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Density of states for high Tc superconductors in the Kronig-Penney superlattice structure
PHYSICS LETTERS A
Physics Letters A 175 (1993) 58—64 North-Holland
Density of states for high T~superconductors in the Kronig—Penney superlattice structure Juh-Tzeng Lue
and Yen-Pang Liang
Department ofPhysics, National Tsing Hua University, Hsinchu, Taiwan, ROC Received 1 September 1992; revised manuscript received 7 January 1993; accepted for publication 19 January 1993 Communicated by A. Lagendijk
The densities of states (DOSs) at various thickness ratios for superlattices with a periodic superconducting/normal material structure are calculated by using Green functions derived in accordance with the Kronig—Penney model. The DOSs sharply increase near the band edges resulting from Fermi-level nesting. Increasing the thickness of the superconducting layers, the corresponding DOS of the superconducting state will dominate that of the normal state showing the competition of Cooper paired electrons and normal electrons manipulated by the proximity effect. The enhanced DOS near the band edges is crucial for the retention of high critical temperatures for the recently fabricated YBaCuO/Dy(Pr)BaCuO superlattices.
I. Introduction The high critical temperature T~of the layered copper oxide superconductors, and the retention of a rather high T~for the recently developed YBaCuO/Dy (Pr )BaCuO [1,2] superlattices in which the thickness of the interposing insulating PrBa2Cu3O7 layer is even greater than the coherence length of the bulk material imply some new physical features for the superlattice structure. The simplified equation for the critical temperature including the non-phonon-mediated mechanism can be expressed by [3] Tc~1.14ODexP(_1*)_1.146DexP~~~)~
(1)
where °D is some characteristic temperature, N( 0) is the density of states near the Fermi energy, e’ (w, q) is the dynamical dielectric constant at some specific electromagnetic frequency w and wavevector q, and 2 and p~are directly connected with the effective interaction between electrons. The superconducting state appears only when the force between electron pairs is attractive inferring e’ (cv, q) <0 in which case T, has an upper limit. In our other papers [4], we have demonstrated that the effective dynamical dielectric function of the superconducting/insulating (S/N) superlattice is largely suppressed to a negative value by the coupled surface plasmon polariton waves in the infrared region. By inspection of eq. (1), one would expect a high T~for the plasmon-mediated Cooper pairing, because the plasma frequency cvi, is larger than the phonon frequency by a factor of M/m*, where M is the ion mass and m* is the effective electron mass. But2F(w) the plasma frequency /co [5], while it would increase reduce the interaction by the the formula relation ~= 2=2~u/[1+ J~dcv~aln (Ef/kBOD)] which dewould theelectron modified Coulomb parameter repulsion ~2 given through grades T~.Therefore a compromise intermediate plasma frequency would be plausible in promoting high T~. To obtain a high T~,~(w, q) should be slightly below zero. The retention of high T~in the S/N superlattice with a large negative (q, cv) (as low as 15 000) may be due to the enhancement of the density of states near the band edge which compensates the large I e (q, cv) I. Van Gelder [6] first showed that a periodic order parameter 4(y) may give rise to an anisotropic band struc—
58
0375-9601/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 175, number 1
PHYSICS LETFERS A
29 March 1993
ture consisting of adjunct energy bands. This model is inadequate for describing temperature dependent phenomena and does not yield details for a different construction of periodic lattices. Compound-resonance effects showing splitting of de Gennes—Saint James bound states and beating of the oscillations ofthe density of states occurring at electronic energies below and above the energy gap, respectively, in the superconducting—normal metal (SN) sandwich structures were discussed by Gallagher [7]. His work has no analogy to the SN superlattice. Tanaka and Tsukada [8] have developed theories for superconducting superlatticesbased on the Kronig— Penney model [9]. They found that the critical temperature oscillates as a function of the length of the unit period and interference of the wave functions arises due to scattering of the multilayered structure. However, their results for layer thicknesses of 5—10 A might be constrained mostly by the proximity effect and the Josephson tunneling effect that were not taken into account. In this work, we explicitly derive the temperature dependence of the density of states (DOS) following the same approach as given in ref. [8] without using the Gorkov equations while choosing the layer thickness to be close to the experimental reports [1,2]. Multiple bandgaps and enhanced DOS near the band edges are illustrated in this calculation. The DOS for superconducting states is larger than those for normal states in the first energy band and then inverted in the higher bands.
