HAL Id: jpa-00216569
https://hal.archives-ouvertes.fr/jpa-00216569
Submitted on 1 Jan 1976
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
STUDY OF THE MOTT TRANSITION IN n-TYPE
CdS BY SPIN FLIP RAMAN SCATTERING AND
SPIN FARADAY ROTATION
S. Geschwind, R. Romestain, G. Devlin
To cite this version:
JOURNAL DE PHYSIQUE Colloque C4, supplkment au no 10, Tome 37, Octobre 1976, page C4-313
STUDY OF
THE MOTT TRANSITION IN n-TYPE
CdS
BY SPIN FLIP RAMAN SCATTERING
AND SPIN FARADAY ROTATION
S. GESCHWIND, R. ROMESTAIN
(t)
and G. E. DEVLIN Bell Laboratories, Murray Hill New Jersey, U. S. A.Rbsumb. - A une concentration de 2,4 x 1 0 1 7 cm-3 CdS dope n subit une transition metal-
isolant. Le caracthre de diffusion des electrons des donneurs est ktudie dans le regime metallique, en observant la largeur de raie Doppler, retrkcie par les collisions de la diffusion Raman par spin flip. La rotation Faraday associee aux klectrons des donneurs est utiliske pour mesurer la susceptibiIit6 de ces electrons ; elle montre, dans le c8te metallique de la region de transition, la fois une compo- sante dependante et une independante de la tempkrature. Nous interpretons nos rksultats comme dus
a
une seule phase d'un gaz d'aectrons fortement corr6les plut6t qu'a un systeme de deux phases. Abstract. - At a concentration of 2.4 X 1017 donors n-type CdS undergoes a metal-insulator transition. The diffusional character of the donor electrons is studied in the metallic regime by observing the collisionally narrowed Doppler linewidth of the spin-flip Raman scattering. The spin Faraday rotation of the donor electrons is used to measure the susceptibility of the electrons which shows in the transition region on the metallic side both temperature dependent and tempe- rature independent components. We interpret our results in terms of a single phase strongly corre- lated electron gas, rather than as a two phase system.1. Introduction.
-
n-type CdS, at a donor concen-tration of
-
2 x 1017, roughly corresponding to the SPIN FLIP R A M A N SCATTERING---
----
Mott criterion, undergoes a metal-non metal (M-NM)of the Mott-Anderson type. This transition has been studied by both transport measurements [I] and NMR measurements [2]. CdS, while difficult to obtain it1 as
high purity as Ge and Si, does have the simplifying
.,,
-~r:~::~~
ki-
-
a)ik,AL%skas
feature of being direct band gap with a spherical
-
RNvvrconduction band. Spin-flip Raman scattering (SFRS) was first proposed by Yafet [3] for conduction elec- a
b gl%H.
trons following a suggestion of P. A. Wolff 141 on ----. Raman scattering from Landau levels of carriers in a
semiconductor. A vast amount of work has been done
in InSb in connection with use of stimulated SFRS in the making of tuneable far infrared sources [5].
Thomas and Hopfield [6] first did SFRS on bound donors in CdS and were followed by Fleury and Scott [7] who studied the diffusional motion of the electrons as a function of temperature. In this paper we wish to report on a more detailed examination, using very high resolution SFRS techniques, of this diffusional
M-NM transition. motion in In a range addition, of samples we also spanning study the the
LC
STOKES LASER . A N T I -5
STOKEStransition via susceptibility data on these samples
n,
- of spin-tlip Raman ScatMng (SFRS). obtained by Faraday rotation.2. Outline of spin flip Raman scattering (SFRS).
-
rise to a Zeeman splitting w, =~ P H ,
of the donor The principle of SFRS is illustrated in figure 1. The electron. Incident light a t frequency oi and wave CdS sample is placed in a magnetic field No, giving vector ki flips a spin and is Raman scattered a t frequency mi-
w, (Stokes line) or wi+
w,
(Anti-(?) Work performed at Bell Laboratories while on leave from Stokes line) with a wave vector k,. The difference in
C. N. R. S. France. intensity shown between Stokes and-anti-Stokes
C4-314 S. GESCHWIND, R. ROMES TAIN AND G. E. DEVLIN
corresponds to the Boltzmann factor between the populations of the Zeeman levels for a typical field of 10
k G
and temperature of 2 K at which many experi- ments were done.The scattering can be represented as the classical radiation from a Raman, dipole
The operator D ( ~ ) is given by [8]
D(')
= (cix
E3
(a e-iw" -t C.C.) (1)where o are the Pauli matrix in the space oT the spin states
I
a>
andI
b>.
