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Electron heat capacity and moments of phonon density of states of HoBa2Cu3O7−δ
PHYSICA ELSEVIER
Physica C 262 (1996) 143-148
Electron heat capacity and moments of phonon density of states of HoBa2Cu3OT_ V.N. Naumov, G.I. Frolova *, E.B. Amitin, V.E. Fedorov, P.P. Samoilov Institute of Inorganic Chemistry, Nouosibirsk, 630090, Russian Federation Received 19 October 1995; revised manuscript received 13 February 1996
Abstract
The heat capacity of superconducting HoBa2Cu307_ , ceramics has been measured in the temperature interval 14--315 K. The electron component of the heat capacity 3'T was separated in the normal-state region (above To). The special technique for separating the heat capacity into phonon and electron components was used. Besides the 3' value, the numerical values of the second moment 192, of the fourth moment 194 and of the moment 19, attributed to the upper limit of the phonon density of states were obtained. The obtained moments appear to be somewhat smaller than the analogous ones for YBCO. This decrease characterizes the phonon spectrum softening owing to the substitution of a light ion, Y, by a heavy ion, Ho. The obtained moments allow one to calculate precisely the heat capacity at constant volume C,(T) in the temperature region T > 19,/2"tr.
1. Introduction
The heat capacity measured in the range between helium and room temperature allows one to determine a number of important characteristics of hightemperature superconductors (HTSC) in the superconducting and normal states [1]. In the work of Ref. [2], for example, a special technique had been proposed for calculation of the Sommerfeld constant 3~ (the electron heat capacity coefficien0 and the moments of the phonon density of states (DOS) from the experimental heat capacity. The main requirements in this way are a high precision of the measurements and a precise specification of the sample. Unfortunately, there are few data which suit these requirements. In particular, because of this there is still no definiteness in the value of 3, for compounds * Corresponding author. Fax: + 7 383 235 59 60.
of the YBCO system with rare earth (RE) substitutions of the Y ion. It seems to be interesting to treat in a satisfying way the above requirement for heat capacity data for HTSC ceramics (RE)BCO by means of the technique of Ref. [2] and to obtain electron and phonon characteristics. In Ref. [3] the electron heat capacity v T and several moments of the phonon DOS were calculated on the basis of the experimental heat capacity of the superconducting ceramics YBa2Cu307_ ~. The technique mentioned is based on the description of the lattice heat capacity by means of a finite number of moments of the phonon DOS in the high-temperature expansion, an infinite number of terms of this expansion being virtually taken into account. We obtained (92 , the second moment, ~94, the fourth moment and ~9., the moment attributed to the upper limit of the phonon spectrum (instead of the traditional Debye temperature (90).
0921-4534/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 3 4 ( 9 6 ) 0 0 1 8 1 - 5
144
V.N. Naumov et al./ Physica C 262 (1996) 143-148
In the present work the heat capacity of superconducting ceramic HoBa2Cu307_ 8 is measured and the values of the Sommerfeld constant 3' and the moments @2, @4 and @, of the phonon DOS are obtained. These characteristics are extracted from the heat capacity in the temperature region of the normal state (above Tc).