2. Green function for the S/N superlattice The periodic square-well potential relevant to the Kronig—Penney model has a periodic array of S and N regions with thicknesses d~and d~,respectively [8,9]. Considering the free motion of carriers in the planes normal to the superlattice, the van Hove singularities of DOS at the band edge will be smeared out and can be neglected in this calculation. The chemical potentials for N and S are 4u~and respectively, which are ~,
measured from the vacuum level to the corresponding Fermi levels. The potential U= p~ ~ is the barrier between the superconductor and the normal metal. In the Kronig—Penneymodel, the electronic wavefunction is composed of the core atomic wavefunction ~J(rj) at site Ti, the 2D Bloch function exp ( ik~r) and the 1 D envelope wavefunction. Here, we only consider the plane wavefunction for the ath band along the coordinate of 1 D symmetry, —
.
çD~(c()=Asm(ax)+Bcos(ax),
—d~
=Csin(flx)+Dcos(flx), 0
The normalized Bloch functions are 2[jz cok(x, a)=M” 1flsin(ax)—u2 cos(ax)], 2[p =M” 1asin(flx)—~u2cos(flx)],
—d~
(4)
where ~
=cos(fld~)—cos(ad~)exp(ika), BF(k,a)(. 6a
~sin(ad~)cos(fld~)+
~u2=asin(fld~)+flsin(ad~)exp(ika), a
.
-~sin(fld~)cos(ad~),,,
F(k, a)=cos(ka) —cos(f3d~)cos(ad5) + a±/3 sin(ad~)sin(/3d~).
(5) (6)
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Volume 175, number 1
PHYSICS LETFERS A
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The Green function for fermions at nonzero temperature with the fundamental Matsubara frequency cv= xkT is [10] ,-,
.
*
~
,~
~-~w ~,X, X ) =
.
I
\ 7;’
I’ I I ~
7;’
a,k 1W—L~r~.) ri~f
where Ea (k) = 1~k~ /2m is the kinetic energy measured from the Fermi level. The summations over k and a are taken within the first Brillouin zone which produces G
~
~ — —
,~a1 2xix Jf .Tf [öF(k,a)/&a}Ø~(s,a)çDk(x’,a)ddk (iw—a2/2m+E~)F(k,a) ‘
—77/a C
mMØ~(x,a)Øk(x’,cx÷) ia+[(a+/fl÷)cos(a+d 5)sin(fl+d~)+sin(a+d~)cos(fl+d~)]sin(~z+)’
(8)
where cos(a+d 2+ cos(a±d~+fl+d~)—y 5—f3+d~) cos(u÷)= l—y÷ 2
—
y+—
a÷—/3+
(9)
,
and the + sign means the value in the upper complex plane. In calculating eq. (8), we have taken the integration over the infinite contour of the upper half-circle including a pole at a~= ,.J2m(Ef + iw). With the substitution of the Bloch wavefunctions çok(x’, a) and ~ a) into eq. (2), we can obtain the Green functions for the propagation of a single particle from position x to x’. They are (i) for —d5
)~i[(/Ifl(d)i(I3d)+i(d)(fld)]i(•)~ mMØ~(x,a+)Øk(x’,a+)
(10)
—
(ii) for na—d~
(11)
(iii) for na
ia~sin(~z+)ly~ (cos(a+x’ —/3+x) [sin(a+d, +fl+d~)—y~ sin(ct~d,—/3~d~)] 2÷)sinCu÷)sin(Ia+x’ —fl+xI) —2y~sin(fl+d~)cos[a÷(x’+d~)+fl÷x]+i(l—y + y~±2y~ [2y~ sin(a+d,) sin(fl~d~)—isin(ji÷)] sin(a~x’)sin(fl÷x)),
(12)
(iv) for na—d~
1 (cos(a+x_p+x’)[sin(a÷d,+p+dn)—y~sin(a÷d~—/3÷d~)] sinCu+) l—y~\ 2
‘/3+
—2y~sin(a+d~)cos[a+x+fl÷(x’~ + Y++2/+
60
[2Y+sin(a+d~)sin(fl+d~)_isin(/L+)]sin(a+x)sin(fl÷x’)).
