CORRESPONDENCE 977

Thermal Noise in the Hot Electron Regime in FETs Abstract-Klaassen has given two different formulas for the

thermal noise in the conducting channel of FETs in the hot electron regime. I t is shown that the first published formula is formally cor- rect. A discussion is given of the equivalent noise temperature of the hot electrons.

Klaassen has given two different formulas for noise in conducting channels due to hot electrons, such as occur in FETs. The first for- mula can be written [l ]

S i ( X , %’,I) = 4kTeg(x‘)G(x’ - x) (1) where T, is the equivalent noise temperature of the electrons in the channel at x’, g(x’) is the actual conductance per unit length a t x‘, and S(x’-x) the Dirac delta function.

The second formula can be written [ 2 ] Si(%, x’, f) = 4kTego(x’)6(x’ - x) (2)

where T, has the same meaning as before, and go(%’) is the low-jield conductance per unit length at x’.

There is no reason why the noise a t x’ should be expressed in terms of the low-jield conductance at x’, whereas it makes consider- able sense to express the noise in terms of the actual conductance at x!. That is, the second formula does not seem appropriate, whereas the first formula is formally correct if T. is properly chosen.

This can be substantiated as follows. The noise enerated in conductors should properly be called diffusion noise [3f One would expect that the derivation of the diffusion noise source should remain valid a t high fields; consequently one may write

Si(%, x‘, f) = 4q2Dnn(x’)A (x’)6(x’ - x) (3)

where q is the electron charge, D, is the diffusion constant, n(x’) is the carrier concentration at x’, and A(%’) is the cross-sectional area of the conducting channel a t x’.

b’e now introduce an equivalent temperature Te(x’), by equating

qDn k T e ( x ’ ) m (4) where pn is the high-field mobility. Substituting into (3) yields

Si(%, x’, f) = 4kTe(x’) [4pL,n(x’)A]8(x’ - x) = 4kTe(x’)g(x’)G(x‘ - x) ( 5 )

which corresponds to thermal noise of the conductance per unit length g ( x ’ ) at the temperature T.(x’), thus verifying (1).

The next question is what expression should be chosen for the equivalent noise temperature T,. Klaassen [l], [2] suggests

T e = To(1 + PE/Ec) (6)

in analogy with published hot electron noise measurements by Erlbach and Gunn [4]. However, their measurements should be represented by

Te = To(1 + p’E2/Ec2) ( 7 ) as long as the field strength E is not too large; only at very high field strengths can one write

Te = To[l +@’(E - E1)/Ec]. (8 )

In these expressions p , p’, p”, and E, are constants and E, is the criti- cal field strength occurring in the expression for the high-field mobility p :

where PO is the low-field mobility. Bougalis and van der Ziel [ 5 ] have shown in their study of hot

electron effects in single-injection space-charge-limited solid-state diodes that ( 7 ) must be used in order to explain their data. It would thus seem that (7) should also be used for hot electron noise in FETs, except erhaps at the highest fields, as already suggested by van der Ziel [6j:

Finally, it should be pointed out that Erlbach and Gunn’s mea- surements [ 4 ] may not be directly applicable to FETs. They mea- sured the noise in a direction perfiendicular to the applied field,

whereas in FETs the noise is measured in a direction parallel to the applied field. The hot electron noise is probably anisotropical; hence, though the functional dependence of T, on E/Ec may not be different, the coefficients p , p’, and p” might not be the same for the two cases. Therefore, these coefficients should be determined from noise mea- surements on the actual devices under consideration.

A. VAN DER ZIEL Dep. Elec. Eng. Univ. Minnesota Minneapolis, Minn. 55455

REFERENCES [l] F. M. Klaassen, “On tpe influence of hot carrier effects on the thermal noise of

field-effect transistors, I E E E T r a n s . Electron Deuices, vol. ED-17. Oct. 1970, pp. 8511-862. __ Electron Dewices (Corresp.), vol. ED-18, Jan. 1971, pp. 74-57.

Comments on hot carrier noise in field-effect transistors.” I E E E T r a n s .

