- Email: [email protected]
Phonon squeezed state in high-Tc superconductors
Physica B 165&166 (1990) 1083-1084 North-Holland
PROHOI SQUEEZED STATE IIf RIGH-Tc SUPERCOlmUCTORS
Takayuki SOTA and Katsuo SUZUKI Department of Electrical Engineering. Waseda University, Shinjuku, Tokyo 169, Japan We study the phonon states in high Tc superconducting materials within a weakcoupling polaron model. It is confirmed that the squeezed state of phonons gives lower ground state energy of the whole system than the two-phonon coherent state and the displaced state proposed previously. We also examine a condition that the squeezed state occurs. It follows that the squeezed state can exist when the interband electron transfer is allowed in many-band system. 1. IIITRODUCTIOH
Previously we perforaed calculations of the hole-concentration dependence of Tc and the isotope shift in high-T c CuO based superconducting aaterials taking account of the quasi-two dillensional Cu02 plane for the electron subsystea and introducing the twophonon coherent state for the phonon subsystem instead of the bare phonon state (1). The two-phonon coherent state proposed by Zheng (2) is a kind of the squeezed state and is also called the displaced-squeezed state (3). In our previous paper we regarded the two-phonon coherent state as given a priori and did not examine whether it gives the lowest ground state energy or not. Recently. for a problem of a tunneling particle coupled to a phonon co-ordinate, Jayannavar(4) pointed out that there exists a variational wave function describing the phonon subsystem which gives a lower value for the ground state energy than the two-phonon coherent state. In this paper, using a simplified model which can be applied to high-T c CuO based superconducting materials, we examine which kind of phonon state is realized and how that state is interpreted physically. We confine ourselves to a study of the normalstate. 2. FORJlULATIOI
2.1. 1lI0del We construct a model for high-Tc CuO based superconducting lIaterials within a weakcoupling polaron 1I0del. Assumptions used are as follows. The electron subsystem consists of two bands where, for simplicity, purely electronic interband hopping processes are neglected. For the phonon subsystem two kinds of phonons are considered. One gives the electron-phonon coupling causing the intraband electron transition and the other does that causing the interband electron 1990 - Elsevier Science Publishers B.V. (North-Holland)
transition. Our 1I0del HllIIiltonian takes the fora Hf=11 H=H O + H'+ H".
(1)
H' =L k[glai,k+qal,k+g2a2,k+qa2.kl x (bi,_q+b1,q)+H.c.,
(3)
'<' [g3al.k+qa2.k+g3a2,kal,k+q + * + 1 H " =Lk
x (b2,_q+b 2 ,q)'
(4)
where e i k is the energy of electrons and W f q (=w j) is the angular frequency of the j -'1;h mode phonon. Here k denotes the wavevector k and the spin a and q denotes the wavevector q and the polarization direction A. al k and ai k (bj Q and b f q )represent the creation and' the ahnihllatron operators of electrons (phonons). gl and g2 are the intraband electron-phonon coupling constants and g3 is the interband one. H.c. indicates Hermitian conjugate. 2.2. Calculations and results In order to obtain an effective Hamiltonian describing the weak-coupling polarons, we successively eliminate terms H' and H" in Eq. (1) using a unitary transformation as follows, Heff=e S" e- R" (e S' e- R' He R' e- S' )e R" e- S" , (5) R'=(ulai.k+qal,k+u2a2,k+qa2.k+H.c.1 x (b1,q-bi,_ql,
(6)
T. Sota., K. Suzuki
1084
(7)
x (b 2 ,q-b2,_q)' S'=7 1(b 1 ,q2_bi._ q2),
(8)
S "-- 7 2 (b 2, q2b+ - 2, -q 2) .
