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Magnetic field dependence of the specific heat of Si:P near the metal-insulator transition
Physica B 165&166 (1990) 285-286 North-Holland
MAGNETIC FIELD DEPENDENCE METAL-INSULATOR TRANSITION
H. v. LGHNEYSEN
OF THE SPECIFIC
HEAT OF Si:P NEAR
THE
and M. LAKNER
Physikalisches Institut der Universitat Karlsruhe, D-7500 Karlsruhe, FRG
The specific heat of uncompensated Si:P with P conceygatiog 0.79 and 4.5. 1018 cm3, i.e. in the vicinity of the critical concentration NC = 3.2. 10 cm , has been measured between 0.05 and 3 K and in fields up to 5.7 T. A magnetic-field induced increase of the number of localized electrons exbibitin a Schottky anomaly is observed together with a concomitant decrease of the contribution from delocalized electrons.
The metal-insulator (MI) transition in Si:P occurring as a function of P concentration N has been studied for more than a decade with various methods. Very recently, we reported ve accurate measurements of the low-temperature spect9 IC heat of uncompensated Si:P [ 11extending earlier work [2-51. In particular, a quantitative analysis in terms of localized and delocalized electrons based on a phenomenological two-component model [2] was given. The localized electrons give rise to an anomalous contribution AC - IIn to the s ecific heat C between -0.05 and -0.5 K. This contra+ ution arises from excitation of exchange-coupled clusters of localized spins. It develops towards a Schottky anomaly in a magnetic field. The delocalized electrons yield a contribution 7T. As observed before [4,5], 7 varies smoothly+j rou_3h the MI transition which occurs at NC = 3.2. 10 cm [6]. There appears to be a finite 7 at concentrations as low as NC/lo. In the present work, we report on a detailed study of the magnetic field dependence of C for two uncompensated samples, one me&lhc yd one insulating, with N = 4.5 and 0.79.10 cm- , in order to elucidate the relative contributions of localized and delocalized electrons in an external magnetic field B.
responding to predominantly ferromagnetic (antiferromagnetic) interactions between localized electrons. The upturn of the data at very low temperat (T < 0.2 K) is attributed to the Zeeman splitting of !@; nuclei [l]. To illustrate the reliability of our data
0
B.15T
??
B=3T
v
B =45T
. B=57T
Fig. 1 shows C vs. T for the insulating sample in various magnetic fields. The anomalous contribution AC (with ~2= - 0.04 for this sample) is suppressed in a magnetic field, and a Schottky anomaly develops. The solid lines are fits to the data (except for B = 0) of the form
CtB,T) = r(B)T +
aT3+ &h(B,T)
(1)
sT3 represents the phonon contribution which will not be considered further. r(B)T represents the contribution due to delocalized electrons in the impurity band and CSc (B,T) is a Schottky anomaly for two nondegenera ‘Ie levels of energy E = gp B ff arising from the Zeeman splitting for localize 8 aectrons. Here B eff = B + Bi IS a fit parameter with Bi > 0 (~0) cor-
$c$i~9he;;l~c;~ Equ. (i) to the data.
0921-4526/90/893.50 @ 1990 - Elsevier Science Publishers B.V. (North-Holland)
FIGURE 1 temptpre T, for Si:P. with So Id lines indicate fits of ’
H. v. Lahneysen, M. Lakner
286
15
1 N :079 . . N = 019 lO"d 0 A N = 65
FIGURE 2 Excess specific heat AC vs. temperature T for the same sample as Fig. 1. Solid lines indicate fits of two-leve! Schottky anomalies with B,ff = 1.38, 2.8, 4.45 and 5.8 T.