I) (13)
Volume 175, number 1
PHYSICS LETFERS A
29 March 1993
3. Density of states The density of states (DOS) near the Fermi level E~at the Matsubara frequency cv for a periodic structure is [11] pw(x)=
(14)
-~-ImG~(x,x’,E~),
which is dependent on temperature, Fermi energy and potential barrier between the metal and the superconductor. We may evaluate the average DOS at superconducting and normal materials, respectively, by neglecting the proximity effect, fls
~
JP~,s(x)dx,
n~= ~—Jp~,~(x)dx.
(15)
Using the Green functions established in eqs. (10)—(13), we can obtain ns=
+
—
~Im[~-sin~+)
l~y~(sina+ds+fl+dn_~2+ sin(a+d7—/3+d~)
~2~—sin(fl+dn)sin(a+ds))],
n~=—
(16)
xlm[ i/i sin(~z)l—y~.(sina+ds+fi÷dn_Y~+ sin(a+d~—I3÷d~)
+fldsin(P+dn)sin(a+ds))],
(17)
The calculation of DOS for U~0 and T 0 is intricate and tedious. Since the Matsubara frequency w= itkT can only take values up to I 0_2 eV even for the highest T~superconductor (T~ 100 K), we can presume that cv is much smaller than E~and U. Conspicuously, only the first order approximation of w/2/3 or w/E~will be taken into account in the calculation of DOS. The implemented parameters are 2(l—2iw/afl), cos(p~)~ 1 1 [cos(a÷d y~ray 7+fi+d~)—y~ cos(a÷d5—/3+d~)] 2(a+d~+fl+d~) +y~sin2(a+d 2(a~d 2(fl+d~)]}112• (18) sin(ji+) = 1 {sin 5—fl+d~)—2y~[sin +sin 7) —y. 4. 2
2
The DOSs are rn
2] aLK5[1+(w/2Ef) in K 6+[cv/2(E~—LJ)]I6+I5[I6—wK6/2(E~—U)]/2K5 2} it /3LK5{1+[cv/2(E~—U)] where ‘
It
( 19 )
‘
L=[sin2(a÷d~)+sin2(fl~d~)]”2, K
2]’ 4=K0—K1 + ad~[l+(w/2E~)
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Volume 175, number 1
PHYSICS LETTERS A
d[l+(/2E)2]’
14b0h1+
29 March 1993
K 5=K~—I~+K~—If—K2, 15=2K010+2K111—12,
2] /3dn(l+cv/2Ee)2 K K K K3+[w/2(Ef—U)]13 13—[w/2(E1—U)]K, fld~[l+cv/2(Ef_U) 1 2[b 0=b0c0, I,=y 1c1—(2cv/a/3)a1d1J 2{a 2+ (a 2— (b 2— (b,c 2]} =2y 2b2c2d2 (2cv/a/3) [(a2d2) 3d3) 2c2) 3) 1 3=2y[a,b2c2d2+a2b,c2d2—(w/afl)(a2d2a3d,—b2c2b3c3)] 2[a K0=a0d0, K1=y 1d1+(2cv/a/3)b1c1] 2[ (a 2+ (a 2+ (b 2— (b 2+ (2w/afl) (a K2=2y 2d2) 3d,) 2c2) 3c3) 2b2c2d2 +a3b3c3d3)] ‘6=’0’1
6
~
‘2
—
a0=sin(ad5+/1d~), a2=sin(ad5) •
,
b0=cos(ad5+/id~),
a,=sin(/3d~)
,
[cv (ad5
/3db
L~
c0 = sinh + Ef • (cvad5\
\1
a1=sin(ad5—/3d~),
b2=cos(ad5) .
d~= cosh
[~
(ad.
+
—
lid,, \
[cv (ad.
L~
U)]’
b3=cos(fld~)
,
c, = sinh ~ (cv /id,, c2=s1nh~~,-~--~_,),c3=s1nh~,,~~2~EU ~—~---
b1=cos(ad5—fld~),
—
Ef
—
u)
.
Ef_U)]’
d~= cosh
[~
(ad.
(cvad5\ (cv fld~ \ d2=cosh~.~~_)~ d3=cosh~EU).
—
Ef_U)]’ (20)
In deriving the above equations, we only keep the positive values of sin (j&~)in order to obtain positive values of n,, and n5.
