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Low-temperature specific heat of isotopically enriched silicon single crystals
15 July 2002
Physics Letters A 299 (2002) 656–659 www.elsevier.com/locate/pla
Low-temperature specific heat of isotopically enriched silicon single crystals Quanli Hu ∗ , Tetsuji Noda, Hiroshi Suzuki, Toshiyuki Hirano, Takenori Numazawa National Institute for Materials Sciences, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan Received 13 May 2002; received in revised form 13 May 2002; accepted 21 May 2002 Communicated by R. Wu
Abstract The single crystal of isotopically enriched silicon (28 Si) has been made by FZ method with 111 growth direction. The specific heat measurement at low temperature was performed at the temperature range of 1–140 K. The low-temperature specific heat of 28 Si diverges from Debye T 3 behavior from 14.5 K, and this is in contrast to natural silicon, where deviations from the T 3 behavior occur already at much lower temperature (7.2 K). In addition, the specific heat of 28 Si becomes obviously higher than that of natural silicon at T > 20 K. 2002 Elsevier Science B.V. All rights reserved.
1. Introduction The thermal physical properties of materials are known to be an important factor in the recent development of semiconductor industries and the nextgeneration computer—the quantum computer. The recent availability of highly enriched isotopes and growth techniques has made possible the study of the thermal physical properties of semiconductor single crystals in which the isotope concentrations can be controlled [1]. In some materials (carbon [2], germanium [3] and boron [4]), the thermal physical properties at low temperature are very sensitive for the variation of isotope concentration. For example, the enriching the isotopic purity of 12 C from 98.9 to 99.9% may
* Corresponding author. JSPS fellow at National Institute for Materials Sciences. E-mail address: [email protected] (Q. Hu).
cause obvious variation on thermal conductivity, specific heat and thermal expansion [5]. In addition, many researchers have predicated the large increase of thermal conductivity of isotopically enriched silicon [6]. Takyu et al. reported the successful growth of an isotopically enriched 28 Si bulk single crystal with the isotopic enrichment of 28 Si (99.924 at%), but unfortunately did not provide results of thermal conductivity measurement [7]. Ruf et al. measured the thermal conductivity of isotopically enriched silicon single crystal (28 Si: 99.8588 at%) between 2 and 310 K and found the maximum value of about 30000 W/m K around 20 K, which is six times larger than that of natural silicon [8]. And they also indicated that in the highly isotopically enriched sample, the T 3 behavior of thermal conductivity can be observed for temperatures up to higher temperature than that of natural silicon [9]. In order to further clarify the isotopic effects on the T 3 behavior of thermal conductivity in isotopically modified silicon, in this study another important thermal
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 6 7 9 - 5
Q. Hu et al. / Physics Letters A 299 (2002) 656–659
parameter, specific heat, was measured at low temperature for isotopically enriched silicon and natural silicon.
Table 1 Impurities in the isotopically enriched 28 Si and natural Nat. Si single crystals C 28 Si (ppm)
2. Experimental Isotopically modified silicon single crystals (28Si and Nat. Si) were formed by the floating-zone method (FZ) at 1673 K, and the 111 growth direction was determined by the X-ray Laue backscattering method. The isotopic composition was measured by secondary-ion-mass spectroscopy (SIMS) and found to be 28 Si (99.88465 at%), 29 Si (0.07689 at%), and 30 Si (0.03845 at%). Also, the ICPMS analysis result of other impurities in the isotopically enriched 28 Si and natural Nat. Si single crystals were shown in Table 1. The calorimetric technique used here is known as the time-constant or relaxation method. The range of the measured temperatures was between 1 and 140 K; these temperatures were reached and maintained by a sorption-pumped 3 He system. To mount the sample, one face of a small sample pellet (mass 2–6 mg, 2 × 2 mm2 ) was bonded with a known amount of low-temperature grease to a sapphire chip on which a serpentine metallic heater was evaporated. A Cernox temperature sensor was attached to the chip with 50 µm gold leads. The chip was suspended by 20 µm tungsten leads to realize electric connection and a thermal link to the calorimeter cell. The heat capacities of the chip and grease were determined separately and subtracted as shown in Ref. [5].
