- Email: [email protected]
Spontaneous spin-flip Raman linewidth and nonlinear processes in InSb
Volume 8, number 3
OPTICS COMMUNICATIONS
SPONTANEOUS
July 1973
SPIN-FLIP RAMAN LINEWIDTH
AND NONLINEAR
PROCESSES
IN InSb *
S.R.J. BRUECK and A. MOORADIAN Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts 021 73, USA Received 3 May 1973
A small-signal gain technique has been used to measure the lineshape of spontaneous spin-flip Raman scattering as a function of magnetic field (H = 0.5-10 kG) for an electron concentration n = 10 is cm -3 at T = 2°K with both photons propagating normal to H. Four-wave mixing processes have been observed for varying carrier concentrations together with an interference between the resonant spin-flip nonlinearity and the nonresonant nonlinearity resulting from conduction electron nonparabolicity.
A direct measurement of the gain of spin-flip Raman scattering has been made by focusing two laser beams at frequencies 661 and 662 into a sample of n-type InSb in a magnetic field and varying the magnetic field through the spin-resonance condition; cos = 6o 1 - 602. If the laser beams are properly polarized there is gain at the Stokes frequency 602 and loss at the pump frequency 661" At sufficiently low pump power levels, this technique allows a direct measurement of the spin-flip Raman lineshape [ 1 ]. This lineshape plays an important role in determining the gain and tuning characteristics [2] of the spin-flip laser as well as in controlling the frequency modulation and fast pulse generation properties of the spinflip laser under the application of small electric fields [3]. Because the spin-flip linewidth is extremely narrow [less than 100 MHz (0.003 cm - 1 ) for some experimental conditions], this lineshape is not accessible by conventional spectroscopic techniques. The degenerate four-wave mixing processes 663 = 2w I -- 602 and 664 = 26°2 - 661 have been observed for varying electron concentrations. There are b o t h resonant and nonresonant contributions to the third order nonlinear susceptibility which describes these mixing processes. An interference phenomenon is ob* This work was sponsored by the Department of the Air Force.
served between the resonant contribution arising from the spin-flip interaction and the nonresonant contribution arising from conduction electron nonparabolicity [4]. Since the concentration dependences of these effects are different, the interference effects are more pronounced for higher electron concentrations. By adjusting one laser to generate two output frequencies on adjacent CO laser transitions w 1 and COl, the nondegenerate four-wave mixing processes t t 603 = 661 -- 602 + 661 and 664 = 661 - 602 + 661 have been observed. These processes allow the observation of the resonant nonlinearity for larger frequency shifts w 1 - 662 under phase matched conditions t since 661 - 661 is small. In the gain experiment, two single frequency TEM00 laser beams are combined with a Ge beam splitter, antireflection coated on one surface. It is important that the two beams be collinear since any component of the scattering wavevector, q = k 1 - k2, parallel to the magnetic field can give rise to substantial broadening of the observed lineshape [5]. In the present experiments, the angle between k 1 and k 2 w a s less than 1 o inside the sample and had a negligible effect on the observed lineshape. The sample was immersed in superfluid helium (T ~ 2°K) in a magnetic field (H ~ 10 kG). The exit face of the sample was ground with a 5/am grit in order to diffusely scatter the laser 263
V o l u m e 8, n u i n b e r 3
OPTICS COMMUNICATIONS
beams and minimize cavity effects. The co 1 laser beam was chopped and the gain was measured by synchronousiy detecting the transtnitted power at co2" This resulted in a signal proportional to {exp[g(H)l] ~:, wheie g(H) is the spin-flip Raman gain which is proportional to the laser power at col, l is the sample leng[h and {J is a field independent parameter somewhat less than umty which indicates that there is a small amount of nrodulated power at co2 for magnetic fields far fron] the resonance conditions. In the small signal reglrne, the signal is linearly related to the gai~] and hence the lmeshape. The experiments reported here were carried out at two power levels of the pump radiauon to insure that they were in the linear regm]e. The result of this experiment for a frequency dift'e~ence co 1 .... co2 = 4.17 cm l is shown in fig. 1. The observed lineshape is symmetric and has a full width at half height of 3 G which corresponds to a frequency width of 200 MHz at the measuring tuning rate of 67.5 MHz/G. The resolution is better than 1 MHz and is limited only by the frequency instabilities betweei1 the two lasers. The variation in this linewidth as a lhnction of magnetic field is shown in fig. 2. At all magnetic fields the lineshape was symrnetric and approximately lorentzian. Because these experiments were carried out close to the interband resonance in InSb (the energy gap at zero field is approxirnately 1888 c m - 1 ) , the measurements were repeated for several values of col • Variation of" the pump photon energy had no effect on the linewidth to within experimental error. At the lowest magnetic fields the measured linewidth is comparable to, but somewhat broader than, linewidths that have been obtained in microwave electron spin resonance measurements [6]. The differing matrix elements for
]oo< •
J u l y 1973
-i wl: 1868.] cm-
.