evaluation, we show in Fig. 2 AC = C - 7T - pT3. For B >_3T AC resembles auite closelv a Schottkv anomah 2;; Beff = B: For T << ~,uBB, AC’iS considerably wider Sch In 15 T. even taking into account the nuclear contribution (see also Fig. 1). The position and width of the anomaly prohibits a determination of 7 for this field. These data are therefore fitted with the zerofield value r(O). Comparison with #T) justifies this analysis (see Fig. 3). The larger width of AC for B 5 1.5 T is simply due to the fact that a small field cannot completely suppress the thermal excitation of exchange-coupled localized electrons (e.g. singlettriplet transitions for antiferromagnetic pairs). The number of localized electrons NScf, as determined from the fit CSc to AC depends on a magnetic field in high fields whtc h was not observet! in previous work [5]. A similar quality of fit was obtained for the metallic sample, although the scatter in AC is large! because of the much larger yT contribution to AC. We note in passing that only for low-h’ samples Beff= B. while for high N, B,ff < B, indicating the importance of antiferromagnetic mteractions among localized electrons Fig. 3 shows the dependence of Y and N magnetic field. The data are normalized 2to 7?sljw;::cti is obtained from a linear fit of C/T vs. T to the zerofield data above 1.5 K. N S,.h(B) varies in the opposite direction as r(B), i.e. a large magnetic field causes a shift from delocalized to localized electrons. This shift is much more pronounced for the insulating than for the metallic sample. Although an explanation for the VT term on the insulating side is lacking, the data clearly point to a magnetic-field induced MI transition observed here in a thermodynamic quantity. The density of delocalized electrons N calculated from 7 and of localized electrons obtaint?d from NSch do not
10'%m-'
OOW BiTI
FIGURE 3 Magnetic-field dependence of Y/ ~(0) jcircies) and NSch/NSch (1.5 T) (triangles) for two Si:P samples.
simply add up to yield the total density N of donor electrons. Hence the phenomenological two-component model [2] appears to be incomplete. An extended review of recent work on the MI transition in Si:P will appear elsewhere 173. We thank Dr. W. Zulehner. Wacker Chemitronic for kindly providing the samples. This work was supported by Deutsche Forschungsgemeinschaft.
REFERENCES [I] M. Lakner and H. v. Lijhneysen, 648 (1989)
Phys. Rev. Lett. @,
[2] M.A. Paalanen, J.E. Graebner, R.N. Bhatt and S. Sachdev, Phys. Rev. Lett. c,l. 597 (1988) [3] J. K. Marko, J.P. Harrison Rev. BlO, 2448 (1974)
and J.D. Vuirt,
Phys.
[4] N. Kobayashi, S. Ikehata, S. Kobayashi and Sasaki, Solid State Commun. 24, 67 (197’7)
\I’.
]5] N. Kobayashi, S. Ikehata, S. Kobayashi and Sasaki, Solid State Commun. 32, 1147 (1979)
W.
[O] Here we use the Thurber scaie, see footnote in Ref. [l]. See, however, also U. Thomanschefsky, J. Appl. Phys. (to be submitted) [7] I-I. v. Lohneysen, in Festkiirperprobleme/Advances in Solid State Physics Vol. 30 (1990), forthcoming
MAGNETIC FIELD DEPENDENCE METAL-INSULATOR TRANSITION
H. v. LGHNEYSEN
OF THE SPECIFIC
HEAT OF Si:P NEAR
THE
and M. LAKNER
Physikalisches Institut der Universitat Karlsruhe, D-7500 Karlsruhe, FRG
The specific heat of uncompensated Si:P with P conceygatiog 0.79 and 4.5. 1018 cm3, i.e. in the vicinity of the critical concentration NC = 3.2. 10 cm , has been measured between 0.05 and 3 K and in fields up to 5.7 T. A magnetic-field induced increase of the number of localized electrons exbibitin a Schottky anomaly is observed together with a concomitant decrease of the contribution from delocalized electrons.
The metal-insulator (MI) transition in Si:P occurring as a function of P concentration N has been studied for more than a decade with various methods. Very recently, we reported ve accurate measurements of the low-temperature spect9 IC heat of uncompensated Si:P [ 11extending earlier work [2-51. In particular, a quantitative analysis in terms of localized and delocalized electrons based on a phenomenological two-component model [2] was given. The localized electrons give rise to an anomalous contribution AC - IIn to the s ecific heat C between -0.05 and -0.5 K. This contra+ ution arises from excitation of exchange-coupled clusters of localized spins. It develops towards a Schottky anomaly in a magnetic field. The delocalized electrons yield a contribution 7T. As observed before [4,5], 7 varies smoothly+j rou_3h the MI transition which occurs at NC = 3.2. 10 cm [6]. There appears to be a finite 7 at concentrations as low as NC/lo. In the present work, we report on a detailed study of the magnetic field dependence of C for two uncompensated samples, one me&lhc yd one insulating, with N = 4.5 and 0.79.10 cm- , in order to elucidate the relative contributions of localized and delocalized electrons in an external magnetic field B.
responding to predominantly ferromagnetic (antiferromagnetic) interactions between localized electrons. The upturn of the data at very low temperat (T < 0.2 K) is attributed to the Zeeman splitting of !@; nuclei [l]. To illustrate the reliability of our data
0
B.15T
??
