Table 1 of the experimental CSI
~(10-24 cm2) a(l/sec
The author is deeply indebted to Dr. W. G. Proctor, Zurich, Switzerland for may stimulating discussions.
(7) 35 1536
where c is the distance between the two ions, ajl,f and aQ are the transition probabilities for S/So =$, per Unit SOUiC VOltage, 'yMiS the hexadecapole antishielding factor calculated by Sternheimer [i’] for Cs. The expected value of Ml6 is of order of 1O-48 cm4. The measured limit is lo-46 cm4, which is not surprising.
M= 71 -___ = 2ii2
12(1 (I +- 2)(2z 1)(2Z +- 3)(2z 1)(21 +- 5,3) ,2 ,2 M2&)
1. A.Kastler, Experientia, 8 (1952) 1. 2. W. G. Proctor and W. N. Tantilla, Phys. Rev. 101 (1956) 175’7. 3. W. G. Proctor and W. Robinson, Phys. Rev. 104 (1956) 1344. 4. D. I. Bolef, Scientific Paper (Westinghouse Research Laboratory), 63-128-108-PL (1963). 5. A. Tzalmona, Ph.D. Thesis (unpublished), submitted to the Feinberg Graduate School, Weizmann Institute of Science, Rehovoth, Israel, November 1966. 6. R. M. Sternheimer, Phys. Rev. 80 (1950) 102. 7. R. M.Sternheimer, Phys. Rev. 146 (1966) 140.
where m depends on the fourth derivatives of the electric potential and is calculated for cubic NaCl model in which longitudinal sound waves are propagating; p(w) is the density of states at four times the Larmor frequency. No saturation effect is obtained for )A m ) = 4 for all cases. The upper limit of the hexadecapole moment is obtained using the ratio of the hexadecapole and quadrupole transition probabilities :
OBSERVATION EMISSION OF
OF THE PHONONS
STIMULATED IN PbTe
C. A. NANNEY Bell
We report the first observation of breaks in the I-V characteristics of a nonpiezoelectric semiconductor which occur when the drift velocity reaches the transverse and the longitudinal sound velocities. The stimulated emission of phonons by electrons is observed to produce a spatial dependence in the electrical conductivity of PbTe.
The electron-phonon coupling mechanism in nonpiezoelectric materials acts through the deformation potential and is considerably weaker than in piezoelectric materials except at very high frequencies. Even for high frequencies, unless the sound wave length, AS, is less than the Thomas-Fermi (or Debye) screening length, AT-F, the effective coupling is greatly reduced as a consequence of space charge effects unless 66
the material is multivalleyed [l]. Multivalleyed materials are, for example, n-Ge and n,p-TbTe which have band minima at the zone edge in the (111) direction. In Ge hot electron effects  complicate the observation of the electron-phonon interaction. Lead telluride in contrast has a large Fermi velocity, a large mobility *, a low trans* Typical V-set
mobilities at 4oK.
are 106 cm2/
18 December 196’7
an effect on the conductivity of a semiconductor as a result of the departure of the phonon bath from thermal equilibrium due to phonon emission by the current carriers and concluded that the effect on carrier transport can be considerable when the drift velocity of the carriers exceeds the sound velocity. Conwell’s expression ** for the generation rate dNi,dt of longitudinal phonons is dN; dt
Te .T+ C
‘“Dqcos 2 k,T
m N;(Z)DCOS x -
Fig. 1. The resistance change near one end of a sample of n-PbTe as a function of current magnitudeand direction. The current is parallel to the (100) direction and the overall length of the sample in 5.29 mm. The characteristics were measured 3.7~s after the beginning of the current pulse. The concentration in the sample is n = 2.0 x 1017/cm3. verse sound velocity only about $ that of Ge and, as well be shown, exhibits no hot electron effects prior to the electron velocity reaching the sound velocity. We have discussed  the observation of breaks in the Z-V characteristics of n-type PbTe at drift velocities equaling the sound velocities at cryogenic temperatures (40K and 77oK). The samples are ohmic until the drift velocity reaches the transverse sound velocity at which point the resistance increases suddenly by about S%, with only a small increase in current. This is followed by another break in the I- V characteristics and a larger increase in resistivity at the longitudinal sound velocity. An explanation of the mechanism of the breaks is the onset of stimulated emission [4,5] of phonons occurring when the electron velocity distribution becomes inverted with respect to the thermal phonon velocities. Conwell  predicted
The first two terms in the brackets correspond to spontaneous emission. The third term which is porportional to the phonon flux q describes stimulated emission or absorption depending upon whether the drift velocity is greater or less than the sound velocity. Fig. 1 shows AR& versus current for the two directions of current flow, when the voltage is measured over approximately one quarter of the sample length aajacent to one end. The conductivity of this sample segment depends markedly on the direction of current flow. This effect reverses with current directions when the voltage is measured at the other end of the sample. The change in resistivity is large when the electrons have drifted a long distance toward the voltage probe and small when they drift only a short distance. The difference between the two curves is attributed to convective amplification of phonons by stimulated emission. The difference must be due to the third term (stimulated phonon emission) of es.(l) since the first two are independent of the background phonon density and lead to a uniform spatial resistance, as in ohmic heating which is also independent of the direction of drift. The author is grateful to D. E. McCumber and H. N. Spector for valuable conversations. The expert technical assistance of J. P. Garno deserves special recognition. ** The reader is referred
to ref. 5 for the meaning of
1. 2. 3. 4. 5.
G. Weinreich et al. , Phys. Rev. 114 (1959) 33. S. H. Koenig et al. , Phys. Rev. 138 (1962) 1668. C. A. Nanney, Bull. Am. Phys. Sot., 11 (1966) 258. A. B. Pippard, Phil. Mag. (GB) 8 (1963) 161. E. Conwell, Phys. Letters 13 (1964) 285.