THEORY -FIXED COLLISION
"~~
. co1 = 1888.3cm-I
PARAMETERS
- w1 = 1893.5cm-
~_.=
l: 0
L
t
1
2
~
±
4
•
t
6
I~_ 1780
L
,
,,
/
I
8
~
To
T h e solid c u r v e s are c a l c u l a t e d l i n e w i d t h s for the f i x e d collision p a r a m e t e r s r s = 2 x 10 9 sec, r p = 4 X 10 12 sec, T e = 1 0 ° K a n d the field d e p e n d e n t collision p a r a m e t e r s r s = 1 X 10 8/H3/2 sec, r p = 3 X 1 0 - 1 3 t t s e c a n d T e = 10 ° K .
the Raman and the spin resonance absorption processes may explain this linewidth difference. The dependence of linewidth on magnetic field is approximately exponential throughout the entire range studied. This is somewhat surprising since the electron gas passes from a degenerate regime (e F > kT) with several Landau levels occupied at H = 500 G into a nondegenerate regime (e F < kT) in the quantum limit (only n = 0 Landau level occupied) for H = 10 kG. Homogeneous spin relaxation broadening and inhomogeneous broadening due to conduction band nonparabolicity both contribute to the lineshape of spin-flip scattering. This nonparabolicity results from
I
i
L
,
1785
~
,
~
t_l
i
t790
I
i
J
I
t795
MAGNETIC FIELD (G) Fig. 1. Gain of spin-flip Rarnan scattering (n = 8 X 1014 264
'~
Fig. 2. L i n e w i d t h o f s p o n t a n e o u s spin-flip R a m a n s c a t t e r i n g vs. m a g n e t i c field s t r e n g t h . (n = 8 X 1014 c m - 3 , T = 2 ° K ) .
Linewidth : 3G <=i>200 MHz
~
~
H (kG)
/ ~ 4
,
T I
i !
~J
1775
]
!
_____~ /
L I
.
THEORY FIELD DEPENDENT COLLISION PARAMETERS
t0t4cm- 3 nT==7-92K x
.
A
wl-lBBO'4cm
30
.
]/
-1 SO
.
crn
3, COl
_
O92
=
4.17 cm -1 , T = 2°K).
