Superconductivity of thin wire networks with very small meshes

Solid State Communications, Printed in Great Britain.

0038-1098/91$3.00+.00

Vol. 78, No. 1, pp. 45-48, 1991.

Pergamon Press plc

SUPERCONDUCTIVITY

OF THIN WIRE NETWORKS WITH VERY SMALL MESHES Yuji Suwa

Department

of Applied Physics, Science University Shinjuku-ku, Tokyo 162, Japan

of Tokyo,

and Masaru Tsuksda Department

of Physics, (Received

University

of Tokyo, Bunkyo-ku,

28 December

Tokyo 113, Japan

1990 by T. Tsuzuki)

For the purpose of the studies of the artificially designed wire networks with very small meshes, a new model of a square lattice wire network is proposed. By solving that model, the single electron states of the wire lattice are clarified. The energy dispersion, the density of states and the analytic expression of the Green’s function are obtained. Superconducting transition temperature T, in the absence of the magnetic field is calculated and it is pointed out that T, oscillates as the function of the size of the meshes.

the two or three dimensional nature according to the network dimensionality. In the case of the superconducting wire networks, two kinds of crossovers are expected due to the existence of the two kinds of characteristic length i.e., the coherence length and mean free path. For the purpose of the theoretical studies of the above problems, a model which can represent both limits of the mesh size by changing a few parameters is required, and it is significant to make clear the single electron states of such a model. In this letter, we propose a new model for the square lattice wire networks which is valid in both macroscopic and microscopic scales. By solving that model of wire lattice, we calculate the energy dispersion of the single electron states and we get the figure of the density of states (DOS). The analytic expression of the Green’s function is also obtained. On the basis of these knowledge of the single electron states, one can calculate any physical quantity. As an example, we calculate the superconducting transition temperature T, in the absence of the magnetic field assuming the eiistence of the pairing interaction. It is shown that T, oscillates as the function of the mesh size L. That is an example of the interference effect of the electronic wave function which is significant when L is as small as the mean free path. An experimental difficulty is that the pattern size presently fabricable is larger tha.n a typical size of an electronic mean free path, but we believe that this problem will be solved by the progress of the micro fabrication technology in the near future. We consider the square lattice of very thin wires, where infinite wires are weakly coupled at each intersecting point. Figure 1 shows the schematic view of our

Recent development of the micro fabrication technology’ 9’ enables us to create a new kind of materials. Artificially designed micro structures are now targets of the basic scientific study. Among such a kind of materials, superconducting wire networks are well studied experimentally3t4v5 and theoretically6~7~8*g*10*11. A very interesting point of those systems is that the interference effect of the order parameter reflecting the network pattern can be observed. The upper critical field shows those effects and it is calculated from the Ginzburg-Landau equation suited to the wire networks,6t7v8 which is a kind of a wave equation treating the order parameter as a wave function. This effect is observable because one can make network patterns smaller than the superconducting coherence length. Owing to the fact that they are artificially designed, one can arbitrary control the basic parameters of the networks such as the symmetries, the periodicity and the lattice constant. Self-similar patterns418 and quasi-periodic patterns3 are well studied. We are interested in controlling the lattice constant from the macroscopic size to the microscopic size. When the mesh size is diminished and becomes smaller than the electronic mean free path, other kinds of interference effects are expected as the consequence of the interference of the single electron’s wave function. They will be seen even in the normal metal wire networks as the modification of the conductivity, Hall coefficient, magneto resistance, etc. It is an interesting point to see the dimensional crossover of these quantities with the change of the mesh size. When the mesh is macroscopic size, the networks will show the nature of the one dimensional wires, and when it is microscopic, the system will show 45

46 model.

THIN WIRE NETWORKS WITH VERY SMALL MESHES Hamiltonian

is represented

as

j-1

Vol. 78, No. 1 j

j+l

j+2

.._

i+l

i-l

A2kZ Eli

=

2m

-

(2)

/h

where a:,, and aojk are the creation and annihilation operator of the electron with spin Q, wave number k, on the wire named 1, respectively. The transfer energy from the k state on the wire i parallel to the s-axis to the k’ state on the wire j parallel to the y-axis is represented as tiklk’

=

ta

_e-‘kz,+lk’yi,

(3)

C

where t, u, t are the transfer integral, the width of the wire, and the size of the specimen, respectively. Throughout this letter, we use the character i as the index corresponding to the wire parallel to the z-axis, the character j as the one parallel to the y-axis, and the character 1as the one of the two kinds [see Fig. 11. This model is analogous to the Cot&on’s free electron network mode112*13, but it is different in the nature of the intersecting point. The former describes the connection as the existence of the transfer energy between the two infinite wires, and the latter treats it as the ends of four wires where wave function must be continuous. The advantage of our model is that the Hamiltonian is simply expressed without any special boundary condition, while the latter model requires special continuity condition between r-wire and y-wire. Owing to this, calculation of the single electron states and the Green’s function is easily performed. This system can be solved by using Bloch’s theorem because it has a translational symmetry of the square lattice. The eigen energy E with crystal momentum (k,, k,) are determined by the following equations,

(4) sinh2 pL (coshpL

- cos k,L)(coshpL

- cos k, L)

for E < 0,

Fig. 1. A schematic view of the model system. The crossing wires have a transfer energy between them at the intersecting point. The wires parallel to the x-axis are numbered as i - 1, i, i + 1 and the wires parallel to the y-axis are numbered as j - 1, j, j + 1 The mesh size is represented by L.

