Photon-induced superconducting phase transition in some cuprates

PHYSICA

Physica C 212 (1993) 128-132 North-Holland

Photon-induced superconducting phase transition in some cuprates K.P. S i n h a Institute of Fundamental Research, NEHU, Shillong- 793022, India and Jawaharlal Nehru Centre for Advanced Scientific Research, Indian Institute of Scwnce Campus, Bangalore-560012, India Received 18 January 1993

Recent observations of photo-induced transient superconductivity in some cuprate systems which are otherwise in a semiconducting or insulating state are explained on the basis of a theoretical model which was suggested by us over two and a half decades ago. In these systems, the radiation field quanta play a dual role. In the first they produce carriers and in the second role they take part in the pairing process along with virtual exchange of other boson modes, i.e. phonons or electronic boson modes. Owing to the two-boson nature the coupling constant 2rb depends on the photon density (photoexcitations) impressed on the system. The relative role of the photoresponse time and the relaxation (recombination) time are important in producing the metallic and superconducting state.

I. Introduction Recent experimental investigations on cuprate systems, having compositions in the semiconductor regime, have shown light-induced changes, which are indicative of photo-induced superconductivity [ 19 ]. In some cases there is an increase of the critical temperature Tc with radiation dosage [ 4 - 6 ] . The most interesting result is the growth of the absolute value of the diamagnetic m o m e n t almost linearly with radiation dosage with saturates beyond a certain stage [4]. The onset of transient photo-induced superconductivity, at high excitation levels, is a real p h e n o m enon [3,7]. Other experiments conclusively demonstrate that photoexcitation induces and enhances superconductivity in thin films of RBa2Cu3Ox with controlled oxygen stoichiometry [8 ]. Also photo-induced transient thermoelectronic measurements indicate an additional superconducting phase transition in YBCO about 35 K [9]. In the context of these results, it is justified to recall our predictions of the possibility of photo-induced superconductivity in a m u l t i b a n d semiconducting system made two and a half decades ago and subsequently followed up in a series of papers [ 1013]. The purpose of the present paper is to develop a

lbrmalism, based on our earlier ideas [ 10-131, to explain the results of radiation-induced superconductivity of those cuprate systems which are otherwise in the semiconducting state.

2. Formalism for the model system The strongest evidence o f p h o t o i n d u c e d superconductivity is found in films of YBa2Cu30,, which is on the border line of i n s u l a t o r - m e t a l transition and is deficient in oxygen [ I - 8 ] . Owing to this, two completely filled degenerate states at F and M points involving Cu d w and d~: and r~-bonding orbitals of oxygen (also coordinated to Ba) atoms appear [14, 15 ]. Although the two CuO2 bands remain the same as in YBa2CU3OT, the Fermi level shifts upwards due to the ionicity of the missing oxygen atoms. This produces significant changes in the Fermi surface and the hole count in the CuO2 bands leading to the insulating state. There are states above the Fermi energy at these points. The corresponding direct optical gap is estimated to be around 2.0 eV [15]. A similar situation is obtained in other cuprates, for example in L a - C u - O , C a - S r - B i - C u - O , B i 2 ( S r 2 C a ) ~ _ , . ( L a 2 Y ) x C u 2 0 : , and Nd2Cu04 sys-

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K.P. Sinha /Photon-induced superconductivity in cuprates tems in the semiconductor/insulator composition range. Thus in a formal treatment of the model, the Hamiltonian will consist of fermion fields (electrons or holes), a real photon field and virtual boson fields (phonons, charged or neutral bosons etc. ) [ 16-21 ]. The presence of a coherent radiation field connects fermionic states of the relevant bands. Further, owing to the use of a laser in the experiments, which has a macroscopic occupation of the radiation field mode, it will suffice to treat this field classically. However, the field intensity will be expressed in terms of the number of quanta nr for the mode in question. The Hamiltonian for the multiband model system in the presence of the radiation field and the virtual boson field is written as [ 13 ] H = Ho + Hrr + Hfb ,

(1)

where no=~

ek, C],,~Ck,,~+nr+nb + n ¢ .

