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The current-voltage characteristics of field-effect transistors with short channels
Solid State Communications, Vol. 19, pp. 471—473, 1976.
Pergamon Press.
Printed in Great Britain
THE CURRENT—VOLTAGE CHARACTERISTICS OF FIELD-EFFECT TRANSISTORS WITh SHORT CHANNELS~ K. Hess Ludwig Boltzmann Institut für Festkorperphysik and Institut für Angewandte Physik der Universität, A-1090 Wien, Strudlhofgasse 4, Austria G. Dorda Forschungslaboratorium der Siemens AG, Munchen, Germany and C.T. Sah Department of Electrical Engineering, University of Illinois, Urbana, IL 61801, U.S.A. (Received 18 December 1975; in revised form 12 February 1976 by N.J. Mayer) The current—voltage characteristics of MOS field-effect transistors is investigated theoretically and experimentally in the region of extremely high drain electric fields ED using the electron temperature concept in the classical three-dimensional theory. It is found that the drain field at which the drain current becomes non-ohmic on the basis of hot electron effects is related to the surface field E~8by ED -..,/E~8.Other expressions for the field dependent mobility are also given which allow the construction of the current—voltage characteristics. THE CURRENT—VOLTAGE characteristics of MOS transistors is dominated by hot electron effects if the drain field ED exceeds a certain temperature dependent value (typically iO~V/cm at 300 K) and the magnitude of the drain voltage UD is much smaller than the gate voltage ~JG. This wastheoretically2 first observed and by Fang and Fowler’ and confirmed later experimentally.3 The theories dealing with the hot electron effect in surface channels were2’4’5 mainly the two-as the of concerned the surfacewith conduction dimensional aspect energy levels of the electrons are grouped into subbands.5 More recently, scattering between the subbands6 has been included in the calculations. However, at very high drain fields ED which are obtained in transistors with channel length below 10 .tm (for lower temperatures also longer channels) the mean energy of the electrons exceeds the spacing of the subbands for the following reason. In the range of interest the mean power loss of the electrons to the lattice (ac/at) is given by the loss to optical phonons.2 Using the two-dimensional theory which takes only one subband into account we have:
‘ae I ~
R
the rate of energy gained by the electrons from the applied field is higher than its loss to the lattice. This situation is unstable as long as two-dimensional aspects dominate and therefore ends in a stationary state where intersubband scattering is very important. EDC was estimated to be of the order of 1 o~ V/cm. Nakamura6 took into account intersubband scattering including up to 10 subbands. However, at fields as high as i04 V/cm, more than 10 subbands are important and three-dimensional effects will dominate. Thus for ED l0~V/cm it seems to be commendable to treat the problem in the classical
—
K~[i
—
eOR/Te_ORI’Te1/[eOR~~T_~
(1) *
where T and Te are the lattice and electron temperature, respectively, °R is the Debye temperature, KR is a constant and the summation goes over all modes of optical phonons. Note that I(ae/at)I has a maximum value of ~ KR. On the other hand, the2electron which ingains the energy range offrom the electric field atevd a rate ejiE high fields equals 8ED, v~being the saturation velocity. This means that above a certain critical field EDC EDC = ~ (KR /ev~) (2)
Work supported by the “Ludwig Boltzmann Gesellschaft zur Forderung der wissenschaftlichen Forschung” and by the “Fonds zur Farderung der schaftlichen Forschung in Osterreich”. ~
manner proposed by Schrieffer,7 incorporating hot electron effects in his theory. The extension of Schrieffer’s theory to the hot electron case is very simple if the following approximations are made. (i) The electron temperature concept is used, i.e. the spherically symmetrical part fo of the distribution 471
472
FIELD-EFFECT TRANSISTORS WITH SHORT CHANNELS
Vol. 19, No. 5
function f is given by: 1. (3) fo = C exp [—(mv2/2 + q~)/kT~ (ii) The momentum relaxation time r which is a function of the carrier energy e is approximatedby: r[e] ~r[(3~’ir/4)2kTe].
