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Experimental tests of localization in semiconductors
Phys~a 117B& 118B(1983) 81-83 North-HollandPubhslungCompany
81
Experimental Tests of Localization in Semiconductors G. A. Thomas Bell Laboratories, Murray Hill, New Jersey The characteristic lengths of the electron wave function in a random three-dimensional system tend to diverge at a critical point at zero temperature and a density n c. The results differ from predictions of Mott and of scaling theories of localization. The lengths associated with electron diffusmn appear to be dominated by Coulomb interactions and electron-electron inelasttc scattenng Measurements of the characteristic lengths of the electron wave function (1) in Si:P are illustrated in Fig. 1. These lengths are defined using the dielectric susceptibility (2,3) 41rx and the electrical conductivity (4-6) ~(o) m the zero temperature limit At n c the typical electron separation, a c - n c 1/3,
which is 20(fl--cm) - l ('--7(fl--cm) -1 for Ge:Sb), so that Eqn. (5) defines a corresponding le.ngth, LMOTT -- 20(~/3f2) I/3 a c '~750A ('--2200A for Ge:Sb). Values of ~(o) measured from a series of samples (4,6) (solid circles) and from samples under applied umaxlal stress (5) (open circles) give the Lo values shown in Fig 1 The solid line through these points ]s a fit to
(1)
L . ffi=L ° ( n / n ¢ - l ) - '
is (3.74 X 1018 cm-3) -I/3 'ffi 64.~. [ac " (0.155 X 1018 cm-3) -I/3 •ffi 190,~ in Ge'~b, see Refs. 6,7]. In the insulator we define the localization length (1) (see insert Fig. 1) L x - k F r ( 4 f X ) ~,
with L ° ,,- a c and ~ == 0.48 ± 0.07, symmetric with L x. The transition in Fig. 1 is extremely sharp, qualitatively in agreement with Mott's suggestions ( Y ).
(2) 10 4
where the Fermi-Thoma~ k F r 2 [12wnp'/{t, is 20~, the effective screening (4) ~' is 2 (1 m wave-length
ask~)J
screening wave length and the Bohr radius a s valley degeneracy for Ge:Sb), and the Fermi
kF-1 ffi (31r2n/p) -1/3,
(6)
I INSULATOR
'
I
SI P M E T A L
(3)
where the degeneracy p is 6 (4 in Ge:Sb), so that ~ F.I.~I7.2~(n/nc)T9 g
io3
•,,.---LX--~
•e - - - Lo. - - - . b
-r
Far-infrared (2) (sohd squares) and cavity resonance (3) measurements (open squares) of 4~rx give the L x values shown. The solid line is a fit to the divergent form L x = L~ ( n e / n - l ) - ' ,
LMOTT
J
!
(4)
where Lax ,~ a¢ and ~ == 0.58 -+ 0.08. In the metal, we define L~, ," ac 01R/¢(o),
=
io 2
(5)
where the conductivity e m " (w/3r2) 1/3 (e2/l~)/ac, which is ~ ¢ ( o ) near the loffe-Reipd criterion (8) ( e r a - 2 2 3 ( f l - c m ) ' l ) . Mott has proposedL9 ) a smaller characteristic conductivity eM -- (e2,/l])/2Oac,
0 378-4363/83/0000-0000/$03.00 © 1983 North-Holland
I 2
L
J 4
j
I 6
DENSITY ( 1 0 Ill c m " s )
1. E~idonce for a critical point at the metal-
insulator transitmn and zero temperature.
82
G A Thomas / Experimental tests o f locahzatlon m semwonductors
A critical point at nc and T •ffi OK is assumed by scaling theories of localization (1), but current versions estimate u --- 1 and a conductivity prefactor simdar to Mott's; i.e., using Eq, 5, L ¢ - - , LMOTT (n/nc--1) -1 (see "LOCAL. X100" in Fig. 3). Mott has suggested (9) the existence of a minimum metallic conductivity, with a slow variation of ¢(o) above it (see dashed line "LMoTr" in Fig. 3). These estimates are apparently unsatisfactory.
We argue that the critical region encompasses at least the range of n in which L x and L , are larger than a c Support for this estimate is shown in Figure 2 where we have plotted the calculated lengths n -1/3, k~-l, and the Boltzman mean free path (4) ~n (dashed line which includes both screening and scattering from the P donors with realistic estimates of the potentials). For comparison we show the usual effective mean free path (8) ~a " or(O) k F n-I(e2/h-)-1.