H 0 2 0 3 , BaCO 3 and CuO taken as the initial substances in appropriate weights. The homogenized mixture was pressed into a tablet and annealed in air at 920°C for 24 h. After cooling in the furnace which had been switched off the tablet was ground, homogenized and pressed anew. The tablet obtained in such a way was annealed at 930°C for 48 h and cooled to 650°C at a rate of 2-3°C/min; then the furnace was switched off. X-ray phase analysis of the sample showed no other phases distinct from the 123 phase. The oxygen content was determined iodometrically [4] with an error of +0.02 to be 6.87 per formula unit. The high-purity
2. Experimental A polycrystalline sample of HoBa2Cu307_ 8 was prepared according to the standard procedure with
Table 1 Heat capacity of HoBa2Cu307_ 8 (M = 740.16 g m o l - i, 6 = 0.13)
T
Cp
T
Cp
T
Cp
T
Cp
(K)
(J tool- J K - l)
(K)
(J mol- 1 K - 1)
(K)
(J mol- i K - i)
(K)
(J mol- 1 K - 1)
14.41 15.11 15.86 16.53 17.30 18.16 20.07 21.52 23.63 25.84 27.59 29.10 31.08 33.96 37.15 40.05 42.61 45.48 48.55 51.87 55.86 59.85 61.52 63.45 65.19 67.16 69.04 70.64 72.87 73.54 75.34 76.02 76.46 76.69
6.296 6.586 6.907 7.369 7.919 8.566 10.29 11.73 14.16 17.01 19.54 22.08 25.69 30.64 35.82 40.78 45.43 50.78 56.36 62.31 69.81 77.25 80.66 83.58 86.43 89.92 93.49 96.06 100.35 101.37 103.88 105.59 106.25 106.75
77.51 78.47 79.42 79.87 81.28 82.19 83.09 83.12 83.98 84.86 85.74 85.76 86.24 86.60 87.46 88.29 88.80 89.11 89.65 89.92 90.72 91.18 91.51 92.30 93.07 93.51 93.85 94.62 95.38 95.79 96.15 96.90 96.97 97.65
107.26 109.18 I 11.30 111.76 114.14 115.61 117.26 117.58 118.50 120.22 121.64 121.64 122.99 123.34 124.73 126.34 127.37 127.74 128.60 129.82 131.15 131.84 132.40 134.71 131.92 132.34 132.06 133.14 134.06 134.95 135.67 136.70 136.27 137.98
98.02 99.10 99.14 100.21 100.46 100.79 101.71 102.36 104.22 108.23 113.35 117.27 121.47 125.60 129.65 133.62 137.53 141.39 145.18 148.94 152.64 155.66 156.31 158.88 159.94 162.06 163.83 165.22 168.36 171.48 174.56 177.86 181.33 184.79
i'37.63 139.74 140.19 140.66 141.06 141.79 142.68 143.52 145.85 151.29 157.78 162.46 167.80 172.70 177.04 182.14 186.32 190.49 194.72 198.51 202.12 204.05 205.30 207.34 208.52 210.05 212.13 212.68 215.46 218.19 220.23 223.48 226.51 229.18
188.22 191.63 195.02 198.40 198.85 201.97 204.18 205.76 209.46 209.53 213.28 217.00 219.92 220.71 224.40 225.09 228.07 230.23 231.73 235.33 235.38 240.40 245.29 250.28 255.53 260.76 265.97 271.16 276.32 280.58 283.34 301.96 314.80
231.97 234.19 236.52 238.61 238.77 241.05 242.41 243.58 245.69 245.86 247.79 249.76 251.77 252.28 253.92 254.47 256.34 257.53 257.91 259.91 260.60 262.27 265.52 267.50 270.33 272.55 274.72 276.11 278.06 280.61 280.74 284.41 289.73
V.N. Naumov et al. / P hysica C 262 (1996) 143-148
composition of the sample could therefore be described by the formula HoBa2Cu306.87 :t: 0.02We determine the molar ratio through this structural unit. The number of atoms n in this structural unit is 12.87, and the corresponding molecular weight M is 740.16 g. The heat capacity was measured by the adiabatic pulse method in the range 14.41-314.80 K. The mass of the sample was 5.4778 g. The method of measurement and the characteristics of the experimental installation are described in Ref. [5]. There were 135 experimental points obtained of the heat capacity. The experimental points are presented in Table 1 and Fig. 1. The average deviation of the experimental points from the smoothed curve is ~ 0.45% in the interval 14-20 K, 0.31% in the interval 20-100 K including the region of phase transition, and 0.24% above 100 K. The superconducting transition temperature occurred at 92.3 K (the temperature where the maximum of the derivative AC/AT is positioned).
3. Calculation of electron and phonon characteristics From the experimental C p ( T ) data, the Sommerfeld constant y, and the moments of the phonon DOS, the second moment g92, the fourth moment (94 and the moment ~9,, attributed to the upper limit of the phonon spectrum, were calculated. The technique
145
of calculating these characteristics had been developed in Ref. [2]. It is based on the high-temperature expansion of the lattice heat capacity. Here we present the formulas which are necessary for the treatment of the experimental data. The lattice heat capacity in the harmonic approximation can be presented by the formula
C(r) 3Nk 02
-1
044
12T2
04
e-Y + T2e----~,[]~ + (1)
, Izl < 2-rr. T Here N is the number of atoms, k is the Boltzmann constant. For metals and superconductors the left side of the equation contains one more term, the electron component, and is of the form (C(T)3,T)/3Nk. (Strictly speaking the term 3,T includes both the electron component and the linear anharmonic one.) It is convenient to introduce the special coordinates Y and X: ~(z) =e=+e-Z-
2,
z=
Y('y, T, C) = 12T2[1 - (C(T) - "yT)/3Nk],
(2)
x(e,,r)=
ie,/T)
1-
.6." .°
• • ° ° ° ¢°~' . o . ' * ° ° ° ° ' ~ *
i. o'°'~ • °
7 200 Q°
///
E 15o
-D
~J100 O_ CD
,o
..¢"
5O
°, •-
50
I00
. (3)
25O
7 -5
r 1
150
200
250
Temperature (K) Fig. 1. Heatcapacityof HoBa2Cu307_ 8 in the interval 14-315 K.