4. Numerical calculations For the thicknesses of superconducting and normal layers d, = d,, =5 A, and a potential barrier U= 0.05 eV, the superlattice builds up a single gap near 0.4 eV for both the superconducting and normal metal states, and the DOS increases sharply near the gap edges. Increasing the potential barrier U to 0.3125 eV, as shown in fig. 1, the position of the bandgap shifts to 0.55 eV and becomeswider. The DOS for n~and n,, below the bandgap indicates a cross over. Increasing the layer thicknesses to d,= 12 A, d,, = 24 A and c15 = 24 A, d,, = 12 A, the DOSs are shown in figs. 2 and 3, respectively, the number of bandgaps increases while the DOS for the thicker superconducting layer is higher. As the thickness of the superconducting layer increases to 48 A and d,, = 12 A, the gap number increases and the gap becomes much narrower. All these figures manifest n~>n,, for d7> d,, and fl5 < n,, for d. < d,,. For d5 = d~,only n5 at the first band is greater than n~,and n5 is always below n,~at higher bands. Near the band edge, the DOS for the superlattice is much greater than the value for bulk materials. The pair potential 4(x) obeys the integral equation [12] 4(x)=V(x)T>~Jdx’4(x’)K~(x,x’), where the kernel is related to the Green function by 62
(21)
Volume 175, number 1
31
PHYSICS LETTERS A
29 March 1993
—
~29~ ns 28’
_____
27’ 26’
-
0
I
I
0,2
Of+
I I 06 06 EK(eV)
I
I
jO
1,2
-
Fig.l.DOSforU=O.3l25eV,T=l0K,withd,=d,~=5A.
I
3029.6 4~29. U,
0
0
EK (ev )
E,~(eV )
Fig. 2. DOS for U=0.3l25eV, T~0K,withd,=12A,d.,=24A.
Fig. 3. DOS for U=O.O5eV, T=OK, with d
5=~24A,
d,= 12 A. (22)
w(X,X’)av,
which defines two-particle propagators. From the integral equation, the critical temperature T, will be dependent on the thickness of each layer of the superlattice as a result of the proximity effect as given by [13] 2~T~ 2x2T,~b,,,,exp(—2d,,k) 23 TC~T~_ 2n 3 ( ) 4(d+b)2+ (d~+b~) ‘
where T~is the original critical temperature of the superconductor, b,~,is an extrapolation length defined by b,= 4(x)/[dA(x)/dx] ~ is the thermal diffusion length, and k1 is the decay length. The decrease of T~is not so apparent if d~is larger than d,,. In conclusion, we have calculated the density of states based on a simple Kronig—Penney model of a superconductor/normal metal superlattice at various thickness ratios and temperatures. Multiple bands are illustrated which relate to the thick layer thickness. The DOS increases sharply near the band edges owing to the versatility of the Peierls instability. As denoted in eq. (1), the DOS is crucial to the critical temperature.
~
,
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The nesting of DOS near the Fermi level can compensate the suppression of the dynamical dielectric constant, therefore it elucidates the retention of superconducting states for the superlattice structure which are experimentally proved [1,2].
Acknowledgement This work was supported by the National Science Council of the Republic of China under contract NSC8 1021 2-M-007.
References Eli J.M. Triscone, M.G. Karbut, L. Antognazza, 0. Brunner and P. Fisher, Phys. Rev. Lett. 63 (1989)1016. [2] J.M. Triscone, P. Fisher, 0. Brunner, L. Antognazza and A.D. Kent, Phys. Rev. Lett. 64 (1990) 804. [3] M. Crisan, Theory of superconductivity (World Scientific, Singapore, 1989) p. 250. E41 J.T. Lue and J.S. Sheng, Phys. Rev. B, in press. [5] P.B. Allenand R.C. Dynes, Phys. Rev. B 12 (1975) 905. [6]A.P. van Odder, Phys. Rev. 181 (1969) 787. [7]W.J. Gallagher, Phys. Rev. B 22 (1980) 1233. [8]Y.TanakaandM. Tsukada, Phys. Rev. B40 (1989) 4482; 39 (1989) 491; 44(1991) 7578; SolidState Commun. 69(1989)195; J. Phys. Soc. Japan 60 (1991) 1327. [9]R.deL. Kronig and W.G. Penney, Proc. R. Soc. A 130 (1931) 499. [10]G.D. Mahan, Many particle physics (Plenum, New York, 1981) ch. 4. Eli] S. Doniach and E.H. Sondheimer, Green’s functions for solid state physicists (Benjamin, New York, 1974) p. 75. [12]L.P. Gorkov, Soy. Phys. JETP 10 (1960) 998. [13]Y. Tanaka and M. Tsukada, Phys. Rev. 37 (1988) 5095.