EL e-twit is the amplitude of the optical electric field.is the sum over excited states
I
n>
of matrix.elements of the electric dipole operator er. The cross section do/dQ N J cc J 2 is strongly enhanced when thefrequency mi approaches an appropriate excited state. This turns out to be the case when 4 880
A
Argon ion laser is used to scatter from localized donors, the significant excited state being then the exciton bound to the neutral donor which lies at 4 870A.
3. Study of the SFRS linewidth.
-
If the donorelectron is localized as it is in the dilute regime on the nonconducting side of the M-NM transition then the optical SFRS linewidth should simply reflect the EPR iinewidth be it due to TI, T, or an inhomogeneous T:. This indeed has been verified, as for example in sample
A listed in table I where the quoted SFRS linewidth of
Avrneas (cm-l)
p (2 K) Aveer (calc.1 0 = 90° N(cm-3) ohm-cm (cm-1) T = 2 K
0.003 cm-
'
is independent of scattering angle and also equal to the observed EPR linewidth. On the other hand, if the electron moves with velocity v, then the Raman line will have an additional Doppler shift given by Av = q.v where q is the scattering vector. For elec-tron velocities of v
-
lo7 cmls, corresponding to Fermi velocities for the concentration used, Av--
1O1%-l. On the other hand, the electrons undergo collisions ata rate approaching 1014 s-l, so that the Doppler shift is collisionally narrowed, givingrise to a diffusiond line whose effective linewidth is given by
Av = 4 z ( A v ~ ~ ~ ) ~ 2, =
(
4
-
sin -3':~.
(3)This diffusional broadening is already clearly pre- sent at a concentration N = 2.4 x 1017 where the SFRS linewidth is shown in figure 2 for forward
n ~ I T I STOKES
FIG. 2.
-
a ) 90' scattering in a sample with N = 2.4 x I01 7 cm-3.The lower portion of the figure 2a shows the Stokes and anti-Stokes lines on an expanded scale. The width of 0.01 cm-1 is due to the diffusional motion of the electrons ; b) Forward scattering. While there is a change of scale from a to b, both were
taken with the same resolution.
scattering (0 = 0, i. e., q = 0) and 0 = 900. At 900, Av = 0,01 cm-' whereas for 6 = 0, Av = 0.002 cm-I including an instrumental width of 0.001 cm-'. This
q 2 dependence of
Av
is further illustrated in figure 3for a more metallic sample with N = 1.4 x loi8 for
scattering angles close to the forward direction.
lnst rumental
width SIN*
2
2FIG. 3.
-
Small angIe forward scattering illustrating the diffu- sional motion of the carriers via the q 2 dependence of theSTUDY O F THE MOTT TRANSITION I N n-TYPE CdS C4-3 1 5
If we make the assumption that v may be related to the concentration, N, as in a free electron Fermi gas, then since p = m*/Ne2z,, eq. (3) may be written for
8 = 900 and m* = 0.2 as
The increased diffusional motion with increasing N and metallic character is illustrated in table I where the results are compared with eq. (4) 191. The agreement is better than one has reason to expect as ' ~ e r m i gas parameters are used in eq. (3), as if it were a free electron gas, when one is undoubtedly dealing with a strongly correlated electron system, especially at the onset of the metallic transition for N = 2.4 x 1017. At a concentration of 10'' and above one would expect eq. (3) to be more applicable. In addition, variations of linewidth of factors of two are observed in different regions of the same sample. This is undoub- tedly due to even slight inhomogeneities in concentra- tion as Av varies so rapidly with N in this regime. It should be emphasized that the SFRS technique probes a region of the sample that in our case is of the order of 100 y. Such variations of N are of course avera- ged in a typical transport measurement. Morever, some of the values of p were taken from reference [I], and may be somewhat different in our samples. Nonetheless, the increasing diffusion with increasing N is clear even though more detailed sample characteri- zation is needed to make more precise quantitative comparisons.
