,.,(-r -5 +u) -Fw(--i
- 01.
P
(15)
10-7
1o-8 9 -0.4
-0.2
0.0
As it had been shown in [17] F, (u, 5) may be directly introduced into the Pierret-Shields formula (11). (A slightly different form of F, (u, 5) in [17] results from a different cause of degeneracy considered.)
3. SIMULATION
0.2
1.0
0.4
VGS -VT[V]
RESULTS
In order to estimate the influence of bandgap narrowing on the I-V characteristics of a MOSFET the original Pierret_Shields[9] model was compared to its both modified versions presented in 2.1 and 2.2 for various approximations of physical and apparent bandgap narrowing. The values of Nc, Nv, and the temperature dependence of EGOwere taken from [ 181. In the original Pierret-Shields model[9] incomplete ionization of impurities was not taken into account. The flat-band voltage and the bulk-to-source voltage were assumed to be zero. In the modified version described in 2.1 the formulae for the apparent bandgap narrowing presented in [lo] 1, (Slotboom) and [I l] In (Klaassen) were used. In the modified version described in 2.2 two formulae for physical bandgap narrowing were used: the one presented in [19] In (Kuzmicz) for all substrate dopings considered, and the one described in [12] In (Jain) for N, > IO’* cmm3. Analytical approximations of F,:, and F3,‘2were taken from [20]. In Fig. 2(a) normalized transfer characteristics according to both the original model In and its modified versions In (Slotboom), In (Klaassen) and In (Kuzmicz) are shown as a function of the difference between the gate-to-source voltage V,, and the threshold voltage VT for substrate doping equal to 5 . 10” cm- ‘. In all calculations the threshold voltage was evaluated using the physical bandgap narrowing approximation described in [19], with the surface potential chosen so that the concentration of minority carriers at the surface was equal to the concentration of ionized purities. In Fig. 2(b) normalized transfer characteristics are shown for substrate doping equal to 2. 10’scm-3. In this figure points corresponding to In (Jain) were added. Figure 3(a) and 3(b) present the relative error of the original model Zn with respect to its modified versions as a function of VGs - VT. Substrate dopings are the
1)l
J )
,* Id -0.4
-0.2
0
0.2
0.4
0.6
0.8
1
VGS-VT(V]
Fig. 2. (a) Normalized drain current as a function of the difference between gate voltage and threshold voltage according to the original model and modified ones (NA = 5. IO” cm-j). (b) Normalized drain current as a function of the difference between gate voltage and threshold voltage according to the original model and modified ones (NA = 2. IO’*cm-‘).
same as in Fig. 2(a) and 2(b). The error decreases with gate voltage due to a quick increase of drain current. Thus the effect of bandgap narrowing is important only in the subthreshold and the near threshold regions of MOS operation. It can be seen that for higher substrate doping the discrepancies between the modified models and the original one are much greater. Figure 4 shows the percentage error of the original model with respect to the modified ones in saturation as a function of gate-oxide. The gate voltage is equal to the threshold voltage. The error slightly increases for thinner oxides. Figure 5 presents the same error as in the previous figure as a function of substrate doping. Obviously the error increases when the impurity concentration grows. Finally, in Fig. 6 the error is displayed as a function of a scaling factor, which is constructed so that the product of oxide thickness and substrate doping of the considered devices is constant (according to constant field scaling rules reduction of oxide thickness by a given factor k must be accompanied by the
Influence of bandgap narrowing on I-V of MOSFET
(a)80
1133
loo1 _-80 -
tox = 10nm VGS = VT
- Id(Kuzmicz) - - Id(Slotboom) tox = 10 nm
Id(Klaassen) - Id(Kuzmicz) ’ Id(Jain) - Id(Slotboom
0-t -0.4
1 -0.2
0.0
0.2
0.4
0.6
0.6
d
Iq(Klaassen)
0
1.0
0.5
1.0
100
_________
601L
Fig. 5. Relative error of the original model with respect to the modified ones as a function of substrate doping (t,, = 10 nm).
Id(Kuzmicz)
. Id(Jain)
Id(Klaassen) 6Om
2 w z 4 :
4om
20
O-0.4
20
- - Id(Slotboom)
.
z 5
10
5.0
SUBSTRATE DOPING [lO’$m’3]
VGS . VT [V]
(b)
2.0
tax = 10nm
VDS = 1V
-0.2
I
0.0
0.2
0.4
0.6
0.8
1.0
than the “nominal” 10 nm. In the case of transistors with higher substrate doping and thinner oxides the error quickly increases. It is easily noticed that the apparent-bandgap-narrowing-based model using the approximation presented in the paper of Klaassen et al.[l l] yields values very close to those obtained using the model based on physical bandgap narrowing. On the other hand if the approximation of apparent bandgap narrowing presented in [lo] is used the two modified models differ from each other considerably.