300
146
V.N. Naumov et al./ Physica C 262 (1996) 143-148 1.0
280 K
+
0.8
.
0.6 0 K >-
04
\ 0.0
i
i
0.0
r
i
I
0.5
i
i
i
i
I
i
i
1.0
i
i
I
i
i
i
1.5
i
I
J
,
,
2.0
,
i
2.5
,
,
, , ~ 1 ,
3.0
X (10 6 K -2) Fig. 2. Heat capacity of HoBa2Cu307 _ ,~ ill terms of the coordinates ( X, Y ) together with data from Ref. [6].
In terms of the Y and X coordinates the temperature dependence of the total heat capacity C(T) is presented by this equation:
Y(y, T, C) = ~92 - ~94X(t9,, T).
(4)
The representation (4) for the heat capacity is true within certain temperature limits. The lower limit 6),/2-rr is determined by the convergence of the high-temperature expansion. The upper limit is the temperature where the anharmonic contribution quadratic in T becomes evident. The anharmonic contribution linear in T, as mentioned above, enters in yT. Ordinarily the characteristic values of these
limits for RBCO compounds are ~ 80-250 K. Within these temperature limits and with the appropriately chosen parameters ~9, and 3/ the experimental points fall on the straight line of Eq. (4). The parameters 6) 2 and ~94 can then easily be determined as the coefficients of this straight line by means of the least-squares method. It should be noted, that the straight line Y(X) describes the lattice heat capacity. The linear components are described by the parameter y. This technique had been used for calculation of both the Sommerfeld constant in the region of the normal state and moments ~92, O 4 and the moment
Table 2 Electron and phonon characteristics of HoBa2Cu307 _ ,s and YBa2Cu307 - 6. Tc is the superconducting transition temperature, n is the number of atoms in one structural unit, y is the electron heat capacity constant, (92, O 4 and O , are the characteristic temperatures connected with moments of the phonon spectrum, OD(~) is the Debye temperature
Tc (K) n
HoBCO a
HoBCO b
YBCO c
YBCO a
92.3 12.87
91.8 12.96
92.2 12.92
92.0 12.96
2.72 ± 0.5 430 + 9 486 ± 15 570 + 30 554 ± 11
2.92 + 0.2 427 ± 3 484 + 4 569 ± 8 551 ± 4
2.26 ± 0.2 443 ± 5 509 ± 10 613 + 22 572 ± 7
2.02 + 0.07 448 :i: 2 517 ± 4 627 ± 8 578 ± 3
~,/n (mJ m o l - ] K - 2) (92 (K) 04 (K) ~9, (K) 69D (oc) (K) a b c a
Our Ref. Ref. Ref.
data. [6] data, our analysis. [3] data (our YBCO). [7] data, our analysis.
147
V.N. Naurnov et al./ Physica C 262 (1996) 143-148
~ 8
~"8 •
T 6 E
] •
"TN--
• ,o •
T 6
. ."'~'.'"
E 3
g•
4
4
J2
J2
.
50
100 150 Temperature (K)
.
.
.
,
50
200
.
.
.
.
i
.
.
.
.
1 O0 Temperature
,
.
150 (K)
.
.
.
,
.
.
.
.
200
Fig. 3. Electron heat capacity of HoBa2Cu307_ ~, our data.
Fig. 4. Electron heat capacity of HoBa2Cu307_ ,~, data from Ref. [6], our analysis.
19, for YBCO in Ref. [3] and for HoBCO in the present work. Our experimental points in the X and Y coordinates at the optimal parameters 19, and 7 are shown in Fig. 2 together with the data of Atake et al. [6] that we have treated according to the described technique. The optimal parameters correspond to the minimum of the functional
ity Co(T)(curve 1) and the experimental heat capacity (curve 2). Curve 1 shows a monotonic increase with T, reaching a constant value (Table 2); it is practically constant in the temperature interval 5 1019,). Curve 2 is systematically lower than curve 1, and above the temperature ~ 100 K shows a drop at high temperature. This behavior is caused by the presence of additional components: an electron one and an anharmonic one.