Volume 8, number 3
OPTICS COMMUNICATIONS
the interaction between the conduction and valence bands and is given to lowest order in kz, the electron momentum along the magnetic field, by cos(kz) = cosO(1-h2k2z/mcEg) where cos0 is the b o t t o m of the band value of the spin energy, m c the effective mass and Eg the energy gap. A phenomenological theory of the spin-flip lineshape involving these two broadening mechanisms has been previously presented [5]. The spin-relaxation gives rise to a simple linewidth broadening characterized by the collision time r s. The nonparabolicity broadening is considerably influenced by the presence of orbital collisions which alter the momentum of a spin excitation. These collisions, characterized by the time rp, result in a motional narrowing of the nonparabolicity contribution to the linewidth. In the limit, valid for this electron concentration, in which these orbital collisions are rapid compared with the extent of the inhomogeneous nonparabolicity broadening, the calculated lineshape of spontaneous spin-flip light scattering is lorentzian and the linewidth is given by P ~ 2/r s + 2rp [(co 2) - (cos)2 ] ,
(1)
where the angular brackets denote averages over the electronic m o m e n t u m k z weighted by the Fermi function. These averages can be readily evaluated for T = 0°K and the resulting expression for the linewidth in the quantum limit is p ~ 2/rs + a2 r pWs0CF/~g ,2 . 2 1 ~ 2 ,
(2)
where the Fermi energy e F is proportional to I / H 2. Thus, the nonparabolicity contribution to the linewidth is proportional to 1/H 2. This results from the increasing density of states at the b o t t o m of the conduction band as the magnetic field is increased. For k T e ~> eF, where T e is the electron temperature t , the spread of k z values no longer decreases as the magnetic field is increased and the linewidth becomes an increasing function of magnetic field. In the limit k T e >> eF, in which the electron distribution is maxwellian, the linewidth expression is 2 2 /Eg2 , P ~ 2/r s + 2rpcoso(kTe) (3) ~ Electrical measurements indicate that the temperature in the interaction volume is raised somewhat above the helium bath temperature, cf. ref. [3].
July 1973
and the nonparabolicity contribution to the linewidth is proportional to H 2. The solid curve in fig. 2 labeled fixed collision parameters shows the theoretical variation in the linewidth for r s -- 2 × 10 9sec, rp = 4 X 10-12sec, and T e = 10°K. The value o f t s was chosen to fit the low H data and rp was picked to force the theory and experiment to agree at H = 7.5 kG. The theoretical lineshape evaluation has been restricted to the quantum limit for simplicity. Clearly, this simple model with magnetic field and energy independent collision times is not adequate to completely explain the data. Complete detailed microscopic theories of the spin-flip Raman line shape are not presently available. The present microscopic theories [7, 8] are limited to the T = 0°K nonparabolicity dominated regime and are not immediately extendable to the present parameter regime. Previous calculations of the magnetic field dependence of the spin-relaxation time and the m o m e n t u m relaxation time indicate that these magnetic field dependences are in the proper direction to fit the data. Pavlov [10] has evaluated the spin-lattice relaxation time, which is a contribution to the r s scattering, due to acoustic phonon scattering and finds 1/Tla c c~ TH3/2. Davies [7] has evaluated the field dependence of r~ at T = 0°K and finds rp oc H for low magnetic ~eields. The second curve in fig. 2 shows the calculated linewidths for r s = 1 X 1 0 - 8 / H 3/2 sec and rp = 3 X 10 -13 H sec. Both of these variations are in the proper direction to fit the data but in the absence of a complete mmroscopic theory which includes the temperature and energy dependences of these parameters they cannot be used to provide more than a rough correction to the model. The results of a degenerate four-wave frequency mixing experiment are shown in fig. 3 for a magnetic field of 535 G with a frequency separation co 1 - °°2 = 1.25 cm -1 in a sample of concentration n = 1 X 1014 cm - 3 . In this experiment, col a n d co 2 are mixed to produce outputs at co3 = 2°°1 - (-02 and 004 = 2co2 - col" These outputs result from the modulation of the InSb polarizability due to the spin-flip process t and are resonant for cos = col - 0°2" The top trace shows the transmitted power on resonance with the two new frequencies 603 and 0°4" The second ~ This modulation of the polarizability has also been utilized to generate electromagnetic radiation at the difference frequency ool - co2 [10]. 265
Volume 8, number 3
OPTICS COMMUNICATIONS
1 × 1016 cm 3. In addition the nondegenerate fourwave mixing processes co 3 = co I - o02 + C°'l and 004 = o0'1 - 0o2 + C°l were observed by adjusting one of the CO lasers so that it oscillated on two adjacent CO laser transitions, co 1 and o0'1- These nonlinear mixing processes will occur during the operation of the spin-flip laser as well as in this experimental configuration. Thus a m u l t i m o d e p u m p laser will result in a m u l t i m o d e spin-flip laser and this must be taken into account in the use of the spin-flip laser as a spectroscopic source.