657
Nat. Si (ppm)
< 165 < 48
H 100 < 144
B
Fe
Mg
0.015 0.02
< 0.3 12
< 0.3 0.2
(a)
3. Results and discussion Figs. 1(a) and (b) show the plot of specific heat divided by T versus T 2 for isotopically enriched 28 Si and natural silicon. The straight lines are the fitting curves according to the data of T < 5 K. The two special temperatures of 7.2 and 14.5 K in Fig. 1 are determined as follows: (1) fitting the initial part of curvature at low temperature (T < 5 K) with linear equation; (2) extending the liner to the high temperature; (3) determining the temperature value when the experimental data starts to deviate from liner relation (relative difference > 5%). It can be seen that the low-temperature specific heat of 28 Si diverges
(b) Fig. 1. Temperature square dependence of specific heat divided by T for natural silicon (a) and isotopically enriched 28 Si (b).
from Debye T 3 beginning at 14.5 K, in contrast to natural silicon, where deviations from the T 3 behavior already occur at a much lower temperature of 7.2 K. However, no obvious difference in specific heat at T < 20 K between the two kinds of samples, such as that which we determined in the case of isotopically modified boron [4], has yet been found.
658
Q. Hu et al. / Physics Letters A 299 (2002) 656–659
Using Debye’s theory [9], one can now calculate the specific heat of a solid, which is proportional to T 3 at a sufficiently low temperature. In other words, the lattice vibration in this assumption is a nearly harmonic vibration. According to Debye’s theory, the atomic system is assumed to be an elastic medium in which only certain frequencies can be excited and maintained at low temperature. In particular, at very low temperatures, the thermal activation of a specific heat measurement can mainly cause excitation of a long acoustic phonon wave. In the two samples with different 28 Si contents, the enriched 28 Si has lower average atomic mass and higher vibration frequency of the lattice than natural silicon. From the viewpoint of the absolute value of specific heat, this would lead to a lower specific heat for isotopically enriched 28 Si at a given temperature compared with natural silicon. Here we are concerned with the isotopic effect on the harmonic lattice vibration or Debye T 3 behavior at low temperature, which was already observed in Fig. 1. Because the isotopically enriched 28 Si crystals already have a decreased minor isotope (29 Si and 30 Si) content compared with natural silicon crystal, this will obviously increase the isotopic homogeneity of the 28 Si crystal lattice from the viewpoint of defect physics. At the same time, the enrichment of 28 Si influences the frequency distribution of harmonic lattice vibration, and decreases or deletes the low-frequency term from 29 Si and 30 Si. Moreover, this enhances the ability to resist anharmonic occurrences so that the near harmonic lattice vibration is maintained at higher temperatures compared with natural silicon. In order to understand the isotopic effect on specific heat at high temperature, the anharmonic effect must be considered. Fig. 2 shows that the specific heat of isotopically enriched silicon (28 Si concentration > 99.88465 at%) starts to deviate from the conventional behavior with increasing temperature. In addition, the specific heat of isotopically enriched silicon becomes significantly higher than that of natural silicon when the temperature is higher than 20 K. This is because when the lattice vibrations with higher frequencies are being excited with increasing temperature, more modes or oscillators will come into operation than the Debye theory predicts. Also, the interaction between atoms will have a marked influence on vibration of the increased modes. Thus, the anharmonic effect is the main cause of the increase of spe-
Fig. 2. Specific heat versus T for isotopically enriched 28 Si and natural silicon.
cific heat for silicon at higher temperatures, which cannot be explained by the change of volume mass alone. Therefore, in some temperature regions (T > 14.5 K for isotopically enriched 28 Si and T > 7.2 K for natural silicon), the thermal energy is sufficient produce anharmonic lattice vibrations. Considering one aspect, the isotopic effect of enriched 28 Si may change the amplitude of lattice vibrations; this kind of change is related not only to average atomic mass, but also to atomic interaction in the isotopically enriched 28 Si lattice. From another aspect, the isotopic effect will influence the zero point energy of the lattice system even at very low temperature; although it should be reflected in the specific heat at low temperature related to the Debye behavior, however, this has not yet been found. According to this analysis, the frequency distribution between ω and ω + dω is proportional, not to ω2 as in Eq. (1), but to a higher power of ω. At higher temperatures, an additional term should be involved in the specific heat,
T C(T ) = 9R D
3 D/T 0
x 2 ex F (ω) dx, (ex − 1)2
(1)
where F (ω) is a frequency distribution which includes an additional term for the higher power of the ω frequency: F (ω) = αω2 + βω4 + cω6 + · · · .