64
Physics Letters A 175 (1993) 58—64 North-Holland
Density of states for high T~superconductors in the Kronig—Penney superlattice structure Juh-Tzeng Lue
and Yen-Pang Liang
Department ofPhysics, National Tsing Hua University, Hsinchu, Taiwan, ROC Received 1 September 1992; revised manuscript received 7 January 1993; accepted for publication 19 January 1993 Communicated by A. Lagendijk
The densities of states (DOSs) at various thickness ratios for superlattices with a periodic superconducting/normal material structure are calculated by using Green functions derived in accordance with the Kronig—Penney model. The DOSs sharply increase near the band edges resulting from Fermi-level nesting. Increasing the thickness of the superconducting layers, the corresponding DOS of the superconducting state will dominate that of the normal state showing the competition of Cooper paired electrons and normal electrons manipulated by the proximity effect. The enhanced DOS near the band edges is crucial for the retention of high critical temperatures for the recently fabricated YBaCuO/Dy(Pr)BaCuO superlattices.
I. Introduction The high critical temperature T~of the layered copper oxide superconductors, and the retention of a rather high T~for the recently developed YBaCuO/Dy (Pr )BaCuO [1,2] superlattices in which the thickness of the interposing insulating PrBa2Cu3O7 layer is even greater than the coherence length of the bulk material imply some new physical features for the superlattice structure. The simplified equation for the critical temperature including the non-phonon-mediated mechanism can be expressed by [3] Tc~1.14ODexP(_1*)_1.146DexP~~~)~
(1)
where °D is some characteristic temperature, N( 0) is the density of states near the Fermi energy, e’ (w, q) is the dynamical dielectric constant at some specific electromagnetic frequency w and wavevector q, and 2 and p~are directly connected with the effective interaction between electrons. The superconducting state appears only when the force between electron pairs is attractive inferring e’ (cv, q) <0 in which case T, has an upper limit. In our other papers [4], we have demonstrated that the effective dynamical dielectric function of the superconducting/insulating (S/N) superlattice is largely suppressed to a negative value by the coupled surface plasmon polariton waves in the infrared region. By inspection of eq. (1), one would expect a high T~for the plasmon-mediated Cooper pairing, because the plasma frequency cvi, is larger than the phonon frequency by a factor of M/m*, where M is the ion mass and m* is the effective electron mass. But2F(w) the plasma frequency /co [5], while it would increase reduce the interaction by the the formula relation ~= 2=2~u/[1+ J~dcv~aln (Ef/kBOD)] which dewould theelectron modified Coulomb parameter repulsion ~2 given through grades T~.Therefore a compromise intermediate plasma frequency would be plausible in promoting high T~. To obtain a high T~,~(w, q) should be slightly below zero. The retention of high T~in the S/N superlattice with a large negative (q, cv) (as low as 15 000) may be due to the enhancement of the density of states near the band edge which compensates the large I e (q, cv) I. Van Gelder [6] first showed that a periodic order parameter 4(y) may give rise to an anisotropic band struc—
58
0375-9601/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume 175, number 1
PHYSICS LETFERS A
29 March 1993
ture consisting of adjunct energy bands. This model is inadequate for describing temperature dependent phenomena and does not yield details for a different construction of periodic lattices. Compound-resonance effects showing splitting of de Gennes—Saint James bound states and beating of the oscillations ofthe density of states occurring at electronic energies below and above the energy gap, respectively, in the superconducting—normal metal (SN) sandwich structures were discussed by Gallagher [7]. His work has no analogy to the SN superlattice. Tanaka and Tsukada [8] have developed theories for superconducting superlatticesbased on the Kronig— Penney model [9]. They found that the critical temperature oscillates as a function of the length of the unit period and interference of the wave functions arises due to scattering of the multilayered structure. However, their results for layer thicknesses of 5—10 A might be constrained mostly by the proximity effect and the Josephson tunneling effect that were not taken into account. In this work, we explicitly derive the temperature dependence of the density of states (DOS) following the same approach as given in ref. [8] without using the Gorkov equations while choosing the layer thickness to be close to the experimental reports [1,2]. Multiple bandgaps and enhanced DOS near the band edges are illustrated in this calculation. The DOS for superconducting states is larger than those for normal states in the first energy band and then inverted in the higher bands.