Goin x t 0
H = 535G
We would like to thank R.W. Davies, P.L. Kelley, and H.J. Zeiger for helpful discussions and D.J. Wells, W. Laswell, and T.A. Lind for e x p e r t technical assistance.
| =
_1 t896
References I
I
1894 FREQUENCY
1
I
t892
L
t890
(cm -t)
Fig. 3. Observation of the degenerate four-wave mixing processes co3 = 2col - °°2 and co4 = 2~32 - COl due to the spinflip nonlinearity (n = 1 X 1014 c m - , COl - CO2 = 1.25 cm -1, T= 2°K). trace was taken with H = 0 and shows no f r e q u e n c y mixing. The b o t t o m trace was again taken at H = 0 at a lower gain level to show the relative intensities at co 1 and co 2. There is a n o n r e s o n a n t nonlinearity even at zero magnetic field in InSb due to the nonparabolicity of the c o n d u c t i o n band [3]. The nonparabolicity induced nonlinearity is p r o p o r t i o n a l to the electron density, n, while the spin-flip nonlinearity at low m a ~ e t i c fields, where several Landau levels are p o p u l a t e d , is p r o p o r t i o n a l to the electron density within k T e of the Fermi surface or n 2/3. T h e r e f o r e the ratio b e t w e e n the n o n p a r a b o l i c i t y and the spinflip nonlinearities should increase as the electron c o n c e n t r a t i o n is increased. The n o n p a r a b o l i c i t y nonlinearity is n o t evident in the data of fig. 3 because of the low e l e c t r o n c o n c e n t r a t i o n ; however, a strong interference t b e t w e e n the resonant and n o n r e s o n a n t c o n t r i b u t i o n s to the third order nonlinear susceptibility was observed in a sample of c o n c e n t r a t i o n n = t Similar interference effects in the nonlinear susceptibility involving cyclotron frequency resonances have been observed by Yablonovitch et al. [ 11 ]. 266
July 197 3
[ 1 ] Preliminary accounts of this work have been presented previously: A. Mooradian, Proc. o f Esfahan Syrup. on Fundamental and Applied Laser Physics, Iran, 1971, M. Feld, A. Javan and N. Kurnit, Eds., (John Wiley, N.Y. - to be published); S.R.J. Brueck, Bull. Amer. Phys. Soc. 18 (1973) 400. [2] S.R.J. Brueck and A. Mooradian, Appl. Phys. Letters 18 (1971) 229. [3] A. Mooradian, S.R.J. Brueck, E.J. Johnson and J.A. Rossi, Appl. Phys. Letters 21 (1972) 482. [4] C.K,N. Patel, R.E. Slusher and P.A. Fleury, Phys. Rev. Letters 17 (1966) 1011; P.A. Wolff and Gary A. Pearson, Phys. Rev. Letters 17 (1966) 1015. [51 S.R.J. Brueck and F.A. Blum, Phys. Rev. Letters 28 (1972) 1458; S.R.J. Brueck, A. Mooradian and F.A. Blum (to be published). [6] R.A. Isaacson, Phys. Rev. 169 (1968) 312; E.M. Gershenzon, N.M. Pevin and M.S. Fogel'son, Fizika Tverdogo Tela 10 (1968) 2880 [Translation: Soviet Physics-Solid State 10 (1969) 2278]. [7] R.W. Davies, Phys. Rev. B (to be published); R.W. Davies, Bull. Amer. Phys. Soc. 17 (1972) 335. [8] Y.C.S, Auyang, Ph.D. Thesis (MIT, 1971) (unpublished). [9] S.T. Pavlov, Fizika Tverdogo Tela 8 fi966) 900 [Translation: Soviet Physics-Solid State 8 (1966) 719]. [10] Van Tran Nguyen and T.J. Bridges, Phys. Rev. Letters 29 (1972) 359; see also Terrence L. Brown and P.A. Wolff, Phys. Rev. Letters 29 (1972) 362. [ 11 ] E. Yablonoviteh, N. Bloembergen and J.J. Wynne, Phys. Rev. B 3 (1971) 2060.