(9)
Here ui (i=I-3) and 7 j (j=1,2) are variational paraaeters which are determined so as to minimize the ground state energy of the whole system. The displaced states are obtained for ui ~O and 7 while the displaced-squeezed states are obtained for ui~O and 7 j~O. The former (the latter) is formally equivalent to the coherent state (the squeezed state) well known in the field of quantum optics. We give a few comments on the variational parameters. To perform the unitary transformation analytically, we impose the following restriction on the variational parameters: u2=u1 or -u1' and u3 is a real number. The former restriction requires g2=gl or -gl' and the latter does g3 a real number. After performing the above mentioned unitary transformation, we evaluate the expectation value of the ground state energy E(u1,u3,71,72) using the norllal-state eigenvector written as I 1l"} = I O} pII k a i.k II l a 2,l I O} e where I O} ( I O) e) represents the phOnon (electronf vacuum state. Using E the variational condit ions 0 E=O determine the var iat ional parameters, namely for the case g2=gl we get u1=gl/wl' u3=g3Iw2' 71=0, 72=0, while for the case g2=-gl we get ul =g11wI' 71 =0,
rO'
u3= (g3 /w 2) x {l+4(g12IW1W2 )e- 47 2e-Z(~-~ )/N, }-1 TIO)
where Z=8u32e-472. Ni (i=1-3) are expressed by a combination of the particle and the hole occupation numbers and teras with Nl and N3 (N 2 ) are interpreted as contributions to a renormal1zation of the electron masses through the interband (intraband) electronphonon coupling in the expression for E. At this stage the following is found. In the case of g2=gl' i.e., when the dilational type deformation of the two bands occurs through the intraband electron-phonon coupling, allowed phonon states are restricted to the displaced states. These states have been used in the formulation of the s.all polaron. On the other, in the case of g2=gl' i.e., when the shear type deformation of
the two bands occurs, allowed state of phonons causing the intraband (interband) electron-phonon coupling is the displaced state (the squeezed state). 3. DISCUSSIONS Here we discuss only the squeezed state. As can be seen from Eqs. (10) - (11), the twophonon coherent state proposed by Zheng (3) is a special case of the squeezed state: u3=g31w2 and 72 is deterllined by Eq. (11) with u3=g3Iw2' To compare the ground state energy of the squeezed state calculated using u3 and 72 obtained by solving Eqs. (10)- (11) self-consistently with that of two-phonon coherent state we have performed simple model numerical calculations though they are not shown here. We find that the former is lower than the latter. Numerical calculations using more realistic model are needed and now in progress. Within our model where purely electronic interband hopping processes are neglected, it is found that the squeezed state can exist. To examine whether our results depend on the lIodel used here or not, we have performed calculations using another three types of model: the first model is the single band model, the second model is an extended version of the present model where purely electronic interband hopping processes are included, and the third model is also an lIodified version of the present model where nei ther purely electronic interband hopping nor phonon assisted interband transition is allowed. In the first model and the third one the squeezed state does not occur, while in the second model all phonon states can be the squeezed states provided that phonons strongly interact with the particles. Therefore it follows that a condition of existence of the squeezed state is that the interband electron transfer is allowed in the many-band system. REFERElCES
(1)
(2) (3) (4)
T. Sota and K. Suzuki, in PHONONS 89 (World Scientific, Singapore, 1990), pp.298-300. Zheng Hang, Phys. Rev. B36 (1987) 8736; Phys. Rev. B38 (1988) 11865; J. Phys. C: Solid State Phys. 21 (1988) 2351. Hong Chen et al., Phys. Rev. B39 (1989) 546; Phys. Rev. B40 (1989) 11326. A.M. Jayannavar, Solid State Com.un. 71 (1989) 689.
PROHOI SQUEEZED STATE IIf RIGH-Tc SUPERCOlmUCTORS
Takayuki SOTA and Katsuo SUZUKI Department of Electrical Engineering. Waseda University, Shinjuku, Tokyo 169, Japan We study the phonon states in high Tc superconducting materials within a weakcoupling polaron model. It is confirmed that the squeezed state of phonons gives lower ground state energy of the whole system than the two-phonon coherent state and the displaced state proposed previously. We also examine a condition that the squeezed state occurs. It follows that the squeezed state can exist when the interband electron transfer is allowed in many-band system. 1. IIITRODUCTIOH
Previously we perforaed calculations of the hole-concentration dependence of Tc and the isotope shift in high-T c CuO based superconducting aaterials taking account of the quasi-two dillensional Cu02 plane for the electron subsystea and introducing the twophonon coherent state for the phonon subsystem instead of the bare phonon state (1). The two-phonon coherent state proposed by Zheng (2) is a kind of the squeezed state and is also called the displaced-squeezed state (3). In our previous paper we regarded the two-phonon coherent state as given a priori and did not examine whether it gives the lowest ground state energy or not. Recently. for a problem of a tunneling particle coupled to a phonon co-ordinate, Jayannavar(4) pointed out that there exists a variational wave function describing the phonon subsystem which gives a lower value for the ground state energy than the two-phonon coherent state. In this paper, using a simplified model which can be applied to high-T c CuO based superconducting materials, we examine which kind of phonon state is realized and how that state is interpreted physically. We confine ourselves to a study of the normalstate. 2. FORJlULATIOI
2.1. 1lI0del We construct a model for high-Tc CuO based superconducting lIaterials within a weakcoupling polaron 1I0del. Assumptions used are as follows. The electron subsystem consists of two bands where, for simplicity, purely electronic interband hopping processes are neglected. For the phonon subsystem two kinds of phonons are considered. One gives the electron-phonon coupling causing the intraband electron transition and the other does that causing the interband electron 1990 - Elsevier Science Publishers B.V. (North-Holland)
transition. Our 1I0del HllIIiltonian takes the fora Hf=11 H=H O + H'+ H".