(4)
(iii) The surface field E~= — a~/az is a constant
denoted and Sah8 extended ~ calculation including The by notation isSchrieffer’s adopted from Schrieffer.7 Pierretthe many valley structure of silicon conduction band. They also showed that the constant-field approximation is not too good. We will make this approximation, however, since it enables us to find very simple relations for the transistor characteristics. As shown by Schrieffer,7 the ratio of effective mobility to bulk mobility is determined by the dimensionless parameter a which is rewritten by using Te for T: a = (qE~ 8r)’\/2mkT~. (5) 9 It follows that below a certain value of a For small values a the effective mobility doescaused not by which we call a~of (a~ 0.2), current saturation depend on r. hot electrons cannot occur. This means that the electron temperature has to exceed the value ‘VTe/lOO K
=
a~qE~8r/’~/2mk(100 K)
(6)
to give essential deviations from Ohm’s law. Te according to equation (6) is typically above 1000K. In the limit of high electron temperatures we have forr:’° r where
4 =
3\/lr
1 2NR + 1) + 1) ‘~JTe/lO0KB{~AR( B
and
=
0
2
4
6
8
ip
12
14
Fig. 1. ID—UD characteristics of the Siemens transistor G237/2, T = 77 K, (100) surface. The inset shows how EDS was determined at 4.2 K. 04
0.31-
*
I
A
I
Fig. 2.
[EDS
A I
IA I
•
A
I
Ezs [V/cm
S
]
1O~
Il0~(V/cm)] / [E~
8/ l0~(V/cm)] 1 /2 vs ~ T = 300 K Siemens transistor G237/2 ; L = 7.5 pm, d0~= 5000 (100) surface. *, T = 300 K, Sato 3L= A, l_3pm,d etal.~ 0~=3100A,(100) r = 300 K, Fang and Fowler;’ L = 10 pm,surface. d0,, = 1pm, (100) surface. a, T = 77 K Transistor G237/2.., T = 4.2K, Transistor TEl07;’~L = 50 pm, d0~= 5000 A, (1l0)surface.
obtain the drain fieldcombining EDS necessary for the (7) implies T~-~ oo) and equations (6)occurrence and (8) we
0.0145(kT/h)
of current saturation:
[EDS/104(V/cm)1
=
2
AR
=
0.1215
(looK~)(4ooK1 DR T
,‘\ 0R
)2
i~1O8eV/cm
and DR is the coupling constant to the optical phonons. To arrive at equation (7) it is assumed that Te °R’ where 0R is the Debye Temperature. On the other hand, the electron temperature is determined by the power balance1’ ~‘
Ia
epeffE~= (
\
~
~ eV/cm/
.