0
(7)
The measured or(o) values determine the solid circles and solid line fitted to ,them. These data fit the ~B curve for n ; ~ 10X1018cm -3, but deviate near ~ = ,-, n -]/3 (where also L~ ~ ac, c.f. Figs. 1 and 2). This critical region ( ~ ~ n -1/3 or n / n o - 1 < 1 is slightly larger than that where k F ~ < 1 and much larger than that where or(o) < orMott (L¢ > LMOTT in Fig. 1). I
si P
nc
so \
I
0.8
S
S
J -J 0 6
®
b ._I
i
O.Ot
LMOTT
hO
$I
0.005
I i DENSITY n/nc-I
Si P
0.4
~
/~~---"
/ //
0.2
LOCAL x I00 ,/
o 2O
zor I 5 DENSITY
k~ I I0 n ( l o l a c m -5)
I t5
2. Estimates of the extent of the critical region. For studies of the region very near n c we have applied uniaxial stress S(± to I I I axis) to barely insulating samples of Si:P and measured (5) or(o) (and L o via Eq 5) with the results shown as open circles in Fzg. I and with an expanded scale in Fig. 3 The principal effect (5,10) of S (see insets to Fig. 3) is to raise the (valley-orbit split) ground state energy, thus expanding the donor wave functions and increasing or(o) or equivalently decreasing n c (see upper scale). We have plotted L~"2, since Eq. 6 gives a acceptable fit with v ffi ½, as the solid line in Fig. 3 indicates. Three samples (5) show the same behavior for ( L J L M o r r ) - 2 ; ~ O.l, so we suggest that u ~ ½ is intrinsic to the random 3-dimensional potential. The tail for S ~; 6.5 klm- in Fig. 3 is exponential in S but does not reproduce in magnitude.
o I 6 STRESS (k bar)
3. Comparison with theoretical descriptions in the region near n c. We obtained the zero T values ¢(o) in Figs. (1)-(3) by extrapolations based on studies (11,12) of or(T) in the range 3 mK ~ T ~ 4K. Samples of Ge:Sb show results (6,7) qualitatively similar to those given for Si:P, with smaller characteristic energies, as expected. The magnitude of the corrections to ¢ (o) m Ge:Sb as a function of n are illustrated in Fig. 4 based on the observed form or(T) f ~ r ( o ) mV~-l-BT. We define a diffusive length arising from Coulomb interactions (13,14) LeT ! CT (e2/l~)/(mV~),
(8)
where the constant CT f f S o u / 2 f 2, and the anisotropic scattering factor (13) S O is 1.85 for Ge:Sb ( ~ 1 for Si:P). Similarly, we define an electronelectron inelastic scattering length within the localization model (1)
L~ = C,r (e21~)I(ST).
(9)
G A Thomas / Experzmental tests oflocahzatzon m semzconductors
I
I
Thanks are due particularly to R. N. Bhatt for helpful comments on this manuscript and also to my other collaborators indicated in the references.
I GO Sb
t1¢
83
10 4
REFERENCES T-IOOmK, INELASTIC
•
L.,/'T
F-,o 3 (.9 Z UJ .J
T '= o K , ELASTIC .~--~0"
iO 2
f
/
/
..,..o. , - ' ' " ' ' U ' ' "
~ ~ ~ "°-
- J~o"
/
I
0.2
I
0.4 DENSITY n (lOMcm"~'}
I
0.6
" . Evidence for inelastic electronic processes in the low temperature diffusion The measured values of m and B give the solid circles in Fig. (4), where the large size of Lv~ and Lee at T •- 100 mK compared to ~r, open ctrcles, (or to Le) illustrate that these represent small corrections to u(o). The predicted variation (14) of LvT divided by a constant is shown as the solid line, where this constant is 2.6 if intervalley scattering is neglected and is smaller with realistic scattering (10). The prediction for Lee for weak scattering (divided by ~2) is also shown. Based on Fig. 4, we suggest that Coulomb interactions and inelastic electronelectron scattering dominate the electron diffusion at low temperatures. Consequently, it appears that such effects should be included to describe the critical point.