OPTICS COMMUNICATIONS
SPONTANEOUS
July 1973
SPIN-FLIP RAMAN LINEWIDTH
AND NONLINEAR
PROCESSES
IN InSb *
S.R.J. BRUECK and A. MOORADIAN Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts 021 73, USA Received 3 May 1973
A small-signal gain technique has been used to measure the lineshape of spontaneous spin-flip Raman scattering as a function of magnetic field (H = 0.5-10 kG) for an electron concentration n = 10 is cm -3 at T = 2°K with both photons propagating normal to H. Four-wave mixing processes have been observed for varying carrier concentrations together with an interference between the resonant spin-flip nonlinearity and the nonresonant nonlinearity resulting from conduction electron nonparabolicity.
A direct measurement of the gain of spin-flip Raman scattering has been made by focusing two laser beams at frequencies 661 and 662 into a sample of n-type InSb in a magnetic field and varying the magnetic field through the spin-resonance condition; cos = 6o 1 - 602. If the laser beams are properly polarized there is gain at the Stokes frequency 602 and loss at the pump frequency 661" At sufficiently low pump power levels, this technique allows a direct measurement of the spin-flip Raman lineshape [ 1 ]. This lineshape plays an important role in determining the gain and tuning characteristics [2] of the spin-flip laser as well as in controlling the frequency modulation and fast pulse generation properties of the spinflip laser under the application of small electric fields [3]. Because the spin-flip linewidth is extremely narrow [less than 100 MHz (0.003 cm - 1 ) for some experimental conditions], this lineshape is not accessible by conventional spectroscopic techniques. The degenerate four-wave mixing processes 663 = 2w I -- 602 and 664 = 26°2 - 661 have been observed for varying electron concentrations. There are b o t h resonant and nonresonant contributions to the third order nonlinear susceptibility which describes these mixing processes. An interference phenomenon is ob* This work was sponsored by the Department of the Air Force.
served between the resonant contribution arising from the spin-flip interaction and the nonresonant contribution arising from conduction electron nonparabolicity [4]. Since the concentration dependences of these effects are different, the interference effects are more pronounced for higher electron concentrations. By adjusting one laser to generate two output frequencies on adjacent CO laser transitions w 1 and COl, the nondegenerate four-wave mixing processes t t 603 = 661 -- 602 + 661 and 664 = 661 - 602 + 661 have been observed. These processes allow the observation of the resonant nonlinearity for larger frequency shifts w 1 - 662 under phase matched conditions t since 661 - 661 is small. In the gain experiment, two single frequency TEM00 laser beams are combined with a Ge beam splitter, antireflection coated on one surface. It is important that the two beams be collinear since any component of the scattering wavevector, q = k 1 - k2, parallel to the magnetic field can give rise to substantial broadening of the observed lineshape [5]. In the present experiments, the angle between k 1 and k 2 w a s less than 1 o inside the sample and had a negligible effect on the observed lineshape. The sample was immersed in superfluid helium (T ~ 2°K) in a magnetic field (H ~ 10 kG). The exit face of the sample was ground with a 5/am grit in order to diffusely scatter the laser 263
V o l u m e 8, n u i n b e r 3
OPTICS COMMUNICATIONS
beams and minimize cavity effects. The co 1 laser beam was chopped and the gain was measured by synchronousiy detecting the transtnitted power at co2" This resulted in a signal proportional to {exp[g(H)l] ~:, wheie g(H) is the spin-flip Raman gain which is proportional to the laser power at col, l is the sample leng[h and {J is a field independent parameter somewhat less than umty which indicates that there is a small amount of nrodulated power at co2 for magnetic fields far fron] the resonance conditions. In the small signal reglrne, the signal is linearly related to the gai~] and hence the lmeshape. The experiments reported here were carried out at two power levels of the pump radiauon to insure that they were in the linear regm]e. The result of this experiment for a frequency dift'e~ence co 1 .... co2 = 4.17 cm l is shown in fig. 1. The observed lineshape is symmetric and has a full width at half height of 3 G which corresponds to a frequency width of 200 MHz at the measuring tuning rate of 67.5 MHz/G. The resolution is better than 1 MHz and is limited only by the frequency instabilities betweei1 the two lasers. The variation in this linewidth as a lhnction of magnetic field is shown in fig. 2. At all magnetic fields the lineshape was symrnetric and approximately lorentzian. Because these experiments were carried out close to the interband resonance in InSb (the energy gap at zero field is approxirnately 1888 c m - 1 ) , the measurements were repeated for several values of col • Variation of" the pump photon energy had no effect on the linewidth to within experimental error. At the lowest magnetic fields the measured linewidth is comparable to, but somewhat broader than, linewidths that have been obtained in microwave electron spin resonance measurements [6]. The differing matrix elements for
]oo< •
J u l y 1973
-i wl: 1868.] cm-
.