E [ r ( 7 , r~, C,) - e~ + O ~ X ( O , , r,)] ~,
(5)
i
(i is the number of the experimental point) and the uncertainty of the parameters in Table 2 is determined on the confidence level 95% according to the Fisher criterion (see Ref. [2]). As can be seen in Fig. 2 in terms of the given coordinates, the points in the temperature interval 100-300 K fall very well on the straight line intersecting the ordinate axis at 1. The found values of 7, 192, O4 and 19, are presented in Table 2 together with data of YBa2Cu307_ 8, taken from previous works [3,7]. The component 7 T of HoBa2Cu307_ 8 is shown in Fig. 3 (our data) and Fig. 4 (data of Atake et al. [6], our analysis). (The superconducting phase transition can be seen in the region ~ 90 K.) It has been obtained as the difference between the total beat capacity and the lattice one. The last in turn has been calculated according to Eq. (1) with the found parameters 192, 194 and O , . It should be noted, that Eq. (1), when the optimal parameters 192, 194 and 19, are found, gives the best description of the heat capacity at constant volume Co(T) above the temperature 1 9 , / 2 w . In Fig. 5 the dependence of the Debye temperature 19D(T) was calculated from both the heat capac-
4. Results
(1) The heat capacity of the single-phase superconducting ceramics HoBa2Cu307_ ~ was measured in the interval 14-315 K. (2) The component 7 T was extracted from the total heat capacity in the region of the normal state
..--..
~( 5 0 0 43
g
/
4(?O .o
~
/
300
I--
gi 200
......... 0
50
' .... 1O0
' ........ 150
Temperature
, ' ....
200
250
'' 300
(K)
Fig. 5. Debye temperature ~D(T) calculated from C,,(T) (curve 1) and from the experimental heat capacity (curve 2).
148
V.N. Naumov et a l . / Physica C 262 (1996) 143-148
(above Tc) from our data and from data due to A take [6]. The results are presented in Table 2. The errors in Table 2 are the uncertainties of the parameters at the minimum of the functional (5) on the confidence level of 95%. According to our evaluation, the linear contribution from the anharmonicity does not exceed 10-15% of the electron one [2]. Hence, the linear contribution y T that we found represents mainly the electron heat capacity. As seen in Table 2, both values Y that we found are in good agreement. These y values exceed the y ' s found from data of Refs. [3] and [7] for YBa2Cu307_ ~ by ~ 25%. (3) The moments ~92, /94, O , of the phonon DOS and the Debye temperature @D(~) were calculated from our data and from data of Ref. [6]. As seen in Table 2, our moments and Debye temperature coincide very well with corresponding values calculated from data in Ref. [6]. The comparison of the measured moments for HoBa2Cu307_ ~ with analogous characteristics for YBa2Cu307-8 [3,7] shows that all moments and the Debye temperature for HoBCO appear to be somewhat smaller than the corresponding values for YBCO. In our opinion this can be explained by the softening of the phonon spectrum due to the substitution of a light Y ion by a heavy Ho ion.
(4) As seen in Table 2 for every sample investigated, the value tg, is appreciably higher than the value ~gD(OC). This is caused by the difference between the real phonon spectrum and the model Debye spectrum. Especially important is the fact that the real spectra of the investigated compounds do not have the high density of states near the upper limit required by the model Debye spectrum. This means that the moment ~9, characterizes the upper boundary of the phonon DOS better than the Debye temperature ~gD(T) does (calculated even on the asymptote).
References [1] J.W. Loram, K.A. Mirza, J.R. Cooper and W.Y. Liang, Phys. Rev. Lett. 71 (1993) 1740. [2] V.N. Naumov, Phys. Rev. B 49 (1994) 13247. [3] V.G. Bessergenev, Yu.A. Kovalevskaya, V.N. Naumov and G.I. Frolova, Physica C 245 (1995) 36. [4] N.F. Zakharchuck, T.P. Fedina and N.S. Borisova, Sverchprovodimost FChT 4 (1991) 1391 (in Russian). [5] V.N. Naumov and V.V. Nogteva, Pribory i technika eksperimenta (1985) 186 (in Russian). [6] T. Atake, Q.Z. Zhang, Y. Takagi and Y. Saito, Report of the research laboratory of engineering materials, Tokyo Institute of Technology (1989) 11. [7] T. Atake, A. Honda and H. Kawaji, Physica C 190 (1991) 70.