4. Measurements of
x
by Faraday rotation. - In addition to the off-diagonal component of the dipoleD(2) in eq. (1) which gives rises to SFRS, there is a
diagonal part associated with o,. This corresponds to a dipole radiating at the same frequency mi but rotated with respect to the incident polarisation, i. e.
This rotated dipole gives rise to a' Faraday rotation [lo] per unit lenght along Z given by
Since
As a can be made extremely large for the donors by choosing near resonant light excitation as described earlier for the SFRS, one has a way of measuring very selectively the
x
of donors independent of other magnetic impurities present in the sample. In figure 4a is shown the transmission of linearly polarized light through the sample as detected through a polaroid. The separation between peaks corresponds to 1800 of rotation. The rotation is plotted versus H in figure 4. There is a small temperature independent rotation, y,(b)
T= 1.63 K 4 8 8 0 A"
SAMPLE THICKNESS- 1.05 m m N = 7 x I O ' ~ / C C
FIG. 4. - Faraday rotation for localized centers. (v-q) follows a Brillouin function.
due to the interband transitions in pure CdS which is subtracted to give the spin Faraday rotation I) = (9-q).
(cp-y) follows a Brillouin function for S = 112 and
g = 1.79, appropriate for donors in CdS, within experimental error of no more than a few per cent. When R is studied as a function of temperature, a plot of 1/R versus T yields an antiferromagnetic Curie-Weiss 0 of 0.3 K (the same results are found
using 4 965
A
excitation although the rotations are smaller [lo]). This would suggest searching for a spin glass at lower temperatures.In the very metallic region, for example N = 6 x lo1',
x
shows no saturation with H and is independent of T as one would expect from a Pauli susceptibility, However, in the metallic transition region,x
displays both temperature dependent as well as temperature independent components. This is illustrated in figure 5 for N = 1.4 x 10''. This temperature dependent component increases with decreasing concentration and is fairly dominant for N = 2.4 x 1017.The variation of R with temperature, T, mirrors the variation of susceptibility with T in a given sample. To determine the absolute value of
X,
however, one'
r
,ab
C
i 6 1 @ 2TEMPERATURE ( K )
FIG. 5. - R = (dv/dH)
-
~0 plotted versus temperature. The ordinate on the left refers to 4 880 A data while that on the right to 4 265 A. The two scales were adjusted relative to each other s oC4-3 16 S. GESCHWIND, R. ROMESTAIN AND G. E. DEVLIN
needs specific knowledge of the quantity a in eq. (2). This may be easily calculated in the insulating limit where there is only one significant excited level corres- ponding to an exciton bound to the neutral donor (I,) whose position and oscillator strength is well known [6]. Using this exciton level, the calculated a agrees very -well with the magnitude of the rotation observed in figure 5. However, as one crosses to the metallic region with increasing concentration, the bound exciton loses meaning and the question arises as to the changing spectrum of the electron-hole excitation. In the case of the very metallic samples, the lowest such excitation corresponds to the energy needed to promote an electron from the top of the valence band to the Fermi level now in the conduction band and the oscillator strength is spread out among the unoccupied states. The most feasible way to determine a in this case would be experimentally by measuring the optical absorption spectrum. However, beyond the transition on the.metallic side even up to N
-
2 x loi8, it is observed that the luminescence feature corresponding to electron-hole recombination does not move and while it broadens somewhat, its width still remains small compared to the 5 meV separation from the laser line. Therefore up toN N 1018 we tentatively take a to be the same as in
the dilute limit for our analysis of absolute
x
in this region.Thus if we compare the absolute magnitude of the temperature dependent component of the rotation for
N = 1.4 x 1018 to that found per localized spin in the insulating region for N = 7 x loL6, we find that the temperature dependent part in figure 5 corresponds to approximately 1016 electrons with free moments or 1
%
of the total. The same figure is arrived at by comparing the ratio of the temperature depe:ldent and temperature independent parts assuming the latter to be Pauli like with a degeneracy temperatureTF = 300
K
corresponding to N = 1.4 x 1018. By contrast, for the N = 2.4 x 10" sample, which shows only a very small temperature independent susceptibi- lity, an analysis of in terms ofX,
+
x,,,.
shows that the number of Curie-Weiss electrons is comparableto the number of Pauli like electrons. Very similar results for
x
were obtained by Quirt and Marko [ l l ] for Si : P.5.