I. E. Abrahams, P.W. Anderson, D, C. Licciardello and T.V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979); Y. Imi'y, Phys. Rev. Lett. 44, 469 (1980); W. L. McMillan, Phys. Rev. B14, 2739 (1981). 2. M. Capizzi, G. A. Thomas, F. DeRosa, R.N. Bhatt and T. M. Rice, Phys. Rev. Lett. 44, 1019 (1980). 3. H. F. Hess, K. DeConde, T. F. Rosenbaum and G.A. Thomas, Phys. Rev. 825, 5585 (1982). 4. T. F. Rosenbaum, K. Andres, G. A. Thomas and R. N. Bhatt, Phys. Rev. Lett. ,/5, 1723 (1980); G. A. Thomas, T. F. Rosenbaum and R.N. Bhatt, ibid. 46, 1435 (1981). 5. M. A. Paalanen, T. F. Rosenhaum, G.A. Thomas and R.N. Bhatt, Phys. Rev Lett. 48, 1284 (1982); G. A. Thomas, A. Kawabata, Y. Ootuka, S. Katsumoto, S. Kobayashi and W. Saski, Phys. Rev. B24, 4886 (1981). 6. G. A. Thomas, Y. Ootuka, S. Katsumoto, S. Kohayashi and W. Sasaki, Phys. Rev. B25, 4288 (1982). 7. G.A. Thomas, Y. Ootuka, S. Katsumoto, S. Kobayashi and W Sasakl, Phys. Rev. B, in press. 8. A.F. Ioffe and A. R. Regel, Prog. in Semiconductors 4, 237 (1960). 9. N. F. Mott, Phil. Mag. 26, 1015 (1972); ibid. B44, 265 (1981). 10. R. N. Bhatt, Phys. Rev. B24, 3630 (1981); ibid., (1982), m press. 11. T. F. Rosenbaum, K. Andres, G. A. Thomas, P. A. Lee, Phys. Rev. Lett. 46 568 (1981). 12. T F. Rosenbaum, R. F. Mdlisan, G.A. Thomas, P.A. Lee, T.V. Ramakrishnan, R.N. Bhatt, K. DeConde, H. Hess and T. Perry, Phys. Rev. Lett (1981). 13. A. Kawabata, Solid State Commun. 34, 431 (1980); J. Phys. Soc. Japan 49, Suppl. A., 375 (1980) 14 B. L. Altshuler, A.G. Aronov and P.A. Lee, Phys. Rev. Lett 44, 1288 (1980); B L. Altshuler, D. Khmelmtzkii, A 1. Larkin and P A. Lee, Phys. Rev. B22, 5142 (1980). 15. J. J. Quinn and R A Ferrell, Phys Rev 112, 812 (1958)
81
Experimental Tests of Localization in Semiconductors G. A. Thomas Bell Laboratories, Murray Hill, New Jersey The characteristic lengths of the electron wave function in a random three-dimensional system tend to diverge at a critical point at zero temperature and a density n c. The results differ from predictions of Mott and of scaling theories of localization. The lengths associated with electron diffusmn appear to be dominated by Coulomb interactions and electron-electron inelasttc scattenng Measurements of the characteristic lengths of the electron wave function (1) in Si:P are illustrated in Fig. 1. These lengths are defined using the dielectric susceptibility (2,3) 41rx and the electrical conductivity (4-6) ~(o) m the zero temperature limit At n c the typical electron separation, a c - n c 1/3,
which is 20(fl--cm) - l ('--7(fl--cm) -1 for Ge:Sb), so that Eqn. (5) defines a corresponding le.ngth, LMOTT -- 20(~/3f2) I/3 a c '~750A ('--2200A for Ge:Sb). Values of ~(o) measured from a series of samples (4,6) (solid circles) and from samples under applied umaxlal stress (5) (open circles) give the Lo values shown in Fig 1 The solid line through these points ]s a fit to
(1)
L . ffi=L ° ( n / n ¢ - l ) - '
is (3.74 X 1018 cm-3) -I/3 'ffi 64.~. [ac " (0.155 X 1018 cm-3) -I/3 •ffi 190,~ in Ge'~b, see Refs. 6,7]. In the insulator we define the localization length (1) (see insert Fig. 1) L x - k F r ( 4 f X ) ~,
with L ° ,,- a c and ~ == 0.48 ± 0.07, symmetric with L x. The transition in Fig. 1 is extremely sharp, qualitatively in agreement with Mott's suggestions ( Y ).
(2) 10 4
where the Fermi-Thoma~ k F r 2 [12wnp'/{t, is 20~, the effective screening (4) ~' is 2 (1 m wave-length
ask~)J
screening wave length and the Bohr radius a s valley degeneracy for Ge:Sb), and the Fermi
kF-1 ffi (31r2n/p) -1/3,
(6)
I INSULATOR
'
I
SI P M E T A L
(3)
where the degeneracy p is 6 (4 in Ge:Sb), so that ~ F.I.~I7.2~(n/nc)T9 g
io3
•,,.---LX--~
•e - - - Lo. - - - . b
-r
Far-infrared (2) (sohd squares) and cavity resonance (3) measurements (open squares) of 4~rx give the L x values shown. The solid line is a fit to the divergent form L x = L~ ( n e / n - l ) - ' ,
LMOTT
J
!