B=3T
v
B =45T
. B=57T
Fig. 1 shows C vs. T for the insulating sample in various magnetic fields. The anomalous contribution AC (with ~2= - 0.04 for this sample) is suppressed in a magnetic field, and a Schottky anomaly develops. The solid lines are fits to the data (except for B = 0) of the form
CtB,T) = r(B)T +
aT3+ &h(B,T)
(1)
sT3 represents the phonon contribution which will not be considered further. r(B)T represents the contribution due to delocalized electrons in the impurity band and CSc (B,T) is a Schottky anomaly for two nondegenera ‘Ie levels of energy E = gp B ff arising from the Zeeman splitting for localize 8 aectrons. Here B eff = B + Bi IS a fit parameter with Bi > 0 (~0) cor-
$c$i~9he;;l~c;~ Equ. (i) to the data.
0921-4526/90/893.50 @ 1990 - Elsevier Science Publishers B.V. (North-Holland)
FIGURE 1 temptpre T, for Si:P. with So Id lines indicate fits of ’
H. v. Lahneysen, M. Lakner
286
15
1 N :079 . . N = 019 lO"d 0 A N = 65
FIGURE 2 Excess specific heat AC vs. temperature T for the same sample as Fig. 1. Solid lines indicate fits of two-leve! Schottky anomalies with B,ff = 1.38, 2.8, 4.45 and 5.8 T.
evaluation, we show in Fig. 2 AC = C - 7T - pT3. For B >_3T AC resembles auite closelv a Schottkv anomah 2;; Beff = B: For T << ~,uBB, AC’iS considerably wider Sch In 15 T. even taking into account the nuclear contribution (see also Fig. 1). The position and width of the anomaly prohibits a determination of 7 for this field. These data are therefore fitted with the zerofield value r(O). Comparison with #T) justifies this analysis (see Fig. 3). The larger width of AC for B 5 1.5 T is simply due to the fact that a small field cannot completely suppress the thermal excitation of exchange-coupled localized electrons (e.g. singlettriplet transitions for antiferromagnetic pairs). The number of localized electrons NScf, as determined from the fit CSc to AC depends on a magnetic field in high fields whtc h was not observet! in previous work [5]. A similar quality of fit was obtained for the metallic sample, although the scatter in AC is large! because of the much larger yT contribution to AC. We note in passing that only for low-h’ samples Beff= B. while for high N, B,ff < B, indicating the importance of antiferromagnetic mteractions among localized electrons Fig. 3 shows the dependence of Y and N magnetic field. The data are normalized 2to 7?sljw;::cti is obtained from a linear fit of C/T vs. T to the zerofield data above 1.5 K. N S,.h(B) varies in the opposite direction as r(B), i.e. a large magnetic field causes a shift from delocalized to localized electrons. This shift is much more pronounced for the insulating than for the metallic sample. Although an explanation for the VT term on the insulating side is lacking, the data clearly point to a magnetic-field induced MI transition observed here in a thermodynamic quantity. The density of delocalized electrons N calculated from 7 and of localized electrons obtaint?d from NSch do not
10'%m-'
OOW BiTI
FIGURE 3 Magnetic-field dependence of Y/ ~(0) jcircies) and NSch/NSch (1.5 T) (triangles) for two Si:P samples.
simply add up to yield the total density N of donor electrons. Hence the phenomenological two-component model [2] appears to be incomplete. An extended review of recent work on the MI transition in Si:P will appear elsewhere 173. We thank Dr. W. Zulehner. Wacker Chemitronic for kindly providing the samples. This work was supported by Deutsche Forschungsgemeinschaft.
REFERENCES [I] M. Lakner and H. v. Lijhneysen, 648 (1989)
Phys. Rev. Lett. @,
[2] M.A. Paalanen, J.E. Graebner, R.N. Bhatt and S. Sachdev, Phys. Rev. Lett. c,l. 597 (1988) [3] J. K. Marko, J.P. Harrison Rev. BlO, 2448 (1974)
and J.D. Vuirt,
Phys.
[4] N. Kobayashi, S. Ikehata, S. Kobayashi and Sasaki, Solid State Commun. 24, 67 (197’7)
\I’.
]5] N. Kobayashi, S. Ikehata, S. Kobayashi and Sasaki, Solid State Commun. 32, 1147 (1979)
W.
[O] Here we use the Thurber scaie, see footnote in Ref. [l]. See, however, also U. Thomanschefsky, J. Appl. Phys. (to be submitted) [7] I-I. v. Lohneysen, in Festkiirperprobleme/Advances in Solid State Physics Vol. 30 (1990), forthcoming