(1)
H' =L k[glai,k+qal,k+g2a2,k+qa2.kl x (bi,_q+b1,q)+H.c.,
(3)
'<' [g3al.k+qa2.k+g3a2,kal,k+q + * + 1 H " =Lk
x (b2,_q+b 2 ,q)'
(4)
where e i k is the energy of electrons and W f q (=w j) is the angular frequency of the j -'1;h mode phonon. Here k denotes the wavevector k and the spin a and q denotes the wavevector q and the polarization direction A. al k and ai k (bj Q and b f q )represent the creation and' the ahnihllatron operators of electrons (phonons). gl and g2 are the intraband electron-phonon coupling constants and g3 is the interband one. H.c. indicates Hermitian conjugate. 2.2. Calculations and results In order to obtain an effective Hamiltonian describing the weak-coupling polarons, we successively eliminate terms H' and H" in Eq. (1) using a unitary transformation as follows, Heff=e S" e- R" (e S' e- R' He R' e- S' )e R" e- S" , (5) R'=(ulai.k+qal,k+u2a2,k+qa2.k+H.c.1 x (b1,q-bi,_ql,
(6)
T. Sota., K. Suzuki
1084
(7)
x (b 2 ,q-b2,_q)' S'=7 1(b 1 ,q2_bi._ q2),
(8)
S "-- 7 2 (b 2, q2b+ - 2, -q 2) .
(9)
Here ui (i=I-3) and 7 j (j=1,2) are variational paraaeters which are determined so as to minimize the ground state energy of the whole system. The displaced states are obtained for ui ~O and 7 while the displaced-squeezed states are obtained for ui~O and 7 j~O. The former (the latter) is formally equivalent to the coherent state (the squeezed state) well known in the field of quantum optics. We give a few comments on the variational parameters. To perform the unitary transformation analytically, we impose the following restriction on the variational parameters: u2=u1 or -u1' and u3 is a real number. The former restriction requires g2=gl or -gl' and the latter does g3 a real number. After performing the above mentioned unitary transformation, we evaluate the expectation value of the ground state energy E(u1,u3,71,72) using the norllal-state eigenvector written as I 1l"} = I O} pII k a i.k II l a 2,l I O} e where I O} ( I O) e) represents the phOnon (electronf vacuum state. Using E the variational condit ions 0 E=O determine the var iat ional parameters, namely for the case g2=gl we get u1=gl/wl' u3=g3Iw2' 71=0, 72=0, while for the case g2=-gl we get ul =g11wI' 71 =0,
rO'
u3= (g3 /w 2) x {l+4(g12IW1W2 )e- 47 2e-Z(~-~ )/N, }-1 TIO)
where Z=8u32e-472. Ni (i=1-3) are expressed by a combination of the particle and the hole occupation numbers and teras with Nl and N3 (N 2 ) are interpreted as contributions to a renormal1zation of the electron masses through the interband (intraband) electronphonon coupling in the expression for E. At this stage the following is found. In the case of g2=gl' i.e., when the dilational type deformation of the two bands occurs through the intraband electron-phonon coupling, allowed phonon states are restricted to the displaced states. These states have been used in the formulation of the s.all polaron. On the other, in the case of g2=gl' i.e., when the shear type deformation of
the two bands occurs, allowed state of phonons causing the intraband (interband) electron-phonon coupling is the displaced state (the squeezed state). 3. DISCUSSIONS Here we discuss only the squeezed state. As can be seen from Eqs. (10) - (11), the twophonon coherent state proposed by Zheng (3) is a special case of the squeezed state: u3=g31w2 and 72 is deterllined by Eq. (11) with u3=g3Iw2' To compare the ground state energy of the squeezed state calculated using u3 and 72 obtained by solving Eqs. (10)- (11) self-consistently with that of two-phonon coherent state we have performed simple model numerical calculations though they are not shown here. We find that the former is lower than the latter. Numerical calculations using more realistic model are needed and now in progress. Within our model where purely electronic interband hopping processes are neglected, it is found that the squeezed state can exist. To examine whether our results depend on the lIodel used here or not, we have performed calculations using another three types of model: the first model is the single band model, the second model is an extended version of the present model where purely electronic interband hopping processes are included, and the third model is also an lIodified version of the present model where nei ther purely electronic interband hopping nor phonon assisted interband transition is allowed. In the first model and the third one the squeezed state does not occur, while in the second model all phonon states can be the squeezed states provided that phonons strongly interact with the particles. Therefore it follows that a condition of existence of the squeezed state is that the interband electron transfer is allowed in the many-band system. REFERElCES
(1)
(2) (3) (4)
T. Sota and K. Suzuki, in PHONONS 89 (World Scientific, Singapore, 1990), pp.298-300. Zheng Hang, Phys. Rev. B36 (1987) 8736; Phys. Rev. B38 (1988) 11865; J. Phys. C: Solid State Phys. 21 (1988) 2351. Hong Chen et al., Phys. Rev. B39 (1989) 546; Phys. Rev. B40 (1989) 11326. A.M. Jayannavar, Solid State Com.un. 71 (1989) 689.