(4)
where Lax ,~ a¢ and ~ == 0.58 -+ 0.08. In the metal, we define L~, ," ac 01R/¢(o),
=
io 2
(5)
where the conductivity e m " (w/3r2) 1/3 (e2/l~)/ac, which is ~ ¢ ( o ) near the loffe-Reipd criterion (8) ( e r a - 2 2 3 ( f l - c m ) ' l ) . Mott has proposedL9 ) a smaller characteristic conductivity eM -- (e2,/l])/2Oac,
0 378-4363/83/0000-0000/$03.00 © 1983 North-Holland
I 2
L
J 4
j
I 6
DENSITY ( 1 0 Ill c m " s )
1. E~idonce for a critical point at the metal-
insulator transitmn and zero temperature.
82
G A Thomas / Experimental tests o f locahzatlon m semwonductors
A critical point at nc and T •ffi OK is assumed by scaling theories of localization (1), but current versions estimate u --- 1 and a conductivity prefactor simdar to Mott's; i.e., using Eq, 5, L ¢ - - , LMOTT (n/nc--1) -1 (see "LOCAL. X100" in Fig. 3). Mott has suggested (9) the existence of a minimum metallic conductivity, with a slow variation of ¢(o) above it (see dashed line "LMoTr" in Fig. 3). These estimates are apparently unsatisfactory.
We argue that the critical region encompasses at least the range of n in which L x and L , are larger than a c Support for this estimate is shown in Figure 2 where we have plotted the calculated lengths n -1/3, k~-l, and the Boltzman mean free path (4) ~n (dashed line which includes both screening and scattering from the P donors with realistic estimates of the potentials). For comparison we show the usual effective mean free path (8) ~a " or(O) k F n-I(e2/h-)-1.
0
(7)
The measured or(o) values determine the solid circles and solid line fitted to ,them. These data fit the ~B curve for n ; ~ 10X1018cm -3, but deviate near ~ = ,-, n -]/3 (where also L~ ~ ac, c.f. Figs. 1 and 2). This critical region ( ~ ~ n -1/3 or n / n o - 1 < 1 is slightly larger than that where k F ~ < 1 and much larger than that where or(o) < orMott (L¢ > LMOTT in Fig. 1). I
si P
nc
so \
I
0.8
S
S
J -J 0 6
®
b ._I
i
O.Ot
LMOTT
hO
$I
0.005
I i DENSITY n/nc-I
Si P
0.4
~
/~~---"
/ //
0.2
LOCAL x I00 ,/
o 2O
zor I 5 DENSITY
k~ I I0 n ( l o l a c m -5)
I t5
2. Estimates of the extent of the critical region. For studies of the region very near n c we have applied uniaxial stress S(± to I I I axis) to barely insulating samples of Si:P and measured (5) or(o) (and L o via Eq 5) with the results shown as open circles in Fzg. I and with an expanded scale in Fig. 3 The principal effect (5,10) of S (see insets to Fig. 3) is to raise the (valley-orbit split) ground state energy, thus expanding the donor wave functions and increasing or(o) or equivalently decreasing n c (see upper scale). We have plotted L~"2, since Eq. 6 gives a acceptable fit with v ffi ½, as the solid line in Fig. 3 indicates. Three samples (5) show the same behavior for ( L J L M o r r ) - 2 ; ~ O.l, so we suggest that u ~ ½ is intrinsic to the random 3-dimensional potential. The tail for S ~; 6.5 klm- in Fig. 3 is exponential in S but does not reproduce in magnitude.
o I 6 STRESS (k bar)
3. Comparison with theoretical descriptions in the region near n c. We obtained the zero T values ¢(o) in Figs. (1)-(3) by extrapolations based on studies (11,12) of or(T) in the range 3 mK ~ T ~ 4K. Samples of Ge:Sb show results (6,7) qualitatively similar to those given for Si:P, with smaller characteristic energies, as expected. The magnitude of the corrections to ¢ (o) m Ge:Sb as a function of n are illustrated in Fig. 4 based on the observed form or(T) f ~ r ( o ) mV~-l-BT. We define a diffusive length arising from Coulomb interactions (13,14) LeT ! CT (e2/l~)/(mV~),
(8)
where the constant CT f f S o u / 2 f 2, and the anisotropic scattering factor (13) S O is 1.85 for Ge:Sb ( ~ 1 for Si:P). Similarly, we define an electronelectron inelastic scattering length within the localization model (1)
L~ = C,r (e21~)I(ST).