Discussion of results. - Some may be inclined to interpret the two components X, and x,.,. that are observed as arising from well defined microscopic regions of metallic and insulating material with diffus- ing and non diffusing electrons respectively, associated with the clustering due to basic statistical randomness in donor positions. However, we will now present evi- dence that there are no non-diffusing spins beyond the metallic transition which seems to argue against the simultaneous coexistence of two such phases. If there were non-diffusing spins imbedded on a metallic matrix we would certainly expect their number to be greatest close to the M-NM transition point as for example in the sample with N = 2.4 x 1017. If one looked at- 90° scattering in such a sample one would expect to see a narrow line with no angular dependence, superimposed on the broader diffusional line with q2 dependence. However one always sees only a single line with q2 dependence as shown in figure 2a and no evidence for non diffusing carriers.This is further illustrated in figure 3 where again the narrow line observed at B = 0 broadens continuously with increasing angle about B = 0, as q 2 at small scattering angle. We are inclined to believe that the temperature dependent component of
x
is coming from electrons in singly occupied sites of the correlated elec- tron motion [l 11. Because of randomness some or all of these singly occupied sites may even have fixed posi- tions in space. However, the electrons in these singly occupied sites are still in dynamic equilibrium with the rest of the electron sea and leave this site to be filled by another carrier replacing it. Equivalently stated, the electrons on singly occupied sites are Ander- son local moments or Friedel virtual bound states which while giving rise to a x,.,. still have diffusive character in the SFRS.Acknowledgments. - We wish to acknowledge very many helpful discussions with P. W. Anderson, T. M. Rice, P. A. Wolff and Y. Yafet.
References
[I] TOYOTOMI, S. and MORIGAKI, K., J. Phys. Soc. Japan 25 (1968) 807.
f2] ADAMS, F. D., LOOK, D. C., BROWN, L. C., LOCKER, D. R.,
Phys. Rev. B 4 (1971) 2115. [3] YAFET, Y., Phys. Rev. 152 (1956) 855.
[4] WOLFF, P. A., Phys. Rev. Lett. 16 (1966) 225.
[5] PATEL, C. K. N., Laser Spectroscopy, edited by R. G. Brewer and A. Mooradian (Plenum Publishing Co, N. Y., hr. Y.) 1974.
[6] THOMAS, D. G. and HOPFIELD, J. J., Phys. Rev. 175 (1968) 1021.
[7] SCOTT, J. F., DAMEN, J. C. and FLEURY, P. A., Phys. Rev. 6 (1972) 3856.
[8] ROMESTAIN, R., GESCHWIND, S., DEVLIN, G. E. and WOLFF, P. A. 33 (1974) 10.
[9] Preliminary results relating to this diffusional width in CdS were given earlier by R. Romestain, S. Geschwind and
P. A. Wolff. International Conference on Physics of Semiconductors, Stuttgart (1974). However, it contains misprints in the quoted values of Av in the table. [lo] ROMESTAIN, R., GESCHWIND, S. and DEVLIN, G. E., Phys.
Rev. Lett. 35 (1975) 803. See also
PERSHAN, VAN DER ZIEL and MALMSTROM, Phys. Rev. 143 (1966) 574 ;
S H E N , ~ . R. and BLOEMBERGEN, N., Phys. Rev. 143 (1966) 372; LE GALL, H., JAMET, J. P. and DESORMI~RES, B., in Light Scattering in Solids, edited by M. Balkanski (Flamma- rion, Paris) 1971.
[Ill QUIRT, J. D. and MARKO, J. R., Phys. Rev. 7 (1973) 3842. [I21 BRINKMAN, W. F. and RICE, T. M., Phys. Rev. B 2 (1970)
4502 ;