Physica C 262 (1996) 143-148
Electron heat capacity and moments of phonon density of states of HoBa2Cu3OT_ V.N. Naumov, G.I. Frolova *, E.B. Amitin, V.E. Fedorov, P.P. Samoilov Institute of Inorganic Chemistry, Nouosibirsk, 630090, Russian Federation Received 19 October 1995; revised manuscript received 13 February 1996
Abstract
The heat capacity of superconducting HoBa2Cu307_ , ceramics has been measured in the temperature interval 14--315 K. The electron component of the heat capacity 3'T was separated in the normal-state region (above To). The special technique for separating the heat capacity into phonon and electron components was used. Besides the 3' value, the numerical values of the second moment 192, of the fourth moment 194 and of the moment 19, attributed to the upper limit of the phonon density of states were obtained. The obtained moments appear to be somewhat smaller than the analogous ones for YBCO. This decrease characterizes the phonon spectrum softening owing to the substitution of a light ion, Y, by a heavy ion, Ho. The obtained moments allow one to calculate precisely the heat capacity at constant volume C,(T) in the temperature region T > 19,/2"tr.
1. Introduction
The heat capacity measured in the range between helium and room temperature allows one to determine a number of important characteristics of hightemperature superconductors (HTSC) in the superconducting and normal states [1]. In the work of Ref. [2], for example, a special technique had been proposed for calculation of the Sommerfeld constant 3~ (the electron heat capacity coefficien0 and the moments of the phonon density of states (DOS) from the experimental heat capacity. The main requirements in this way are a high precision of the measurements and a precise specification of the sample. Unfortunately, there are few data which suit these requirements. In particular, because of this there is still no definiteness in the value of 3, for compounds * Corresponding author. Fax: + 7 383 235 59 60.
of the YBCO system with rare earth (RE) substitutions of the Y ion. It seems to be interesting to treat in a satisfying way the above requirement for heat capacity data for HTSC ceramics (RE)BCO by means of the technique of Ref. [2] and to obtain electron and phonon characteristics. In Ref. [3] the electron heat capacity v T and several moments of the phonon DOS were calculated on the basis of the experimental heat capacity of the superconducting ceramics YBa2Cu307_ ~. The technique mentioned is based on the description of the lattice heat capacity by means of a finite number of moments of the phonon DOS in the high-temperature expansion, an infinite number of terms of this expansion being virtually taken into account. We obtained (92 , the second moment, ~94, the fourth moment and ~9., the moment attributed to the upper limit of the phonon spectrum (instead of the traditional Debye temperature (90).
0921-4534/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 3 4 ( 9 6 ) 0 0 1 8 1 - 5
144
V.N. Naumov et al./ Physica C 262 (1996) 143-148
In the present work the heat capacity of superconducting ceramic HoBa2Cu307_ 8 is measured and the values of the Sommerfeld constant 3' and the moments @2, @4 and @, of the phonon DOS are obtained. These characteristics are extracted from the heat capacity in the temperature region of the normal state (above Tc).
H 0 2 0 3 , BaCO 3 and CuO taken as the initial substances in appropriate weights. The homogenized mixture was pressed into a tablet and annealed in air at 920°C for 24 h. After cooling in the furnace which had been switched off the tablet was ground, homogenized and pressed anew. The tablet obtained in such a way was annealed at 930°C for 48 h and cooled to 650°C at a rate of 2-3°C/min; then the furnace was switched off. X-ray phase analysis of the sample showed no other phases distinct from the 123 phase. The oxygen content was determined iodometrically [4] with an error of +0.02 to be 6.87 per formula unit. The high-purity
2. Experimental A polycrystalline sample of HoBa2Cu307_ 8 was prepared according to the standard procedure with
Table 1 Heat capacity of HoBa2Cu307_ 8 (M = 740.16 g m o l - i, 6 = 0.13)
T
Cp
T
Cp
T
Cp
T
Cp
(K)
(J tool- J K - l)
(K)