2. Green function for the S/N superlattice The periodic square-well potential relevant to the Kronig—Penney model has a periodic array of S and N regions with thicknesses d~and d~,respectively [8,9]. Considering the free motion of carriers in the planes normal to the superlattice, the van Hove singularities of DOS at the band edge will be smeared out and can be neglected in this calculation. The chemical potentials for N and S are 4u~and respectively, which are ~,
measured from the vacuum level to the corresponding Fermi levels. The potential U= p~ ~ is the barrier between the superconductor and the normal metal. In the Kronig—Penneymodel, the electronic wavefunction is composed of the core atomic wavefunction ~J(rj) at site Ti, the 2D Bloch function exp ( ik~r) and the 1 D envelope wavefunction. Here, we only consider the plane wavefunction for the ath band along the coordinate of 1 D symmetry, —
.
çD~(c()=Asm(ax)+Bcos(ax),
—d~
=Csin(flx)+Dcos(flx), 0
The normalized Bloch functions are 2[jz cok(x, a)=M” 1flsin(ax)—u2 cos(ax)], 2[p =M” 1asin(flx)—~u2cos(flx)],
—d~
(4)
where ~
=cos(fld~)—cos(ad~)exp(ika), BF(k,a)(. 6a
~sin(ad~)cos(fld~)+
~u2=asin(fld~)+flsin(ad~)exp(ika), a
.
-~sin(fld~)cos(ad~),,,
F(k, a)=cos(ka) —cos(f3d~)cos(ad5) + a±/3 sin(ad~)sin(/3d~).
(5) (6)
59
Volume 175, number 1
PHYSICS LETFERS A
29 March 1993
The Green function for fermions at nonzero temperature with the fundamental Matsubara frequency cv= xkT is [10] ,-,
.
*
~
,~
~-~w ~,X, X ) =
.
I
\ 7;’
I’ I I ~
7;’
a,k 1W—L~r~.) ri~f
where Ea (k) = 1~k~ /2m is the kinetic energy measured from the Fermi level. The summations over k and a are taken within the first Brillouin zone which produces G
~
~ — —
,~a1 2xix Jf .Tf [öF(k,a)/&a}Ø~(s,a)çDk(x’,a)ddk (iw—a2/2m+E~)F(k,a) ‘
—77/a C
mMØ~(x,a)Øk(x’,cx÷) ia+[(a+/fl÷)cos(a+d 5)sin(fl+d~)+sin(a+d~)cos(fl+d~)]sin(~z+)’
(8)
where cos(a+d 2+ cos(a±d~+fl+d~)—y 5—f3+d~) cos(u÷)= l—y÷ 2
—
y+—
a÷—/3+
(9)
,
and the + sign means the value in the upper complex plane. In calculating eq. (8), we have taken the integration over the infinite contour of the upper half-circle including a pole at a~= ,.J2m(Ef + iw). With the substitution of the Bloch wavefunctions çok(x’, a) and ~ a) into eq. (2), we can obtain the Green functions for the propagation of a single particle from position x to x’. They are (i) for —d5
)~i[(/Ifl(d)i(I3d)+i(d)(fld)]i(•)~ mMØ~(x,a+)Øk(x’,a+)
(10)
—
(ii) for na—d~
(11)
(iii) for na
ia~sin(~z+)ly~ (cos(a+x’ —/3+x) [sin(a+d, +fl+d~)—y~ sin(ct~d,—/3~d~)] 2÷)sinCu÷)sin(Ia+x’ —fl+xI) —2y~sin(fl+d~)cos[a÷(x’+d~)+fl÷x]+i(l—y + y~±2y~ [2y~ sin(a+d,) sin(fl~d~)—isin(ji÷)] sin(a~x’)sin(fl÷x)),
(12)
(iv) for na—d~
1 (cos(a+x_p+x’)[sin(a÷d,+p+dn)—y~sin(a÷d~—/3÷d~)] sinCu+) l—y~\ 2
‘/3+
—2y~sin(a+d~)cos[a+x+fl÷(x’~ + Y++2/+
60
[2Y+sin(a+d~)sin(fl+d~)_isin(/L+)]sin(a+x)sin(fl÷x’)).
I) (13)
Volume 175, number 1
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3. Density of states The density of states (DOS) near the Fermi level E~at the Matsubara frequency cv for a periodic structure is [11] pw(x)=
(14)
-~-ImG~(x,x’,E~),
which is dependent on temperature, Fermi energy and potential barrier between the metal and the superconductor. We may evaluate the average DOS at superconducting and normal materials, respectively, by neglecting the proximity effect, fls
~
JP~,s(x)dx,
n~= ~—Jp~,~(x)dx.