(4

(b)

TXM

(070)

(18)

(171)

(cl

r TXM

r

(W

Fig. 2. The band structure of the wire lattice with the mesh size (a) L = 12 a.u., (b) L = 24 a.u. and (c) L = 48 a.u., where 1 a.u. is equal to the Bohr radius. X and M point corresponds to (1,O) and (1,l) point of the first Brillouin zone, respectively. E is marked so that the lowest energy of the wire in the absence of the inter-wire transfer is equal to zero.

(5)

sin2 pL (cospL

- cos k,L)(cospL

- cos k,L)

for E > 0.

(6)

From the above equations, it is concluded that one negative energy state may exist in addition to the infinite numbers of the positive energy states. The negative energy solution, which can not be seen in the system with no inter-wire transfer, corresponds to a state localized to a pomt of intersection. Figure 2 shows the energy dispersions for several systems with different L. (a), (b) and (c) corresponds to L = 12 a.u., L = 24 a.u. and L = 48 a.u., respectively, where 1 a.u. is equal to the Bohr radius. Energy E is marked so that the lowest energy in the absence of the inter-wire transfer is equal to zero. It should be noted

that the band structure at different values of L can not be related by a trivial scale transformation. The density of states (DOS) of these systems are plotted in Fig. 3. The structure of the DOS’s in Fig. 3 can be interpreted as the summation of the DOS of the twodimensional cosine-like bands. One dimensional nature can be seen as the envelope fnnction of the DOS which is proportional to I/&?. The system with larger L has more peaks of DOS than the system with smaller L, due to the folding of the Brillouin zone. Here only the positive energy region is plotted. In the negative energy region, there is only one sharp peak. There is an energy gap between the negative energy state and the lowest positive energy state in every figures. The small energy gaps also can be seen between the positive energy states

Vol. 78, No. 1

in Figs. 3(b) and 3(c). The existence of these energy gaps suggests the possibility of the metal-insulator or the metal-semiconductor transition due to the change of L. For this model system, the Green’s function is obtained as follows. The definition of the Green’s function is Gyk,ek’(~ - T’) = -(Trab,k(7)a~,,k,(7’)). (7)

(16) By making a standard approximation for the anomalous Green’s function, a gap equation linearized for the order parameter is obtained as A;

The equations of motion for the Fourier transform with respect to 7 become GP,il k~(%)

=

gkbit’6k -gk

=

c

1,

Ii;, k’ =

k,G;,

k, ,Id“h),

(9)

kl

calculation,

(10) one can obtain

where

-iA.!?

1 - &L

h2y (1 - d-f-'+)(I_

&+k)L)j

(13) (14)

Superconducting transition temperature can be calculated by using this Green’s function if we assume the superconducting Hamiltonian as

H = Ho+ He-e,

4.0

\

(17)

~~~c~k~~k-*~-w.)Gp.-k~,-k+,(w.).

is given by the highest temperature which makes the eigenVdUe Of the matrix Kkk8 unity. Figure 4 shows the L dependence of T, for the two values of the fermi energy. Oscillating behavior of T, as the function of L can be seen in both figures. The wavelength of the oscillation is constant in each figure. It is roughly equal to the fermi wavelength. This behavior of T, is explained by the behavior of the DOS. When L is increased smoothly, the peaks of the DOS move toward the point of E = 0 [see Fig. 31. The peaks of the DOS pass the fixed fermi energy successively when L increases. T, is strongly affected by the DOS at the fermi energy, so it oscillates as the function of L. The small cusps at each bottom of the waves in Fig. 4(b) are due to the lower cusps in the DOS curve. The amplitude of the oscillation is larger when L is smaller. This is because the oscillation effect originates the interference of the electronic wave functions in the network, and larger L results in the decrease of the phase coherence by the scattering. In this letter, we proposed a model for the thin wire square lattice networks which is applicable even when the meshes are smalter than the mean free path. Physical quantities can be calculated by using the exactly obtained Green’s function in any size of meshes, as far as it is larger than the atomic scale. As an example of the physical quantity, we calculated the superconducting transition temperature in the absence of the magnetic field by using Green’s function. We pointed out that T, oscillates as the

T,

kl

gk = (iw,, - +_)-l.

E

Ickk’A;,,

(8)

where

fk

c

(18) ttkn

-gkCCtikJIkrG~,krJ1kl(W,),

After some straightforward

=

of Go

kl c 31

G!kpk~(“‘n)

47

THIN WIRE NETWORKS WITH VERY SMALL MESHES

Cc) L = 48 a.“.

1 I I 2.0

3. The densities of states per unit length of the wire. Only the positive energy region is plotted

hgeSt

(4

.lO

EF = 0.2

eV

.05

.OO .lO

(b) Ep = 0.5 ev

.05

.OC

20

40

60 L (a.u.)

80

100

Fig. 4. T, vs. the mesh size L. The wavelength of the oscillation is roughly equal to the fermi wavelength in each figure.

48

Vol. 78, No. 1

THIN WIRE NETWORKS WITH VERY SMALL MESHES

function of the size of meshes L. It might be observed by the measurement of T, for a number of samples ,with different size of small meshes. Acknowledgement - We would like to thank Mr. K. Kobayashi and Mr. A. A. Yamaguchi for useful discus-

sions. This work was partly supported by the Grantin-Aid for Special Distinguished Research (No.02102002) from the Ministry of Education, Science and Culture of Japan.

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