THEORY -FIXED COLLISION
"~~
. co1 = 1888.3cm-I
PARAMETERS
- w1 = 1893.5cm-
~_.=
l: 0
L
t
1
2
~
±
4
•
t
6
I~_ 1780
L
,
,,
/
I
8
~
To
T h e solid c u r v e s are c a l c u l a t e d l i n e w i d t h s for the f i x e d collision p a r a m e t e r s r s = 2 x 10 9 sec, r p = 4 X 10 12 sec, T e = 1 0 ° K a n d the field d e p e n d e n t collision p a r a m e t e r s r s = 1 X 10 8/H3/2 sec, r p = 3 X 1 0 - 1 3 t t s e c a n d T e = 10 ° K .
the Raman and the spin resonance absorption processes may explain this linewidth difference. The dependence of linewidth on magnetic field is approximately exponential throughout the entire range studied. This is somewhat surprising since the electron gas passes from a degenerate regime (e F > kT) with several Landau levels occupied at H = 500 G into a nondegenerate regime (e F < kT) in the quantum limit (only n = 0 Landau level occupied) for H = 10 kG. Homogeneous spin relaxation broadening and inhomogeneous broadening due to conduction band nonparabolicity both contribute to the lineshape of spin-flip scattering. This nonparabolicity results from
I
i
L
,
1785
~
,
~
t_l
i
t790
I
i
J
I
t795
MAGNETIC FIELD (G) Fig. 1. Gain of spin-flip Rarnan scattering (n = 8 X 1014 264
'~
Fig. 2. L i n e w i d t h o f s p o n t a n e o u s spin-flip R a m a n s c a t t e r i n g vs. m a g n e t i c field s t r e n g t h . (n = 8 X 1014 c m - 3 , T = 2 ° K ) .
Linewidth : 3G <=i>200 MHz
~
~
H (kG)
/ ~ 4
,
T I
i !
~J
1775
]
!
_____~ /
L I
.
THEORY FIELD DEPENDENT COLLISION PARAMETERS
t0t4cm- 3 nT==7-92K x
.
A
wl-lBBO'4cm
30
.
]/
-1 SO
.
crn
3, COl
_
O92
=
4.17 cm -1 , T = 2°K).