(15)
Using the Green functions established in eqs. (10)—(13), we can obtain ns=
+
—
~Im[~-sin~+)
l~y~(sina+ds+fl+dn_~2+ sin(a+d7—/3+d~)
~2~—sin(fl+dn)sin(a+ds))],
n~=—
(16)
xlm[ i/i sin(~z)l—y~.(sina+ds+fi÷dn_Y~+ sin(a+d~—I3÷d~)
+fldsin(P+dn)sin(a+ds))],
(17)
The calculation of DOS for U~0 and T 0 is intricate and tedious. Since the Matsubara frequency w= itkT can only take values up to I 0_2 eV even for the highest T~superconductor (T~ 100 K), we can presume that cv is much smaller than E~and U. Conspicuously, only the first order approximation of w/2/3 or w/E~will be taken into account in the calculation of DOS. The implemented parameters are 2(l—2iw/afl), cos(p~)~ 1 1 [cos(a÷d y~ray 7+fi+d~)—y~ cos(a÷d5—/3+d~)] 2(a+d~+fl+d~) +y~sin2(a+d 2(a~d 2(fl+d~)]}112• (18) sin(ji+) = 1 {sin 5—fl+d~)—2y~[sin +sin 7) —y. 4. 2
2
The DOSs are rn
2] aLK5[1+(w/2Ef) in K 6+[cv/2(E~—LJ)]I6+I5[I6—wK6/2(E~—U)]/2K5 2} it /3LK5{1+[cv/2(E~—U)] where ‘
It
( 19 )
‘
L=[sin2(a÷d~)+sin2(fl~d~)]”2, K
2]’ 4=K0—K1 + ad~[l+(w/2E~)
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Volume 175, number 1
PHYSICS LETTERS A
d[l+(/2E)2]’
14b0h1+
29 March 1993
K 5=K~—I~+K~—If—K2, 15=2K010+2K111—12,
2] /3dn(l+cv/2Ee)2 K K K K3+[w/2(Ef—U)]13 13—[w/2(E1—U)]K, fld~[l+cv/2(Ef_U) 1 2[b 0=b0c0, I,=y 1c1—(2cv/a/3)a1d1J 2{a 2+ (a 2— (b 2— (b,c 2]} =2y 2b2c2d2 (2cv/a/3) [(a2d2) 3d3) 2c2) 3) 1 3=2y[a,b2c2d2+a2b,c2d2—(w/afl)(a2d2a3d,—b2c2b3c3)] 2[a K0=a0d0, K1=y 1d1+(2cv/a/3)b1c1] 2[ (a 2+ (a 2+ (b 2— (b 2+ (2w/afl) (a K2=2y 2d2) 3d,) 2c2) 3c3) 2b2c2d2 +a3b3c3d3)] ‘6=’0’1
6
~
‘2
—
a0=sin(ad5+/1d~), a2=sin(ad5) •
,
b0=cos(ad5+/id~),
a,=sin(/3d~)
,
[cv (ad5
/3db
L~
c0 = sinh + Ef • (cvad5\
\1
a1=sin(ad5—/3d~),
b2=cos(ad5) .
d~= cosh
[~
(ad.
+
—
lid,, \
[cv (ad.
L~
U)]’
b3=cos(fld~)
,
c, = sinh ~ (cv /id,, c2=s1nh~~,-~--~_,),c3=s1nh~,,~~2~EU ~—~---
b1=cos(ad5—fld~),
—
Ef
—
u)
.
Ef_U)]’
d~= cosh
[~
(ad.
(cvad5\ (cv fld~ \ d2=cosh~.~~_)~ d3=cosh~EU).
—
Ef_U)]’ (20)
In deriving the above equations, we only keep the positive values of sin (j&~)in order to obtain positive values of n,, and n5.