(2)
Q. Hu et al. / Physics Letters A 299 (2002) 656–659
Table 2 The coefficients in Eq. (3) |a| (J g−1 K−4 ) |b| (J g−1 K−5 ) |c| (J g−1 K−7 )
28 Si
Nat. Si
8.5903 × 10−7
8.3784 × 10−7 6.8135×10−11 1.8255×10−15
6.7007×10−11 1.7826×10−15
Furthermore, the corresponding expansion of the specific heat is C(T ) = aT 3 + bT 5 + cT 7 + · · · .
(3)
The determination of the coefficients a, b and c may be obtained through the fitting process of the experimental results of specific heat. The comparison of relative coefficients for isotopically enriched 28 Si and natural silicon is a direct way of investigating the isotopic effect on high power terms of temperature in the specific heat expression. Table 2 shows that the contribution of the T 3 term to specific heat in isotopically enriched 28 Si is greater than that in natural silicon, even though the high power terms of temperature (T 5 and T 7 ) in isotopically enriched 28 Si have a smaller contribution than that from natural silicon. It is considered that in the present temperature region (1–140 K), the effect of harmonic lattice vibrations dominates the behavior of specific heat in isotopically enriched 28 Si. The main reason is that the enriched 28 Si lattice system can more effectively suppress the anharmonic effect than natural silicon by increasing the isotopic homogeneity of the lattice. Therefore, the difference in specific heat at high temperature (T > 20 K) can be observed in Fig. 2.
4. Conclusion In summary, at low temperature (T < 20 K), enriched 28 Si single crystals have an isotopic effect
659
on specific heat with different average lattice atomic mass, which changes the frequency of harmonic lattice vibrations and maintains Debye T 3 behavior at T < 14.5 K, as compared with natural silicon (T < 7.2). At high temperature (T > 20 K), the contribution to anharmonic lattice vibrations or additional terms in isotopically enriched 28 Si is smaller than that in natural silicon, yielding the higher specific heat.
Acknowledgement The authors are grateful to Dr. Y. Morishita of AIST for the measurements of silicon isotope abundance using SIMS.
References [1] J. Spitzer, T. Ruf, M. Cardona, W. Dondl, R. Schorer, G. Abstreiter, E.E. Haller, Phys. Rev. Lett. 72 (1994) 1565. [2] L. Wei, P.K. Kuo, R.L. Thomas, Phys. Rev. Lett. 70 (1993) 3764. [3] M. Asen-Palmer, K. Bartkowski, E. Gmelin, M. Cardona, A.P. Zhernov, A.V. Inyushkin, A. Taldenkov, V.I. Ozhogin, K.M. Itoh, E.E. Haller, Phys. Rev. B 56 (1997) 9431. [4] Q. Hu, T. Noda, H. Suzuki, T. Numazawa, O. Arai, N. Nogi, S. Tanaka, J. Appl. Phys. 90 (2001) 728. [5] A.K. McCurdy, H.J. Maris, C. Elbaum, Phys. Rev. B 2 (1970) 4077. [6] E.E. Haller, J. Appl. Phys. 77 (1995) 2857. [7] K. Takyu, K.M. Itoh, K. Oka, N. Saito, V.I. Ozhogin, Jpn. J. Appl. Phys. 38 (1999) L1493. [8] T. Ruf, R.W. Henn, M. Asen-Palmer, E. Gmelin, M. Cardona, H.-J. Pohl, G.G. Devyatych, P.G. Sennikov, Solid State Commun. 115 (2000) 243. [9] P. Debye, Ann. Phys. 39 (1912) 789.