Volume 8, number 3
OPTICS COMMUNICATIONS
the interaction between the conduction and valence bands and is given to lowest order in kz, the electron momentum along the magnetic field, by cos(kz) = cosO(1-h2k2z/mcEg) where cos0 is the b o t t o m of the band value of the spin energy, m c the effective mass and Eg the energy gap. A phenomenological theory of the spin-flip lineshape involving these two broadening mechanisms has been previously presented [5]. The spin-relaxation gives rise to a simple linewidth broadening characterized by the collision time r s. The nonparabolicity broadening is considerably influenced by the presence of orbital collisions which alter the momentum of a spin excitation. These collisions, characterized by the time rp, result in a motional narrowing of the nonparabolicity contribution to the linewidth. In the limit, valid for this electron concentration, in which these orbital collisions are rapid compared with the extent of the inhomogeneous nonparabolicity broadening, the calculated lineshape of spontaneous spin-flip light scattering is lorentzian and the linewidth is given by P ~ 2/r s + 2rp [(co 2) - (cos)2 ] ,
(1)
where the angular brackets denote averages over the electronic m o m e n t u m k z weighted by the Fermi function. These averages can be readily evaluated for T = 0°K and the resulting expression for the linewidth in the quantum limit is p ~ 2/rs + a2 r pWs0CF/~g ,2 . 2 1 ~ 2 ,
(2)
where the Fermi energy e F is proportional to I / H 2. Thus, the nonparabolicity contribution to the linewidth is proportional to 1/H 2. This results from the increasing density of states at the b o t t o m of the conduction band as the magnetic field is increased. For k T e ~> eF, where T e is the electron temperature t , the spread of k z values no longer decreases as the magnetic field is increased and the linewidth becomes an increasing function of magnetic field. In the limit k T e >> eF, in which the electron distribution is maxwellian, the linewidth expression is 2 2 /Eg2 , P ~ 2/r s + 2rpcoso(kTe) (3) ~ Electrical measurements indicate that the temperature in the interaction volume is raised somewhat above the helium bath temperature, cf. ref. [3].
July 1973
and the nonparabolicity contribution to the linewidth is proportional to H 2. The solid curve in fig. 2 labeled fixed collision parameters shows the theoretical variation in the linewidth for r s -- 2 × 10 9sec, rp = 4 X 10-12sec, and T e = 10°K. The value o f t s was chosen to fit the low H data and rp was picked to force the theory and experiment to agree at H = 7.5 kG. The theoretical lineshape evaluation has been restricted to the quantum limit for simplicity. Clearly, this simple model with magnetic field and energy independent collision times is not adequate to completely explain the data. Complete detailed microscopic theories of the spin-flip Raman line shape are not presently available. The present microscopic theories [7, 8] are limited to the T = 0°K nonparabolicity dominated regime and are not immediately extendable to the present parameter regime. Previous calculations of the magnetic field dependence of the spin-relaxation time and the m o m e n t u m relaxation time indicate that these magnetic field dependences are in the proper direction to fit the data. Pavlov [10] has evaluated the spin-lattice relaxation time, which is a contribution to the r s scattering, due to acoustic phonon scattering and finds 1/Tla c c~ TH3/2. Davies [7] has evaluated the field dependence of r~ at T = 0°K and finds rp oc H for low magnetic ~eields. The second curve in fig. 2 shows the calculated linewidths for r s = 1 X 1 0 - 8 / H 3/2 sec and rp = 3 X 10 -13 H sec. Both of these variations are in the proper direction to fit the data but in the absence of a complete mmroscopic theory which includes the temperature and energy dependences of these parameters they cannot be used to provide more than a rough correction to the model. The results of a degenerate four-wave frequency mixing experiment are shown in fig. 3 for a magnetic field of 535 G with a frequency separation co 1 - °°2 = 1.25 cm -1 in a sample of concentration n = 1 X 1014 cm - 3 . In this experiment, col a n d co 2 are mixed to produce outputs at co3 = 2°°1 - (-02 and 004 = 2co2 - col" These outputs result from the modulation of the InSb polarizability due to the spin-flip process t and are resonant for cos = col - 0°2" The top trace shows the transmitted power on resonance with the two new frequencies 603 and 0°4" The second ~ This modulation of the polarizability has also been utilized to generate electromagnetic radiation at the difference frequency ool - co2 [10]. 265
Volume 8, number 3
OPTICS COMMUNICATIONS
1 × 1016 cm 3. In addition the nondegenerate fourwave mixing processes co 3 = co I - o02 + C°'l and 004 = o0'1 - 0o2 + C°l were observed by adjusting one of the CO lasers so that it oscillated on two adjacent CO laser transitions, co 1 and o0'1- These nonlinear mixing processes will occur during the operation of the spin-flip laser as well as in this experimental configuration. Thus a m u l t i m o d e p u m p laser will result in a m u l t i m o d e spin-flip laser and this must be taken into account in the use of the spin-flip laser as a spectroscopic source.