(9)
A better approximation for Peff in equation (8) at high electron temperature may be obtained using Schrieffer’s results and expanding for large ci:
Peff
=
er / rn ~l
q rE~~\ ~/2irmkTe)
(10)
10’°WAJ~?1~~ This equation shows that in contrast to the bulk case no 100 saturation of the drift velocity can be expected. To 2 arrive at these formulas, values for the effective masses \ (8) m 3 haveu1 = ~(DR 1 = 0.19m0 and mt = 0.9m0, the sound velocity 108 eV/cmi 9 x l0~cm/sec and the density p = 2.33 g/cm Approximating i.zeff by Pbulk for the moment (this been used. If one does not make the constant field =
at Iopt/
1.4643
AJ0.063aC[EZS/105(V/cm)]
2<
Vol. 19, No. 5
FIELD-EFFECT TRANSISTORS WITh SHORT CHANNELS
approximation, the parameters 13 and B given by Schrieffer determine the ratio !ieff/Pbuik. Following the arguments given above and using 13 instead of a one arrives at the same proportionality of EDS to ~ as shown in equation (9) but with a much more complicated proportionality constant which depends slightly on temperature. In summary the above results predict the following features of the transistor characteristics: The ID—UD curve will fall below the ohmic line at approximately constant values of the ratio ED/\/EZ 8. ‘D will never saturate, in contrast to the “normal” space-chargelimited case. These features were found to be reflected by the experiments. Figure 1 shows the 77K ID—UD curve of a transistor having a channel length L of 7.5 pm, and width of W = 50 pm. The oxide thickness c1~~ was 5000 A. The data were taken using short pulses lOnsec to lOpsec long in order to avoid heating of the lattice. No time dependence was observed within the above time limits. The transistor shows a similar behavior at 300K. Note that the current is nearly saturated long before UD reaches the value of U0 (the threshold voltage was below 3 V) which cannot be explained in terms of channel pinch off. Measurements at 4.2 K with transistors having longer channels (L = 50 p) have been made previously.’2 Fang and Fowler1 and Sato eta!.3 reported also ID—UD curves at various temperatures and various transistor geometries. Figure 2 shows E~s/~/E; 8 (EDS/ l0~V/cm)/
473
\/E~3/(105V/cm) which should be constant according to equation (9). There is some arbitrariness in the determination of EDS (the field at which strong nonlinearities begin). The typical shape of the transistor characteristics is as follows: At very low values of UD the current is ohmic. At higher UD values superlinear deviations from Ohm’s Law occur especially at low temperatures, which usually cannot be seen on a scale as rough as shown in Fig. 1. At still higher values of UD the characteristics tends towards saturation following again a straight line above a certain value of UD (indicated by the arrow in the inset of Fig. 1). EDS was determined to be the ratio UD/L at the crossing point of the ohmic line and the “saturation line”. This is indicated in Fig. I by the dashed lines and by the arrows. In the case of T = 4.2 K the values of EDS were chosen as shown in the inset of Fig. I since at this temperature two-dimensional 2’12aspects E~ dominate until the current begins to saturate. 8 was taken to be U0 /d0~.The results in Fig. 2 show convincingly that the ratio E~s/..JE8is nearly independent of temperature and transistor geometry. Equation (9) allows in principle a determination of the sum of the coupling constants to the optical phonons. Using E~5/.~/E8 = 0.2 and a~ = 0.2 one obtains ‘.IER D~= 2’10 considering 1.8 arbitrariness x 108 eV/cm.inThis reasonable the the value choiceis of EDS and a~and the weakness of the constant field approximation. Acknowledgement — One of the authors (K.H) would like to express his sincere appreciation to Prof. K. Seeger for his continuous interest and support.
1.
REFERENCES FANG F.F. & FOWLERA.B.,J. App!. Phys. 41, 1825 (1969).
2.
HESS K. & SAH C.T.,Phys. Rev. BlO, 8,3375 (1974).
3.
SATO T., TAKEISHI Y., TANG! H., OHNUMA H. & OKAMOTO Y.,J. Phys. Soc. Japan 31, 1846 (1971).
4. 5.
KROWNE C.M. & HOLM-KENNEDY J.W., Surf Sci. 46,232(1974). STERN F.,Phys. Rev. BS, 4891 (1972).
6.
NAKAMURA K., Proc. 1st mt. Conf Two Dimensional Systems. Brown University (1975).
7. 8.
SCHRIEFFER J.R., Ploys. Rev. 97, 3, 641 (1955). PIERRET R.F. & SAH C.T., Solid State Electron. 11,279 (1968).
9.
See equation (16) in reference 7.
10. 11. 12.
CONWELL E.M., Solid State Physics (Edited by SEITZ F., TURNBULL D. & EHRENREICH H.), Suppl. 9. Academic Press, New York (1967). The derivation of this formula follows the usual lines: multiplying the Boltzmann equation by the energy e and averaging over all values of k one obtains equation (8). HESS K., NEUGROSCHEL A., SHIUE C.C. & SAH C.T., J. AppL Phys~46,4, 1721 (1975).