A. van der Ziel Noise: Sources Characterization Measuvement. Englewood Cliffs, N. J.: Predtice-Hall. 1970, Gh. 5.

Phys. Rev. Lett. vol. 8, Apr. 1962, pp. 280-282. E. Erlbach and J. B. Gunn, “Noise temperature of hot electrons in germanium,”

D. N. Bougalis and A. van der Ziel “Hot electron effects in single injection silicon SCL diodes,” Solid-State Elect;on., vol. 14, Apr. 1971, pp. 265-272. A. van der Ziel. ”Noise resistance of FET’s in the hot electron regime,” Solid- State Electuon.. vol. 14, Apr. 1971,pp. 347-350.

Comments on “Electronic Image Storage Utilizing a Silicon Dioxide Target”

Abstract-It is shown that, contrary to the statements in the above paper,’ the secondary emission yield of thermally grown SiOz does not saturate. Rather, the results reported fall on a yield curve which peaks at a yield of 7.4 at a primary electron energy of 500 eV.

In the above paper,’ the authors indicate the “saturation” of the secondary electron emission yield of a thermally grown Si02 layer for primary electron energies around 100 eV. I t is our experience that such saturation does not occur, but rather that the yield is well char- acterized by the types of semiempirical expressions presented by Dekker [ l ] , Hachenberg and Brauer [2], and Sternglass [3].

As an example, Fig. 1 shows the secondary yield of a 1.5-pm-thick Si02 layer thermally grown on n-type (111) Si (essentially identical results are achieved with 1.0-pm-thick oxides). The dots are our ex- perimental points, and the crosses are Silver and Luedicke’s data from Fig. 4 of their paper. The solid curve is Dekker’s “constant loss” expression [ l , eq. 7.3 1, fitted for a maximum yield 6, = 7.4 at a primary electron energy EO, = 500 eV. Not only does the theoretical curve fit our data well, but it appears that the data of Silver and Luedicke also fall on the same curve.

8 r

7L

8[ 4

3 J /

0 0 8

I -

O 100 200 XK, 400 500 600 PRIMARY ELECTRON ENERGY

Fig. 1. The average secondary electron emission yield of thermally grown SiOp. Dots: author‘s results; crosses: Silver and Luedicke’s results; solid curve: Dek- ker’s “constant loss’ expression I l l .

Manuscript received March 23,1971.

Manuscript received May 3. 1971. R. S. Silver and E. Luedicke, I E E E T r a n s . Electron Deuices, vol. ED-18, Apr.

1971, pp. 229-235.

IEEE TRANSACTIONS ON ELECTRON DEVICES, OCTOBER 1971

I t should be noted that the differences in substrate orientation and conductivity type between our work and that of Silver and Luedicke cannot provide the explanation for the differences in result:, since typical escape depths for secondaries are on the order of 100 A , some two orders of magnitude below the oxide thicknesses considered here.

FRANK L. THIEL Res. and Develop. Labs Corning Glass Works Corning, N. Y . 14830

REFERENCES

[31

A. .T. Dekker, “Secondary e!ectron emission,” in Sol id State Physics, vol. 6, F. Seitz and D. Turnbull, Eds. New York: Academic Press, 1958, pp. 251-311.

A d v a w e s in Electronics and Electvon Physics. vol. 11, L. Marton, Ed. h-ew 0. Hachenberg and W . Brauer. “Secondary electron emission from solids,” in

York: Academic F’Fess, 1959. pp. 413-499.

bardment.” Westinghoune Res. Labs. Sci. Paper 6-94410-2-P9, July 23, 1957. E. J. Sternglass, Theory of secondary electron emission under electron bom-

Authors’ Reply2 We had not intended that our curves of secondary emission be

interpreted as the true secondary emission yield of silicon dioxide; in fact, the designation bg in Fig. 4 was chosen to indicate an effective secondary emission under the normal operating conditions of an actual tube. Unfortunately this distinction was omitted in the final condensation of the paper.

We agree completely with the secondary emission values quoted by Thiel and have obtained similar results under conditions in which the true secondary emission of silicon dioxide was measured.

R. S. SILVER E. LUEDICKE RCA Labs. Princeton, N. J. 08540