Calculation of the Transition Temperature of High Temperature and Bulk Superconductors H.P. Roeser, F. Huber, M. von Schoenermark, A.S. Nikoghosyan
Calculation of the Transition Temperature of High Temperature and Bulk Superconductors1 H. P. Roeser2, F. Huber3, M. von Schoenermark, A.S. Nikoghosyan4 It is well known and it has been shown experimental evidence that the critical transition temperature Tc of any type of superconductor depends among other criteria on the crystal structure. Very good examples are thallium, mercury, lanthanum, tin and gallium. Also in high temperature superconductor (HTSC) cuprates a relation between the geometry of the crystal and the transition temperature has been found. This investigation of HTSC’s with the model of a resonance effect between the de Broglie wavelength of paired current carrying particles and the dimension of the solid state structure forming a quantum well, has been suggested by the authors in a previous paper [1]. In this model the dimension is given by the distance of the atoms providing the current carrying particles and at the same time the structure acts as a resonator stimulating a coherent phase transition from an electron or particle gas to a condensate. In the following we will show that for HTSC’s working with Cu3+-ions [2] and one component bulk materials having an s2-electron configuration there exists a linear behaviour between the geometry parameter (2x)2 and the inverse of the critical transition temperature Tc. The value x describes the nearest distance between two participating atoms equidistantly lined up on a straight line. Fig. 1 shows the result of the HTSC materials. The value n denotes the number of superconducting and well separated copper-oxygen planes in the unit cell. As an example, the distance x is calculated for the HTSC material La1.85Sr0.15CuO4. Here the dopant atom Sr2+ replaces La3+ resulting in a missing electron in the crystal structure. The undoped and pure compound is an insulator. Each strontium atom soaks up one extra electron from a copper atom leaving a hole in the copper-oxygen plane, which is in the microscopic region of the crystal structure providing the perfect superconducting highway for electrons or holes to travel. Four oxygen atoms need in total 8 electrons which are provided by La3+ + Sr2+ + Cu3+. Therefore the carrier distance will be determined by the doping concentration of the strontium atoms and the current will be provided by Cu3+ in the copper-oxygen plane creating a hole with the mass of two electrons. With the distance between two Cu-atoms of 0.37 nm and an optimum doping concentration Sr0.15/La1.85 = 8.1 % one obtains a carrier distance between two Cu3+-holes of x = 0.37/0.081 nm = 4.57 nm. Obviously the data points are very well fitted by a straight line with a slope of m1 = 2.80 • 10-15 m2K, which clearly demonstrates the linear function between crystal structures of these HTSC’s and their transition temperature. For the investigation of bulk superconductors we have selected materials with an s2-electron configuration. In addition they don’t have a completely filled dn-shell (n>2) or they have an empty f-shell. As an example we take vanadium to demonstrate the determination of x. Vanadium has 2 + 8 + 11 + 2 electrons per shell in an electron configuration of 3d3 4s2 with 23 protons and 48 neutrons. The crystal structure of vanadium has a body-centred-cubic form (bcc) with the cell parameters a = b = c = 0.303 nm and α = β = γ = 90.00. The current flow 1
Paper first received 2nd July
2
Röser and Schönermark are with the Institute of Space Systems, Universitaet Stuttgart, Pfaffenwaldring 31,70569 Stuttgart, Germany; E-mail:
[email protected]
3 4
Steinbeis-Transferzentrum Raumfahrt, Roetestr. 15, 71126 Gaeufelden, Germany Department of Microwave and Telecommunication, Yerevan State University, Alex Manoogian 1, Yerevan, 375025 Armenia
direction is across the centred atom along the diagonal lines which is the shortest distance given by x = a • (3/4)½ = 0.262 nm. Fig. 2 shows the result for the bulk materials. The data points can be fitted by a straight line with a slope of m2 = 1.55 • 10-18 m2K, which is about 3 orders of magnitude smaller compared with HTSC’s in Fig. 1. The curve in Fig. 1 can be very well described by [(2x)2 n-2/3 • 2Meff] • [πkTc] = h2
equ. (1)
which has already been suggested by the authors [1]. The cuprates in Fig. 1 are “using” Cu3+ions with holes and an effective mass of Meff = 2me. The ratio of the slopes m1/m2 = 1,808 is very close to the ratio mp/me = 1,836! This might suggest that for s2-bulk superconductors an effective mass of Meff = 2mp is involved. The data points in Fig. 2 fitted by a straight line could be described by equation (1) too but with n = 1 and Meff = 2mp. Using the geometry parameter x and the effective masses involved we could also estimate the pairing energies which are very close to the experimentally available data for bulk and high temperature superconductors, which are in the THz spectral range (1-40 meV). References [1] H.P. Roeser, F.M. Huber, M.F. von Schoenermark, A.S. Nikoghosyan, “Electron Pairs in a Quantum Well in the THz Region …”, submitted for publication, 2007 [2] P.J. Ford, G.A. Saunders, The Rise of the Superconductors: CRC Press, 2005
[m²] n
-2/3
120
-15 m1 = 2.80 . 10 [m²K] -18 2 z = m1 .y + 1.2 .10 [m ]
100
La1.85Ba0.15CuO4
z = (2x)² 10
-18
.
La1.85Sr0.15CuO4
80
60
Tl-1201
La1.85Sr0.15CaCu2O6
.
40
Tl-1212 Y123
Tl-1223 Tl-1234
20
HTSC Fig. 1
0 0
5
10
15
20
y=T 60
[m²] -20
z = (2x)² 10
25 -3
30
35
40
45
-1
10 [K ]
-18
m2 = 1.55 . 10 m²K -20 2 z = m2 . y - 1.44 . 10 [m ]
50
.
-1 .
In In Sn(β β)
Hg(β β) Hg(α α) Ta
40
30
V
20
s²- BULK SC Fig. 2
10
0 0
0,05
0,1
0,15
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y=T
0,25 -1
-1
[K ]
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0,35
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0,45