Pergamon Press.
Printed in Great Britain
THE CURRENT—VOLTAGE CHARACTERISTICS OF FIELD-EFFECT TRANSISTORS WITh SHORT CHANNELS~ K. Hess Ludwig Boltzmann Institut für Festkorperphysik and Institut für Angewandte Physik der Universität, A-1090 Wien, Strudlhofgasse 4, Austria G. Dorda Forschungslaboratorium der Siemens AG, Munchen, Germany and C.T. Sah Department of Electrical Engineering, University of Illinois, Urbana, IL 61801, U.S.A. (Received 18 December 1975; in revised form 12 February 1976 by N.J. Mayer) The current—voltage characteristics of MOS field-effect transistors is investigated theoretically and experimentally in the region of extremely high drain electric fields ED using the electron temperature concept in the classical three-dimensional theory. It is found that the drain field at which the drain current becomes non-ohmic on the basis of hot electron effects is related to the surface field E~8by ED -..,/E~8.Other expressions for the field dependent mobility are also given which allow the construction of the current—voltage characteristics. THE CURRENT—VOLTAGE characteristics of MOS transistors is dominated by hot electron effects if the drain field ED exceeds a certain temperature dependent value (typically iO~V/cm at 300 K) and the magnitude of the drain voltage UD is much smaller than the gate voltage ~JG. This wastheoretically2 first observed and by Fang and Fowler’ and confirmed later experimentally.3 The theories dealing with the hot electron effect in surface channels were2’4’5 mainly the two-as the of concerned the surfacewith conduction dimensional aspect energy levels of the electrons are grouped into subbands.5 More recently, scattering between the subbands6 has been included in the calculations. However, at very high drain fields ED which are obtained in transistors with channel length below 10 .tm (for lower temperatures also longer channels) the mean energy of the electrons exceeds the spacing of the subbands for the following reason. In the range of interest the mean power loss of the electrons to the lattice (ac/at) is given by the loss to optical phonons.2 Using the two-dimensional theory which takes only one subband into account we have:
‘ae I ~
R
the rate of energy gained by the electrons from the applied field is higher than its loss to the lattice. This situation is unstable as long as two-dimensional aspects dominate and therefore ends in a stationary state where intersubband scattering is very important. EDC was estimated to be of the order of 1 o~ V/cm. Nakamura6 took into account intersubband scattering including up to 10 subbands. However, at fields as high as i04 V/cm, more than 10 subbands are important and three-dimensional effects will dominate. Thus for ED l0~V/cm it seems to be commendable to treat the problem in the classical
—
K~[i
—
eOR/Te_ORI’Te1/[eOR~~T_~
(1) *
where T and Te are the lattice and electron temperature, respectively, °R is the Debye temperature, KR is a constant and the summation goes over all modes of optical phonons. Note that I(ae/at)I has a maximum value of ~ KR. On the other hand, the2electron which ingains the energy range offrom the electric field atevd a rate ejiE high fields equals 8ED, v~being the saturation velocity. This means that above a certain critical field EDC EDC = ~ (KR /ev~) (2)
Work supported by the “Ludwig Boltzmann Gesellschaft zur Forderung der wissenschaftlichen Forschung” and by the “Fonds zur Farderung der schaftlichen Forschung in Osterreich”. ~
manner proposed by Schrieffer,7 incorporating hot electron effects in his theory. The extension of Schrieffer’s theory to the hot electron case is very simple if the following approximations are made. (i) The electron temperature concept is used, i.e. the spherically symmetrical part fo of the distribution 471
472
FIELD-EFFECT TRANSISTORS WITH SHORT CHANNELS
Vol. 19, No. 5
function f is given by: 1. (3) fo = C exp [—(mv2/2 + q~)/kT~ (ii) The momentum relaxation time r which is a function of the carrier energy e is approximatedby: r[e] ~r[(3~’ir/4)2kTe].