(2)

The first term denotes the sum of single particle fermion energies, c] .... ck,,~ being the fermion (creation, annihilation) operators in the Bloch state, Ik n , a ) , k is the wave vector, n the band index and tr the spin, Ek, is the corresponding single particle energy; Hr and Hb represent the operators for the energies of the radiation and the boson fields respectively and H~ is the screened Coulomb interaction between fermions (electrons or holes here). These terms are not written explicitly as they are well known. The interaction terms Hr~ and Hro can be expressed as Hf~ = ~ O[c~,,,~ck,,~ exp( -- ig2t) + C],,oCkm,~ exp (ig2t) ] ,

(3)

where £2 is the radiation frequency and D = (2nV)-t/2(g2/c)Aodm~,

(4)

Ao = [ ( 27thc2 /I2) n~] L/2 ,

(5)

d,,,=2rr( Om ler'elO~) .

(6)

Here d,,, is the dipole matrix element connecting the Wannier states 0,~ and 0,; E is the polarization vector of the radiation field and V is the volume; c and h are the usual universal constants and Ao2 gives the field intensity and depends on the number of quanta

129

nr of the radiation field; it also gives a measure of the photoexcitations in the system. Further Hro = ~ Bq( C~_q.... ck,,~bq + h.c. ) ,

(7)

where b~, bq are the boson operators of the mode q and B~ is the electron-boson coupling constant. By a suitable unitary transformation the Hamiltonian can be brought to a time-independent form [ 13 ]. This leads to shift in the energy of the fermions in the relevant bands and also in the boson energy. We shall denote the renormalized energies as g,,,/~q, h~hq etc. The time-independent/~fr i.e.

nfr=Z D(c~m,~c,°,o+hc.)

(8)

will produce interband mixing of states of bands rn and n. However, by an appropriate canonical transformation, the mixing terms can again be eliminated and the band states diagonalized [ 13 ]. After this diagonalization, we confine our attention to the lowest renormalized band populated by the new fermion operators a]~, ak~ interacting via modified fermionboson processes. H = Z Ekaa~ak~ B , D (a~_q,~ak,ob~ +h.c.) ,

(9)

°

where Eo = [ (~lffd-Emn) 2 +DZl ,/2,

(10)

E,~, being the energy difference involved in the interband transition and Eka is the single particle energy of the new fermion states Ik a ) . Note that the modified fermion-boson interaction now carries the factor D/Eo and depends on the amplitude Ao of the radiation field and hence x//~. We have now a metastable situation in the presence of the electromagnetic field. In this non-equilibrium situation it is expedient to define an effective temperature T. The fermions and bosons have their usual distribution function but at temperature ~. Having defined this non-equilibrium situation, we can follow either the Bardeen-Cooper-Schrieffer (BCS) method or the Gorkov-Nambu-Eliashberg formalism as modified by Keldysh [ 10 ]. The BCS type reduced Hamiltonian for general

130

1(2P. Sinha / Photon-induced superconductivio, in cupraws

spin configuration in the non-equilibrium state can be written as FI~.= Z Ek~a~oak~ - ~ Uk,,,a~, ~a*_k.,,a k , , a , , ,

value when R ( n r / V ) > > 1. Beyond a certain level of radiation dosage, the heating effect may interfere with the appearance of an ordered superconducting state.