Physics Letters A 299 (2002) 656–659 www.elsevier.com/locate/pla
Low-temperature specific heat of isotopically enriched silicon single crystals Quanli Hu ∗ , Tetsuji Noda, Hiroshi Suzuki, Toshiyuki Hirano, Takenori Numazawa National Institute for Materials Sciences, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan Received 13 May 2002; received in revised form 13 May 2002; accepted 21 May 2002 Communicated by R. Wu
Abstract The single crystal of isotopically enriched silicon (28 Si) has been made by FZ method with 111 growth direction. The specific heat measurement at low temperature was performed at the temperature range of 1–140 K. The low-temperature specific heat of 28 Si diverges from Debye T 3 behavior from 14.5 K, and this is in contrast to natural silicon, where deviations from the T 3 behavior occur already at much lower temperature (7.2 K). In addition, the specific heat of 28 Si becomes obviously higher than that of natural silicon at T > 20 K. 2002 Elsevier Science B.V. All rights reserved.
1. Introduction The thermal physical properties of materials are known to be an important factor in the recent development of semiconductor industries and the nextgeneration computer—the quantum computer. The recent availability of highly enriched isotopes and growth techniques has made possible the study of the thermal physical properties of semiconductor single crystals in which the isotope concentrations can be controlled [1]. In some materials (carbon [2], germanium [3] and boron [4]), the thermal physical properties at low temperature are very sensitive for the variation of isotope concentration. For example, the enriching the isotopic purity of 12 C from 98.9 to 99.9% may
* Corresponding author. JSPS fellow at National Institute for Materials Sciences. E-mail address: [email protected] (Q. Hu).
cause obvious variation on thermal conductivity, specific heat and thermal expansion [5]. In addition, many researchers have predicated the large increase of thermal conductivity of isotopically enriched silicon [6]. Takyu et al. reported the successful growth of an isotopically enriched 28 Si bulk single crystal with the isotopic enrichment of 28 Si (99.924 at%), but unfortunately did not provide results of thermal conductivity measurement [7]. Ruf et al. measured the thermal conductivity of isotopically enriched silicon single crystal (28 Si: 99.8588 at%) between 2 and 310 K and found the maximum value of about 30000 W/m K around 20 K, which is six times larger than that of natural silicon [8]. And they also indicated that in the highly isotopically enriched sample, the T 3 behavior of thermal conductivity can be observed for temperatures up to higher temperature than that of natural silicon [9]. In order to further clarify the isotopic effects on the T 3 behavior of thermal conductivity in isotopically modified silicon, in this study another important thermal
0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 6 7 9 - 5
Q. Hu et al. / Physics Letters A 299 (2002) 656–659
parameter, specific heat, was measured at low temperature for isotopically enriched silicon and natural silicon.
Table 1 Impurities in the isotopically enriched 28 Si and natural Nat. Si single crystals C 28 Si (ppm)
2. Experimental Isotopically modified silicon single crystals (28Si and Nat. Si) were formed by the floating-zone method (FZ) at 1673 K, and the 111 growth direction was determined by the X-ray Laue backscattering method. The isotopic composition was measured by secondary-ion-mass spectroscopy (SIMS) and found to be 28 Si (99.88465 at%), 29 Si (0.07689 at%), and 30 Si (0.03845 at%). Also, the ICPMS analysis result of other impurities in the isotopically enriched 28 Si and natural Nat. Si single crystals were shown in Table 1. The calorimetric technique used here is known as the time-constant or relaxation method. The range of the measured temperatures was between 1 and 140 K; these temperatures were reached and maintained by a sorption-pumped 3 He system. To mount the sample, one face of a small sample pellet (mass 2–6 mg, 2 × 2 mm2 ) was bonded with a known amount of low-temperature grease to a sapphire chip on which a serpentine metallic heater was evaporated. A Cernox temperature sensor was attached to the chip with 50 µm gold leads. The chip was suspended by 20 µm tungsten leads to realize electric connection and a thermal link to the calorimeter cell. The heat capacities of the chip and grease were determined separately and subtracted as shown in Ref. [5].