VGS-VT [V]
Fig. 3. (a) Relative error of the original model with respect to the modified ones as a function of the difference between gate voltage and threshold voltage (NA = 5 . IO” cmm3). (b) Relative error of the original model with respect to the modified ones as a function of the difference between gate voltage and threshold voltage (NA = 2. IO’* cm-9. increase of substrate doping by the same factor k). Values of the scaling factor k > 1 correspond to devices in which substrate doping is higher than the “nominal” 5 x 10” cmm3. and oxide thickness is less
4. SUMMARY
The Pierret-Shields[9] model was modified so that the influence of bandgap narrowing on the I-V acteristics of MOS transistor be investigated The presented may use apparent or bandgap narrowing. The results of simulation indicate that in the case of MOSFETs with high doping concentration the neglect of bandgap narrowing effect leads to considerable errors especially in the subthreshold and the
Na =
lO”*k [cni’]
tox =
VGS
[nm]
V-f
.
20 50’ 2.5
- Id(Slotboom
I 5
10
25
50
100 0.1
OXIDE THICKNESS [nm]
Fig. 4. Relative error of the original model with respect to the modified ones as a function of gate oxide thickness (NA = 5 IO” cm-‘).
0.2
0.4 SCALING
Fig.
Relative error the modified
10
1.0 k
the original with respect as a of scaling
A. JAKUBOWSKIand L. LUKASIAK
1134 near-threshold mations
regions.
semiconductors. greater
Bandgap
used for analysis due
observed is greater
Therefore to
the
fact
in the space-charge
narrowing
approxi-
were intended
for neutral
these errors
may be still
that
bandgap
region
than that in the neutral
narrowing
of MOS
regions
devices
of bipolar
devices[6]. REFERENCES 1. D. D. Tang, IEEE Trans. Electron. Devices ED-27, 563 ( 1980). 2. b. E.‘Possin, M. S. Adler and B. Jayant Baliga, IEEE Trans. Electron. Devices ED-27. 983 (1980). 3. A. Neugroschel, S. C. Pao and’F. A.‘Lindholm, IEEE Trans. %ectran. Devices ED-29, 894 (1982). 4. M. S. Mock, Solid-St. Electron. 16. 1251 (1973). 5. P. P. Mertens, J. L. van Meerberger, J. F. Nijs and R. J. van Gverstraeten, IEEE Trans. Electron. Devices ED-27, 949 (1980). 6. H. C. Chen, S. S. Li and K. W. Teng, Solid-St. Electron. 32, 339 (1989). 7. J. R. Brews, in High-Speed Semiconductor Devices (Edited by S. M. Sze), pp. 139-209. Wiley, New York (1990).
8. J. R. Brews, W. Fichtner, E. Nicollian and S. M. Sze, IEDM Tech. Din., p. 10, (1979); IEEE Trans. Electron. Devices Left. EfiG1, 2 (i980): 9. R. F. Pierret and J. A. Shields. Solid-St. Elecfron. 26. 143 (1983). 10. J. W. Slotboom and H. C. de Graaff, Solid-St. Electron. 19, 857 (1976). J. W. Slotboom and H. C. de 11. D. B. M. Klaassen, Graaff, Solid-St. Elecfron. 35, 125 (1992). Solid-St. Electron. 34. 12. S. C. Jain and J. D. Rot&ton. 453 (1991). 13. M. Shur, Physics of Semiconductor Devices. PrenticeHall. Enalewood Cliffs. N.J. (1990). 14. T. F. Lee and T. C. McGill, J. appl. Phys. 46, 373 (1975). 15. R. Seiwatz and M. Green, J. appl. Phys. 29, 1034 (1958). 16. R. H. Kingston and S. F. Neustadter, J. appl. Phys. 26, 718 (1955). A. Jakubowski and L. tukasiak, IEEE 17. B. Majkusiak, Trans. Electron. Devices ED-34, 2560 (1987). IS. M. A. Green, J. appl. Phys. 67, 2944 (1990). in Physics qt 19. W. Kuzmicz and W. Zagoidion-Wosik, Semiconductor Devices, Proc. II Int. Workshop, Delhi, India. Tata-McGraw-Hill, New York (1983). F. Serra-Mestres and J. Millan, 20. X. Aymerich-Humet, J. appl. Phys. 54, 28501 (1983).