(11) 3. Estimates and discussion

where u**, = U~b(k, k ' ) - Uc(k. k' )

( 12 )

with /~2_k, D 2 Urb(k , k') =

Eo~

2 (hoOk_k,) [ (~&~_k)~-

(E~. - E~.~,) ~ ]

(13) denoting the radiation admixed boson mediated attractive (for Ek~ -- Ek,, < h&~-k, ) pairing interaction and which dominates the screened Coulomb interaction U~ beyond a certain photon density. A formalism, using Green's function method, yields the superconducting gap function as . [

coseeh

8n 3

UV iF

( 14 )

where W~ is a BCS-type energy cut-off within which Ukk'= U is constant and zero outside; N ( E F ) is the density of states at the Fermi level E~. The critical temperature T~ turns out to be T( = 1.13(W¢/ks) e x p [ - 1/,~rb] ,

(15)

where (16)

)LbR(nr/l/)

2rb-- 1 + R ( n f f V ) R=

"

4d~,(ht2) ( hff2- Emn ) 2 '

2b = ( B 2 / h d ) ) N ( E v ) •

(17) (18)

In writing the above, the screened Coulomb interaction has been ignored and the renormalized energy of the boson mode and the coupling constant i.e. h~b and/~, have been taken to be independent of wave vectors. The important point to note here is that the coupling constant 2~b depends explicitly on the density of quanta (or excitations) produced by the radiation field i.e. nff V. Thus in the regime R ( n r / V) < I, 2~b will increase almost linearly with photoexcitation density and it will reach a saturation

We shall now give a rough estimate of I., and its dependence on the photon (photoexcitation) density for the system YBa2Cu30,,3 in which transient photo-induced superconductivity has been observed [2,3]. The values of the relevant parameters are: for the nitrogen laser used in the experiment h,Q---3.7 e V = 6 × 1 0 -~2 erg (pulse width 600 ps). For the pumped dye laser h ~ = 2 . 6 e V = 4 . 1 6 × 1 0 -j2 erg (pulse width 30 ps). We shall use results of the former. The dipole matrix element d , , n - 6 X 10 -~7 (erg cm 3) ~/2 for an intra-atomic allowed electric dipole transition: we shall, however, choose d,,m= 10 w (erg cm3) ~/2 only which corresponds to an oscillator strength of 0.17, a small fraction of unity. The value of h-Q- Em~ will depend on how close E,~n is to h(2. It may be in the range 10 J2-10-13 erg. We take the value 6X 10 -~3 crg. The parameter 2b (cf. eq. (18) ) depends on the boson exchange mechanism connecting the states of bands m and n. it may involve high-frequency phonons or some high-energy electronic boson exchange [16-21]. The advantage of the involvement of photons is that bands with a high density of states may become available in the nonequilibrium state. Thus we expect N ( E v ) to be sufficiently high. Thus a choice of2b>~ 1 may not be unreasonable. However, we shall take a constant density of states at E~ in view of the quasi-twodimensional nature of the system and take the conservative value 2~= 0.5. The value of r/r/l/ has to be extracted from the experimental data of the present system. From refs. [ 2,3 ], we find that the optical absorption depth is of the order of 300/~. Thus for light intensity, IL= 1016 photons/cm 2, the photon density will be ~3X102~ photons/cm 3 in the absorption depth. The authors of refs. [2,3] observed photo-induced transition to metallic behavior at an intensity above 5X 1015 photons/cm 2, i.e, when the photon density exceeds 102~ photons/cm 3 in the absorption depth. If the quantum efficiency is taken to be unity, the photo-induced charge carrier will be of the same