(9)
G A Thomas / Experzmental tests oflocahzatzon m semzconductors
I
I
Thanks are due particularly to R. N. Bhatt for helpful comments on this manuscript and also to my other collaborators indicated in the references.
I GO Sb
t1¢
83
10 4
REFERENCES T-IOOmK, INELASTIC
•
L.,/'T
F-,o 3 (.9 Z UJ .J
T '= o K , ELASTIC .~--~0"
iO 2
f
/
/
..,..o. , - ' ' " ' ' U ' ' "
~ ~ ~ "°-
- J~o"
/
I
0.2
I
0.4 DENSITY n (lOMcm"~'}
I
0.6
" . Evidence for inelastic electronic processes in the low temperature diffusion The measured values of m and B give the solid circles in Fig. (4), where the large size of Lv~ and Lee at T •- 100 mK compared to ~r, open ctrcles, (or to Le) illustrate that these represent small corrections to u(o). The predicted variation (14) of LvT divided by a constant is shown as the solid line, where this constant is 2.6 if intervalley scattering is neglected and is smaller with realistic scattering (10). The prediction for Lee for weak scattering (divided by ~2) is also shown. Based on Fig. 4, we suggest that Coulomb interactions and inelastic electronelectron scattering dominate the electron diffusion at low temperatures. Consequently, it appears that such effects should be included to describe the critical point.
I. E. Abrahams, P.W. Anderson, D, C. Licciardello and T.V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979); Y. Imi'y, Phys. Rev. Lett. 44, 469 (1980); W. L. McMillan, Phys. Rev. B14, 2739 (1981). 2. M. Capizzi, G. A. Thomas, F. DeRosa, R.N. Bhatt and T. M. Rice, Phys. Rev. Lett. 44, 1019 (1980). 3. H. F. Hess, K. DeConde, T. F. Rosenbaum and G.A. Thomas, Phys. Rev. 825, 5585 (1982). 4. T. F. Rosenbaum, K. Andres, G. A. Thomas and R. N. Bhatt, Phys. Rev. Lett. ,/5, 1723 (1980); G. A. Thomas, T. F. Rosenbaum and R.N. Bhatt, ibid. 46, 1435 (1981). 5. M. A. Paalanen, T. F. Rosenhaum, G.A. Thomas and R.N. Bhatt, Phys. Rev Lett. 48, 1284 (1982); G. A. Thomas, A. Kawabata, Y. Ootuka, S. Katsumoto, S. Kobayashi and W. Saski, Phys. Rev. B24, 4886 (1981). 6. G. A. Thomas, Y. Ootuka, S. Katsumoto, S. Kohayashi and W. Sasaki, Phys. Rev. B25, 4288 (1982). 7. G.A. Thomas, Y. Ootuka, S. Katsumoto, S. Kobayashi and W Sasakl, Phys. Rev. B, in press. 8. A.F. Ioffe and A. R. Regel, Prog. in Semiconductors 4, 237 (1960). 9. N. F. Mott, Phil. Mag. 26, 1015 (1972); ibid. B44, 265 (1981). 10. R. N. Bhatt, Phys. Rev. B24, 3630 (1981); ibid., (1982), m press. 11. T. F. Rosenbaum, K. Andres, G. A. Thomas, P. A. Lee, Phys. Rev. Lett. 46 568 (1981). 12. T F. Rosenbaum, R. F. Mdlisan, G.A. Thomas, P.A. Lee, T.V. Ramakrishnan, R.N. Bhatt, K. DeConde, H. Hess and T. Perry, Phys. Rev. Lett (1981). 13. A. Kawabata, Solid State Commun. 34, 431 (1980); J. Phys. Soc. Japan 49, Suppl. A., 375 (1980) 14 B. L. Altshuler, A.G. Aronov and P.A. Lee, Phys. Rev. Lett 44, 1288 (1980); B L. Altshuler, D. Khmelmtzkii, A 1. Larkin and P A. Lee, Phys. Rev. B22, 5142 (1980). 15. J. J. Quinn and R A Ferrell, Phys Rev 112, 812 (1958)