(J mol- 1 K - 1)
(K)
(J mol- i K - i)
(K)
(J mol- 1 K - 1)
14.41 15.11 15.86 16.53 17.30 18.16 20.07 21.52 23.63 25.84 27.59 29.10 31.08 33.96 37.15 40.05 42.61 45.48 48.55 51.87 55.86 59.85 61.52 63.45 65.19 67.16 69.04 70.64 72.87 73.54 75.34 76.02 76.46 76.69
6.296 6.586 6.907 7.369 7.919 8.566 10.29 11.73 14.16 17.01 19.54 22.08 25.69 30.64 35.82 40.78 45.43 50.78 56.36 62.31 69.81 77.25 80.66 83.58 86.43 89.92 93.49 96.06 100.35 101.37 103.88 105.59 106.25 106.75
77.51 78.47 79.42 79.87 81.28 82.19 83.09 83.12 83.98 84.86 85.74 85.76 86.24 86.60 87.46 88.29 88.80 89.11 89.65 89.92 90.72 91.18 91.51 92.30 93.07 93.51 93.85 94.62 95.38 95.79 96.15 96.90 96.97 97.65
107.26 109.18 I 11.30 111.76 114.14 115.61 117.26 117.58 118.50 120.22 121.64 121.64 122.99 123.34 124.73 126.34 127.37 127.74 128.60 129.82 131.15 131.84 132.40 134.71 131.92 132.34 132.06 133.14 134.06 134.95 135.67 136.70 136.27 137.98
98.02 99.10 99.14 100.21 100.46 100.79 101.71 102.36 104.22 108.23 113.35 117.27 121.47 125.60 129.65 133.62 137.53 141.39 145.18 148.94 152.64 155.66 156.31 158.88 159.94 162.06 163.83 165.22 168.36 171.48 174.56 177.86 181.33 184.79
i'37.63 139.74 140.19 140.66 141.06 141.79 142.68 143.52 145.85 151.29 157.78 162.46 167.80 172.70 177.04 182.14 186.32 190.49 194.72 198.51 202.12 204.05 205.30 207.34 208.52 210.05 212.13 212.68 215.46 218.19 220.23 223.48 226.51 229.18
188.22 191.63 195.02 198.40 198.85 201.97 204.18 205.76 209.46 209.53 213.28 217.00 219.92 220.71 224.40 225.09 228.07 230.23 231.73 235.33 235.38 240.40 245.29 250.28 255.53 260.76 265.97 271.16 276.32 280.58 283.34 301.96 314.80
231.97 234.19 236.52 238.61 238.77 241.05 242.41 243.58 245.69 245.86 247.79 249.76 251.77 252.28 253.92 254.47 256.34 257.53 257.91 259.91 260.60 262.27 265.52 267.50 270.33 272.55 274.72 276.11 278.06 280.61 280.74 284.41 289.73
V.N. Naumov et al. / P hysica C 262 (1996) 143-148
composition of the sample could therefore be described by the formula HoBa2Cu306.87 :t: 0.02We determine the molar ratio through this structural unit. The number of atoms n in this structural unit is 12.87, and the corresponding molecular weight M is 740.16 g. The heat capacity was measured by the adiabatic pulse method in the range 14.41-314.80 K. The mass of the sample was 5.4778 g. The method of measurement and the characteristics of the experimental installation are described in Ref. [5]. There were 135 experimental points obtained of the heat capacity. The experimental points are presented in Table 1 and Fig. 1. The average deviation of the experimental points from the smoothed curve is ~ 0.45% in the interval 14-20 K, 0.31% in the interval 20-100 K including the region of phase transition, and 0.24% above 100 K. The superconducting transition temperature occurred at 92.3 K (the temperature where the maximum of the derivative AC/AT is positioned).
3. Calculation of electron and phonon characteristics From the experimental C p ( T ) data, the Sommerfeld constant y, and the moments of the phonon DOS, the second moment g92, the fourth moment (94 and the moment ~9,, attributed to the upper limit of the phonon spectrum, were calculated. The technique
145
of calculating these characteristics had been developed in Ref. [2]. It is based on the high-temperature expansion of the lattice heat capacity. Here we present the formulas which are necessary for the treatment of the experimental data. The lattice heat capacity in the harmonic approximation can be presented by the formula
C(r) 3Nk 02
-1
044
12T2
04
e-Y + T2e----~,[]~ + (1)
, Izl < 2-rr. T Here N is the number of atoms, k is the Boltzmann constant. For metals and superconductors the left side of the equation contains one more term, the electron component, and is of the form (C(T)3,T)/3Nk. (Strictly speaking the term 3,T includes both the electron component and the linear anharmonic one.) It is convenient to introduce the special coordinates Y and X: ~(z) =e=+e-Z-
2,
z=
Y('y, T, C) = 12T2[1 - (C(T) - "yT)/3Nk],
(2)
x(e,,r)=
ie,/T)
1-
.6." .°
• • ° ° ° ¢°~' . o . ' * ° ° ° ° ' ~ *
i. o'°'~ • °
7 200 Q°
///
E 15o
-D
~J100 O_ CD
,o
..¢"
5O
°, •-
50
I00
. (3)
25O
7 -5
r 1
150
200
250
Temperature (K) Fig. 1. Heatcapacityof HoBa2Cu307_ 8 in the interval 14-315 K.