657
Nat. Si (ppm)
< 165 < 48
H 100 < 144
B
Fe
Mg
0.015 0.02
< 0.3 12
< 0.3 0.2
(a)
3. Results and discussion Figs. 1(a) and (b) show the plot of specific heat divided by T versus T 2 for isotopically enriched 28 Si and natural silicon. The straight lines are the fitting curves according to the data of T < 5 K. The two special temperatures of 7.2 and 14.5 K in Fig. 1 are determined as follows: (1) fitting the initial part of curvature at low temperature (T < 5 K) with linear equation; (2) extending the liner to the high temperature; (3) determining the temperature value when the experimental data starts to deviate from liner relation (relative difference > 5%). It can be seen that the low-temperature specific heat of 28 Si diverges
(b) Fig. 1. Temperature square dependence of specific heat divided by T for natural silicon (a) and isotopically enriched 28 Si (b).
from Debye T 3 beginning at 14.5 K, in contrast to natural silicon, where deviations from the T 3 behavior already occur at a much lower temperature of 7.2 K. However, no obvious difference in specific heat at T < 20 K between the two kinds of samples, such as that which we determined in the case of isotopically modified boron [4], has yet been found.
658
Q. Hu et al. / Physics Letters A 299 (2002) 656–659
Using Debye’s theory [9], one can now calculate the specific heat of a solid, which is proportional to T 3 at a sufficiently low temperature. In other words, the lattice vibration in this assumption is a nearly harmonic vibration. According to Debye’s theory, the atomic system is assumed to be an elastic medium in which only certain frequencies can be excited and maintained at low temperature. In particular, at very low temperatures, the thermal activation of a specific heat measurement can mainly cause excitation of a long acoustic phonon wave. In the two samples with different 28 Si contents, the enriched 28 Si has lower average atomic mass and higher vibration frequency of the lattice than natural silicon. From the viewpoint of the absolute value of specific heat, this would lead to a lower specific heat for isotopically enriched 28 Si at a given temperature compared with natural silicon. Here we are concerned with the isotopic effect on the harmonic lattice vibration or Debye T 3 behavior at low temperature, which was already observed in Fig. 1. Because the isotopically enriched 28 Si crystals already have a decreased minor isotope (29 Si and 30 Si) content compared with natural silicon crystal, this will obviously increase the isotopic homogeneity of the 28 Si crystal lattice from the viewpoint of defect physics. At the same time, the enrichment of 28 Si influences the frequency distribution of harmonic lattice vibration, and decreases or deletes the low-frequency term from 29 Si and 30 Si. Moreover, this enhances the ability to resist anharmonic occurrences so that the near harmonic lattice vibration is maintained at higher temperatures compared with natural silicon. In order to understand the isotopic effect on specific heat at high temperature, the anharmonic effect must be considered. Fig. 2 shows that the specific heat of isotopically enriched silicon (28 Si concentration > 99.88465 at%) starts to deviate from the conventional behavior with increasing temperature. In addition, the specific heat of isotopically enriched silicon becomes significantly higher than that of natural silicon when the temperature is higher than 20 K. This is because when the lattice vibrations with higher frequencies are being excited with increasing temperature, more modes or oscillators will come into operation than the Debye theory predicts. Also, the interaction between atoms will have a marked influence on vibration of the increased modes. Thus, the anharmonic effect is the main cause of the increase of spe-
Fig. 2. Specific heat versus T for isotopically enriched 28 Si and natural silicon.
cific heat for silicon at higher temperatures, which cannot be explained by the change of volume mass alone. Therefore, in some temperature regions (T > 14.5 K for isotopically enriched 28 Si and T > 7.2 K for natural silicon), the thermal energy is sufficient produce anharmonic lattice vibrations. Considering one aspect, the isotopic effect of enriched 28 Si may change the amplitude of lattice vibrations; this kind of change is related not only to average atomic mass, but also to atomic interaction in the isotopically enriched 28 Si lattice. From another aspect, the isotopic effect will influence the zero point energy of the lattice system even at very low temperature; although it should be reflected in the specific heat at low temperature related to the Debye behavior, however, this has not yet been found. According to this analysis, the frequency distribution between ω and ω + dω is proportional, not to ω2 as in Eq. (1), but to a higher power of ω. At higher temperatures, an additional term should be involved in the specific heat,
T C(T ) = 9R D
3 D/T 0
x 2 ex F (ω) dx, (ex − 1)2
(1)
where F (ω) is a frequency distribution which includes an additional term for the higher power of the ω frequency: F (ω) = αω2 + βω4 + cω6 + · · · .