4. Numerical calculations For the thicknesses of superconducting and normal layers d, = d,, =5 A, and a potential barrier U= 0.05 eV, the superlattice builds up a single gap near 0.4 eV for both the superconducting and normal metal states, and the DOS increases sharply near the gap edges. Increasing the potential barrier U to 0.3125 eV, as shown in fig. 1, the position of the bandgap shifts to 0.55 eV and becomeswider. The DOS for n~and n,, below the bandgap indicates a cross over. Increasing the layer thicknesses to d,= 12 A, d,, = 24 A and c15 = 24 A, d,, = 12 A, the DOSs are shown in figs. 2 and 3, respectively, the number of bandgaps increases while the DOS for the thicker superconducting layer is higher. As the thickness of the superconducting layer increases to 48 A and d,, = 12 A, the gap number increases and the gap becomes much narrower. All these figures manifest n~>n,, for d7> d,, and fl5 < n,, for d. < d,,. For d5 = d~,only n5 at the first band is greater than n~,and n5 is always below n,~at higher bands. Near the band edge, the DOS for the superlattice is much greater than the value for bulk materials. The pair potential 4(x) obeys the integral equation [12] 4(x)=V(x)T>~Jdx’4(x’)K~(x,x’), where the kernel is related to the Green function by 62
(21)
Volume 175, number 1
31
PHYSICS LETTERS A
29 March 1993
—
~29~ ns 28’
_____
27’ 26’
-
0
I
I
0,2
Of+
I I 06 06 EK(eV)
I
I
jO
1,2
-
Fig.l.DOSforU=O.3l25eV,T=l0K,withd,=d,~=5A.
I
3029.6 4~29. U,
0
0
EK (ev )
E,~(eV )
Fig. 2. DOS for U=0.3l25eV, T~0K,withd,=12A,d.,=24A.
Fig. 3. DOS for U=O.O5eV, T=OK, with d
5=~24A,
d,= 12 A. (22)
w(X,X’)
which defines two-particle propagators. From the integral equation, the critical temperature T, will be dependent on the thickness of each layer of the superlattice as a result of the proximity effect as given by [13] 2~T~ 2x2T,~b,,,,exp(—2d,,k) 23 TC~T~_ 2n 3 ( ) 4(d+b)2+ (d~+b~) ‘
where T~is the original critical temperature of the superconductor, b,~,is an extrapolation length defined by b,= 4(x)/[dA(x)/dx] ~ is the thermal diffusion length, and k1 is the decay length. The decrease of T~is not so apparent if d~is larger than d,,. In conclusion, we have calculated the density of states based on a simple Kronig—Penney model of a superconductor/normal metal superlattice at various thickness ratios and temperatures. Multiple bands are illustrated which relate to the thick layer thickness. The DOS increases sharply near the band edges owing to the versatility of the Peierls instability. As denoted in eq. (1), the DOS is crucial to the critical temperature.
~
,
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Volume 175, number 1
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29 March 1993
The nesting of DOS near the Fermi level can compensate the suppression of the dynamical dielectric constant, therefore it elucidates the retention of superconducting states for the superlattice structure which are experimentally proved [1,2].
Acknowledgement This work was supported by the National Science Council of the Republic of China under contract NSC8 1021 2-M-007.
References Eli J.M. Triscone, M.G. Karbut, L. Antognazza, 0. Brunner and P. Fisher, Phys. Rev. Lett. 63 (1989)1016. [2] J.M. Triscone, P. Fisher, 0. Brunner, L. Antognazza and A.D. Kent, Phys. Rev. Lett. 64 (1990) 804. [3] M. Crisan, Theory of superconductivity (World Scientific, Singapore, 1989) p. 250. E41 J.T. Lue and J.S. Sheng, Phys. Rev. B, in press. [5] P.B. Allenand R.C. Dynes, Phys. Rev. B 12 (1975) 905. [6]A.P. van Odder, Phys. Rev. 181 (1969) 787. [7]W.J. Gallagher, Phys. Rev. B 22 (1980) 1233. [8]Y.TanakaandM. Tsukada, Phys. Rev. B40 (1989) 4482; 39 (1989) 491; 44(1991) 7578; SolidState Commun. 69(1989)195; J. Phys. Soc. Japan 60 (1991) 1327. [9]R.deL. Kronig and W.G. Penney, Proc. R. Soc. A 130 (1931) 499. [10]G.D. Mahan, Many particle physics (Plenum, New York, 1981) ch. 4. Eli] S. Doniach and E.H. Sondheimer, Green’s functions for solid state physicists (Benjamin, New York, 1974) p. 75. [12]L.P. Gorkov, Soy. Phys. JETP 10 (1960) 998. [13]Y. Tanaka and M. Tsukada, Phys. Rev. 37 (1988) 5095.
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