300
146
V.N. Naumov et al./ Physica C 262 (1996) 143-148 1.0
280 K
+
0.8
.
0.6 0 K >-
04
\ 0.0
i
i
0.0
r
i
I
0.5
i
i
i
i
I
i
i
1.0
i
i
I
i
i
i
1.5
i
I
J
,
,
2.0
,
i
2.5
,
,
, , ~ 1 ,
3.0
X (10 6 K -2) Fig. 2. Heat capacity of HoBa2Cu307 _ ,~ ill terms of the coordinates ( X, Y ) together with data from Ref. [6].
In terms of the Y and X coordinates the temperature dependence of the total heat capacity C(T) is presented by this equation:
Y(y, T, C) = ~92 - ~94X(t9,, T).
(4)
The representation (4) for the heat capacity is true within certain temperature limits. The lower limit 6),/2-rr is determined by the convergence of the high-temperature expansion. The upper limit is the temperature where the anharmonic contribution quadratic in T becomes evident. The anharmonic contribution linear in T, as mentioned above, enters in yT. Ordinarily the characteristic values of these
limits for RBCO compounds are ~ 80-250 K. Within these temperature limits and with the appropriately chosen parameters ~9, and 3/ the experimental points fall on the straight line of Eq. (4). The parameters 6) 2 and ~94 can then easily be determined as the coefficients of this straight line by means of the least-squares method. It should be noted, that the straight line Y(X) describes the lattice heat capacity. The linear components are described by the parameter y. This technique had been used for calculation of both the Sommerfeld constant in the region of the normal state and moments ~92, O 4 and the moment
Table 2 Electron and phonon characteristics of HoBa2Cu307 _ ,s and YBa2Cu307 - 6. Tc is the superconducting transition temperature, n is the number of atoms in one structural unit, y is the electron heat capacity constant, (92, O 4 and O , are the characteristic temperatures connected with moments of the phonon spectrum, OD(~) is the Debye temperature
Tc (K) n
HoBCO a
HoBCO b
YBCO c
YBCO a
92.3 12.87
91.8 12.96
92.2 12.92
92.0 12.96
2.72 ± 0.5 430 + 9 486 ± 15 570 + 30 554 ± 11
2.92 + 0.2 427 ± 3 484 + 4 569 ± 8 551 ± 4
2.26 ± 0.2 443 ± 5 509 ± 10 613 + 22 572 ± 7
2.02 + 0.07 448 :i: 2 517 ± 4 627 ± 8 578 ± 3
~,/n (mJ m o l - ] K - 2) (92 (K) 04 (K) ~9, (K) 69D (oc) (K) a b c a
Our Ref. Ref. Ref.
data. [6] data, our analysis. [3] data (our YBCO). [7] data, our analysis.
147
V.N. Naurnov et al./ Physica C 262 (1996) 143-148
~ 8
~"8 •
T 6 E
] •
"TN--
• ,o •
T 6
. ."'~'.'"
E 3
g•
4
4
J2
J2
.
50
100 150 Temperature (K)
.
.
.
,
50
200
.
.
.
.
i
.
.
.
.
1 O0 Temperature
,
.
150 (K)
.
.
.
,
.
.
.
.
200
Fig. 3. Electron heat capacity of HoBa2Cu307_ ~, our data.
Fig. 4. Electron heat capacity of HoBa2Cu307_ ,~, data from Ref. [6], our analysis.
19, for YBCO in Ref. [3] and for HoBCO in the present work. Our experimental points in the X and Y coordinates at the optimal parameters 19, and 7 are shown in Fig. 2 together with the data of Atake et al. [6] that we have treated according to the described technique. The optimal parameters correspond to the minimum of the functional
ity Co(T)(curve 1) and the experimental heat capacity (curve 2). Curve 1 shows a monotonic increase with T, reaching a constant value (Table 2); it is practically constant in the temperature interval 5 1019,). Curve 2 is systematically lower than curve 1, and above the temperature ~ 100 K shows a drop at high temperature. This behavior is caused by the presence of additional components: an electron one and an anharmonic one.