(2)
Q. Hu et al. / Physics Letters A 299 (2002) 656–659
Table 2 The coefficients in Eq. (3) |a| (J g−1 K−4 ) |b| (J g−1 K−5 ) |c| (J g−1 K−7 )
28 Si
Nat. Si
8.5903 × 10−7
8.3784 × 10−7 6.8135×10−11 1.8255×10−15
6.7007×10−11 1.7826×10−15
Furthermore, the corresponding expansion of the specific heat is C(T ) = aT 3 + bT 5 + cT 7 + · · · .
(3)
The determination of the coefficients a, b and c may be obtained through the fitting process of the experimental results of specific heat. The comparison of relative coefficients for isotopically enriched 28 Si and natural silicon is a direct way of investigating the isotopic effect on high power terms of temperature in the specific heat expression. Table 2 shows that the contribution of the T 3 term to specific heat in isotopically enriched 28 Si is greater than that in natural silicon, even though the high power terms of temperature (T 5 and T 7 ) in isotopically enriched 28 Si have a smaller contribution than that from natural silicon. It is considered that in the present temperature region (1–140 K), the effect of harmonic lattice vibrations dominates the behavior of specific heat in isotopically enriched 28 Si. The main reason is that the enriched 28 Si lattice system can more effectively suppress the anharmonic effect than natural silicon by increasing the isotopic homogeneity of the lattice. Therefore, the difference in specific heat at high temperature (T > 20 K) can be observed in Fig. 2.
4. Conclusion In summary, at low temperature (T < 20 K), enriched 28 Si single crystals have an isotopic effect
659
on specific heat with different average lattice atomic mass, which changes the frequency of harmonic lattice vibrations and maintains Debye T 3 behavior at T < 14.5 K, as compared with natural silicon (T < 7.2). At high temperature (T > 20 K), the contribution to anharmonic lattice vibrations or additional terms in isotopically enriched 28 Si is smaller than that in natural silicon, yielding the higher specific heat.
Acknowledgement The authors are grateful to Dr. Y. Morishita of AIST for the measurements of silicon isotope abundance using SIMS.
References [1] J. Spitzer, T. Ruf, M. Cardona, W. Dondl, R. Schorer, G. Abstreiter, E.E. Haller, Phys. Rev. Lett. 72 (1994) 1565. [2] L. Wei, P.K. Kuo, R.L. Thomas, Phys. Rev. Lett. 70 (1993) 3764. [3] M. Asen-Palmer, K. Bartkowski, E. Gmelin, M. Cardona, A.P. Zhernov, A.V. Inyushkin, A. Taldenkov, V.I. Ozhogin, K.M. Itoh, E.E. Haller, Phys. Rev. B 56 (1997) 9431. [4] Q. Hu, T. Noda, H. Suzuki, T. Numazawa, O. Arai, N. Nogi, S. Tanaka, J. Appl. Phys. 90 (2001) 728. [5] A.K. McCurdy, H.J. Maris, C. Elbaum, Phys. Rev. B 2 (1970) 4077. [6] E.E. Haller, J. Appl. Phys. 77 (1995) 2857. [7] K. Takyu, K.M. Itoh, K. Oka, N. Saito, V.I. Ozhogin, Jpn. J. Appl. Phys. 38 (1999) L1493. [8] T. Ruf, R.W. Henn, M. Asen-Palmer, E. Gmelin, M. Cardona, H.-J. Pohl, G.G. Devyatych, P.G. Sennikov, Solid State Commun. 115 (2000) 243. [9] P. Debye, Ann. Phys. 39 (1912) 789.