Goin x t 0
H = 535G
We would like to thank R.W. Davies, P.L. Kelley, and H.J. Zeiger for helpful discussions and D.J. Wells, W. Laswell, and T.A. Lind for e x p e r t technical assistance.
| =
_1 t896
References I
I
1894 FREQUENCY
1
I
t892
L
t890
(cm -t)
Fig. 3. Observation of the degenerate four-wave mixing processes co3 = 2col - °°2 and co4 = 2~32 - COl due to the spinflip nonlinearity (n = 1 X 1014 c m - , COl - CO2 = 1.25 cm -1, T= 2°K). trace was taken with H = 0 and shows no f r e q u e n c y mixing. The b o t t o m trace was again taken at H = 0 at a lower gain level to show the relative intensities at co 1 and co 2. There is a n o n r e s o n a n t nonlinearity even at zero magnetic field in InSb due to the nonparabolicity of the c o n d u c t i o n band [3]. The nonparabolicity induced nonlinearity is p r o p o r t i o n a l to the electron density, n, while the spin-flip nonlinearity at low m a ~ e t i c fields, where several Landau levels are p o p u l a t e d , is p r o p o r t i o n a l to the electron density within k T e of the Fermi surface or n 2/3. T h e r e f o r e the ratio b e t w e e n the n o n p a r a b o l i c i t y and the spinflip nonlinearities should increase as the electron c o n c e n t r a t i o n is increased. The n o n p a r a b o l i c i t y nonlinearity is n o t evident in the data of fig. 3 because of the low e l e c t r o n c o n c e n t r a t i o n ; however, a strong interference t b e t w e e n the resonant and n o n r e s o n a n t c o n t r i b u t i o n s to the third order nonlinear susceptibility was observed in a sample of c o n c e n t r a t i o n n = t Similar interference effects in the nonlinear susceptibility involving cyclotron frequency resonances have been observed by Yablonovitch et al. [ 11 ]. 266
July 197 3
[ 1 ] Preliminary accounts of this work have been presented previously: A. Mooradian, Proc. o f Esfahan Syrup. on Fundamental and Applied Laser Physics, Iran, 1971, M. Feld, A. Javan and N. Kurnit, Eds., (John Wiley, N.Y. - to be published); S.R.J. Brueck, Bull. Amer. Phys. Soc. 18 (1973) 400. [2] S.R.J. Brueck and A. Mooradian, Appl. Phys. Letters 18 (1971) 229. [3] A. Mooradian, S.R.J. Brueck, E.J. Johnson and J.A. Rossi, Appl. Phys. Letters 21 (1972) 482. [4] C.K,N. Patel, R.E. Slusher and P.A. Fleury, Phys. Rev. Letters 17 (1966) 1011; P.A. Wolff and Gary A. Pearson, Phys. Rev. Letters 17 (1966) 1015. [51 S.R.J. Brueck and F.A. Blum, Phys. Rev. Letters 28 (1972) 1458; S.R.J. Brueck, A. Mooradian and F.A. Blum (to be published). [6] R.A. Isaacson, Phys. Rev. 169 (1968) 312; E.M. Gershenzon, N.M. Pevin and M.S. Fogel'son, Fizika Tverdogo Tela 10 (1968) 2880 [Translation: Soviet Physics-Solid State 10 (1969) 2278]. [7] R.W. Davies, Phys. Rev. B (to be published); R.W. Davies, Bull. Amer. Phys. Soc. 17 (1972) 335. [8] Y.C.S, Auyang, Ph.D. Thesis (MIT, 1971) (unpublished). [9] S.T. Pavlov, Fizika Tverdogo Tela 8 fi966) 900 [Translation: Soviet Physics-Solid State 8 (1966) 719]. [10] Van Tran Nguyen and T.J. Bridges, Phys. Rev. Letters 29 (1972) 359; see also Terrence L. Brown and P.A. Wolff, Phys. Rev. Letters 29 (1972) 362. [ 11 ] E. Yablonoviteh, N. Bloembergen and J.J. Wynne, Phys. Rev. B 3 (1971) 2060.