E [ r ( 7 , r~, C,) - e~ + O ~ X ( O , , r,)] ~,
(5)
i
(i is the number of the experimental point) and the uncertainty of the parameters in Table 2 is determined on the confidence level 95% according to the Fisher criterion (see Ref. [2]). As can be seen in Fig. 2 in terms of the given coordinates, the points in the temperature interval 100-300 K fall very well on the straight line intersecting the ordinate axis at 1. The found values of 7, 192, O4 and 19, are presented in Table 2 together with data of YBa2Cu307_ 8, taken from previous works [3,7]. The component 7 T of HoBa2Cu307_ 8 is shown in Fig. 3 (our data) and Fig. 4 (data of Atake et al. [6], our analysis). (The superconducting phase transition can be seen in the region ~ 90 K.) It has been obtained as the difference between the total beat capacity and the lattice one. The last in turn has been calculated according to Eq. (1) with the found parameters 192, 194 and O , . It should be noted, that Eq. (1), when the optimal parameters 192, 194 and 19, are found, gives the best description of the heat capacity at constant volume Co(T) above the temperature 1 9 , / 2 w . In Fig. 5 the dependence of the Debye temperature 19D(T) was calculated from both the heat capac-
4. Results
(1) The heat capacity of the single-phase superconducting ceramics HoBa2Cu307_ ~ was measured in the interval 14-315 K. (2) The component 7 T was extracted from the total heat capacity in the region of the normal state
..--..
~( 5 0 0 43
g
/
4(?O .o
~
/
300
I--
gi 200
......... 0
50
' .... 1O0
' ........ 150
Temperature
, ' ....
200
250
'' 300
(K)
Fig. 5. Debye temperature ~D(T) calculated from C,,(T) (curve 1) and from the experimental heat capacity (curve 2).
148
V.N. Naumov et a l . / Physica C 262 (1996) 143-148
(above Tc) from our data and from data due to A take [6]. The results are presented in Table 2. The errors in Table 2 are the uncertainties of the parameters at the minimum of the functional (5) on the confidence level of 95%. According to our evaluation, the linear contribution from the anharmonicity does not exceed 10-15% of the electron one [2]. Hence, the linear contribution y T that we found represents mainly the electron heat capacity. As seen in Table 2, both values Y that we found are in good agreement. These y values exceed the y ' s found from data of Refs. [3] and [7] for YBa2Cu307_ ~ by ~ 25%. (3) The moments ~92, /94, O , of the phonon DOS and the Debye temperature @D(~) were calculated from our data and from data of Ref. [6]. As seen in Table 2, our moments and Debye temperature coincide very well with corresponding values calculated from data in Ref. [6]. The comparison of the measured moments for HoBa2Cu307_ ~ with analogous characteristics for YBa2Cu307-8 [3,7] shows that all moments and the Debye temperature for HoBCO appear to be somewhat smaller than the corresponding values for YBCO. In our opinion this can be explained by the softening of the phonon spectrum due to the substitution of a light Y ion by a heavy Ho ion.
(4) As seen in Table 2 for every sample investigated, the value tg, is appreciably higher than the value ~gD(OC). This is caused by the difference between the real phonon spectrum and the model Debye spectrum. Especially important is the fact that the real spectra of the investigated compounds do not have the high density of states near the upper limit required by the model Debye spectrum. This means that the moment ~9, characterizes the upper boundary of the phonon DOS better than the Debye temperature ~gD(T) does (calculated even on the asymptote).
References [1] J.W. Loram, K.A. Mirza, J.R. Cooper and W.Y. Liang, Phys. Rev. Lett. 71 (1993) 1740. [2] V.N. Naumov, Phys. Rev. B 49 (1994) 13247. [3] V.G. Bessergenev, Yu.A. Kovalevskaya, V.N. Naumov and G.I. Frolova, Physica C 245 (1995) 36. [4] N.F. Zakharchuck, T.P. Fedina and N.S. Borisova, Sverchprovodimost FChT 4 (1991) 1391 (in Russian). [5] V.N. Naumov and V.V. Nogteva, Pribory i technika eksperimenta (1985) 186 (in Russian). [6] T. Atake, Q.Z. Zhang, Y. Takagi and Y. Saito, Report of the research laboratory of engineering materials, Tokyo Institute of Technology (1989) 11. [7] T. Atake, A. Honda and H. Kawaji, Physica C 190 (1991) 70.