(4)
(iii) The surface field E~= — a~/az is a constant
denoted and Sah8 extended ~ calculation including The by notation isSchrieffer’s adopted from Schrieffer.7 Pierretthe many valley structure of silicon conduction band. They also showed that the constant-field approximation is not too good. We will make this approximation, however, since it enables us to find very simple relations for the transistor characteristics. As shown by Schrieffer,7 the ratio of effective mobility to bulk mobility is determined by the dimensionless parameter a which is rewritten by using Te for T: a = (qE~ 8r)’\/2mkT~. (5) 9 It follows that below a certain value of a For small values a the effective mobility doescaused not by which we call a~of (a~ 0.2), current saturation depend on r. hot electrons cannot occur. This means that the electron temperature has to exceed the value ‘VTe/lOO K
=
a~qE~8r/’~/2mk(100 K)
(6)
to give essential deviations from Ohm’s law. Te according to equation (6) is typically above 1000K. In the limit of high electron temperatures we have forr:’° r where
4 =
3\/lr
1 2NR + 1) + 1) ‘~JTe/lO0KB{~AR( B
and
=
0
2
4
6
8
ip
12
14
Fig. 1. ID—UD characteristics of the Siemens transistor G237/2, T = 77 K, (100) surface. The inset shows how EDS was determined at 4.2 K. 04
0.31-
*
I
A
I
Fig. 2.
[EDS
A I
IA I
•
A
I
Ezs [V/cm
S
]
1O~
Il0~(V/cm)] / [E~
8/ l0~(V/cm)] 1 /2 vs ~ T = 300 K Siemens transistor G237/2 ; L = 7.5 pm, d0~= 5000 (100) surface. *, T = 300 K, Sato 3L= A, l_3pm,d etal.~ 0~=3100A,(100) r = 300 K, Fang and Fowler;’ L = 10 pm,surface. d0,, = 1pm, (100) surface. a, T = 77 K Transistor G237/2.., T = 4.2K, Transistor TEl07;’~L = 50 pm, d0~= 5000 A, (1l0)surface.
obtain the drain fieldcombining EDS necessary for the (7) implies T~-~ oo) and equations (6)occurrence and (8) we
0.0145(kT/h)
of current saturation:
[EDS/104(V/cm)1
=
2
AR
=
0.1215
(looK~)(4ooK1 DR T
,‘\ 0R
)2
i~1O8eV/cm
and DR is the coupling constant to the optical phonons. To arrive at equation (7) it is assumed that Te °R’ where 0R is the Debye Temperature. On the other hand, the electron temperature is determined by the power balance1’ ~‘
Ia
epeffE~= (
\
~
~ eV/cm/
.
(9)
A better approximation for Peff in equation (8) at high electron temperature may be obtained using Schrieffer’s results and expanding for large ci:
Peff
=
er / rn ~l
q rE~~\ ~/2irmkTe)
(10)
10’°WAJ~?1~~ This equation shows that in contrast to the bulk case no 100 saturation of the drift velocity can be expected. To 2 arrive at these formulas, values for the effective masses \ (8) m 3 haveu1 = ~(DR 1 = 0.19m0 and mt = 0.9m0, the sound velocity 108 eV/cmi 9 x l0~cm/sec and the density p = 2.33 g/cm Approximating i.zeff by Pbulk for the moment (this been used. If one does not make the constant field =
at Iopt/
1.4643
AJ0.063aC[EZS/105(V/cm)]
2<
Vol. 19, No. 5
FIELD-EFFECT TRANSISTORS WITh SHORT CHANNELS
approximation, the parameters 13 and B given by Schrieffer determine the ratio !ieff/Pbuik. Following the arguments given above and using 13 instead of a one arrives at the same proportionality of EDS to ~ as shown in equation (9) but with a much more complicated proportionality constant which depends slightly on temperature. In summary the above results predict the following features of the transistor characteristics: The ID—UD curve will fall below the ohmic line at approximately constant values of the ratio ED/\/EZ 8. ‘D will never saturate, in contrast to the “normal” space-chargelimited case. These features were found to be reflected by the experiments. Figure 1 shows the 77K ID—UD curve of a transistor having a channel length L of 7.5 pm, and width of W = 50 pm. The oxide thickness c1~~ was 5000 A. The data were taken using short pulses lOnsec to lOpsec long in order to avoid heating of the lattice. No time dependence was observed within the above time limits. The transistor shows a similar behavior at 300K. Note that the current is nearly saturated long before UD reaches the value of U0 (the threshold voltage was below 3 V) which cannot be explained in terms of channel pinch off. Measurements at 4.2 K with transistors having longer channels (L = 50 p) have been made previously.’2 Fang and Fowler1 and Sato eta!.3 reported also ID—UD curves at various temperatures and various transistor geometries. Figure 2 shows E~s/~/E; 8 (EDS/ l0~V/cm)/
473
\/E~3/(105V/cm) which should be constant according to equation (9). There is some arbitrariness in the determination of EDS (the field at which strong nonlinearities begin). The typical shape of the transistor characteristics is as follows: At very low values of UD the current is ohmic. At higher UD values superlinear deviations from Ohm’s Law occur especially at low temperatures, which usually cannot be seen on a scale as rough as shown in Fig. 1. At still higher values of UD the characteristics tends towards saturation following again a straight line above a certain value of UD (indicated by the arrow in the inset of Fig. 1). EDS was determined to be the ratio UD/L at the crossing point of the ohmic line and the “saturation line”. This is indicated in Fig. I by the dashed lines and by the arrows. In the case of T = 4.2 K the values of EDS were chosen as shown in the inset of Fig. I since at this temperature two-dimensional 2’12aspects E~ dominate until the current begins to saturate. 8 was taken to be U0 /d0~.The results in Fig. 2 show convincingly that the ratio E~s/..JE8is nearly independent of temperature and transistor geometry. Equation (9) allows in principle a determination of the sum of the coupling constants to the optical phonons. Using E~5/.~/E8 = 0.2 and a~ = 0.2 one obtains ‘.IER D~= 2’10 considering 1.8 arbitrariness x 108 eV/cm.inThis reasonable the the value choiceis of EDS and a~and the weakness of the constant field approximation. Acknowledgement — One of the authors (K.H) would like to express his sincere appreciation to Prof. K. Seeger for his continuous interest and support.
1.
REFERENCES FANG F.F. & FOWLERA.B.,J. App!. Phys. 41, 1825 (1969).
2.
HESS K. & SAH C.T.,Phys. Rev. BlO, 8,3375 (1974).
3.
SATO T., TAKEISHI Y., TANG! H., OHNUMA H. & OKAMOTO Y.,J. Phys. Soc. Japan 31, 1846 (1971).
4. 5.
KROWNE C.M. & HOLM-KENNEDY J.W., Surf Sci. 46,232(1974). STERN F.,Phys. Rev. BS, 4891 (1972).
6.
NAKAMURA K., Proc. 1st mt. Conf Two Dimensional Systems. Brown University (1975).
7. 8.
SCHRIEFFER J.R., Ploys. Rev. 97, 3, 641 (1955). PIERRET R.F. & SAH C.T., Solid State Electron. 11,279 (1968).
9.
See equation (16) in reference 7.
10. 11. 12.
CONWELL E.M., Solid State Physics (Edited by SEITZ F., TURNBULL D. & EHRENREICH H.), Suppl. 9. Academic Press, New York (1967). The derivation of this formula follows the usual lines: multiplying the Boltzmann equation by the energy e and averaging over all values of k one obtains equation (8). HESS K., NEUGROSCHEL A., SHIUE C.C. & SAH C.T., J. AppL Phys~46,4, 1721 (1975).