K.P. Sinha ~Photon-induced superconductivity in cuprates

order. Thus for our computation we shall use a value of nr/V in the range (2-7.5) × 1021 photons/cm 3. This is already in the high photoexcitation regime and we should expect leveling effects as discussed above in the last section. The energy cut-off in eq. ( 15 ), i.e. W¢/kB, is taken at about 725 K, for the boson mode in question. The computed values of ]r using the above parameters, for some range of r / r / V are given in table 1. Experimentally, it is found that at this level of photoexcitation, Tc does not change appreciably with photon density. This is expected from the radiationinduced factor R ( n~/ V) / [ 1 + R ( nr/ V) ] in eq. (16). The above choice of parameters is taken to demonstrate the trend of the ~¢ change as a function of radiation quanta (photoexcitations). The increase of T~ v e r s u s r / r / V is similar to the observed behaviour as indicated by the onset of photo-induced transient superconductivity [2,3 ]. We shall make a few remarks on the decay of photo-induced conductivity in Y B a 2 C u 3 0 6 . 4. In the low pump-light intensity, when the system is in the semiconducting state, the time decay of conductivity follows a power law [ 3 ]. At strong intensity when the system is driven to the metallic regime the lifetime increases by three orders of magnitude [ 3 ]. We interpret it as a metastability occurring because of hole-hole pairing interaction (induced) which competes and suppresses electronhole recombination. In this composition, the electron is believed to be trapped by defects and impurities before recombination. From the experimental work reported [ 1-8], we expect n h to be in the range of 1021 t o 5 Y 1021 c m -3. This is in the metallic regime. The corresponding hole concentration Nh in the two-dimentional sheets will be given by Nh = nhLs, where Ls is the thickness of the layer. With Ls= 2.5 × 10 -7 cm, this gives Nh= 1.25 × 1015 cm -2. This will place the Cu20 layers in the metallic regime. The delayed hole-electron recombination (due to Table 1

nr/V (cm-3)

~rc(K)

2× 1021 3 × 1021 4X 1021 5X 1021 7.5X 1021

95 I00 102.5 104 106.6

131

electron trapping) not only facilitates the build up of the carriers (holes here) towards metallicity, but also of photoexcitations which virtually remain frozen during this period. The mechanism discussed above involves a dual role of the radiation field. It creates carriers (holes here) in unison with traps (which imprison electrons for sufficient length of time below some temperature) to render the material metallic. The second role is the involvement in the pairing mechanism so that the coupling constant ~-rb depends on nr/V. Thus, as seen from eq. (16), 2rb will increase with increasing nr/V and then tend to saturate beyond a certain limit. This kind of behaviour is indeed reflected by the observed photo-induced transient local superconductivity observed [ 1-3,7 ].

4. Concluding remarks Based on our previous models of photo-induced superconductivity in an otherwise semiconducting system, we have presented a theoretical discussion of this phenomenon for cuprate systems. The theory takes cognizance of the twin role of the radiation field. It causes interband transitions and makes available an adequate number of carriers (electrons and holes). If the system contains traps for one kind of carriers (say electrons), the metallic state results from the build up of the carriers (holes) with increasing photon flux beyond a threshold. Below a critical temperature the pairing of the mobile cartiers can take place through exchange of bosons (phonons or electronic boson modes) [ 16-21 ] and the interband mixing effects of the radiation field. The pairing processes involve two boson fields namely photons and boson modes of the system. It is because of the two-boson character of the interaction that the coupling constant 2rb depends on the density of the photons (or photoexcitations) (nr/V), impressed on the system. It places at our disposal the means of enhancing and controlling Tc via nr/V. This is the advantage of the non-equilibrium pairing process. Another important condition for the mechanism to work is that the photoresponse time and the electron-hole recombination (relaxation) times of the carrier should be radically different. We most have z(photoresponse) << z(recombination). If they are

132

K.P. Sinha ~Photon-induced superconductivity in cuprate~'

comparable then a larger intensity of the radiation field will be required which may lead to heating of the sample. Furthermore, the quasi-particle lifetime Zqpof the carrier arising from various scattering processes must be larger than the pair relaxation time r(pair) which is related to the superconducting gap or the critical temperature Tc [ 13 ]. Thus we see that one faces several dynamical processes in realising superconductivity in the non-equilibrium state. Several conditions have to be satisfied for the superconducting state to appear. It appears that the complex systems such as cuprates, in which the onset of photo-induced transient superconductivity has been seen, meet the appropriate requirements. Thus further research work in such systems is desirable to get the photo-induced superconducting state at high enough To. In another paper, we shall discuss the mechanism and conditions for the realisation of metastable superconductivity induced and enhanced by photoexcitation [4-6,8 ].

Acknowledgement The author would like to thank Dr. A.V. Narlikar for drawing his attention to the experimental papers on photo-induced superconductivity in cuprates. The support of the Indian National Science Academy, New Delhi is appreciated.

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