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The superconducting energy gap in 2H-NbSe2 and its coupling to charge density waves
Physica 105B (1981) 422-427 North-Holland Publishing Company
THE S U P E R C O N D U C T I N G E N E R G Y GAP IN 2H-NbSe2 AND ITS C O U P L I N G TO C H A R G E DENSITY WAVES R. S O O R Y A K U M A R and M.V. K L E I N Department of Physics and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana IL. 61801, USA Invited paper At 33 K 2H-NbSe2 distorts into an incommensurate charge density wave (CDW) state. The Raman spectrum shows amplitude modes of A and E symmetrynear 40 cm-I induced by the CDW. At 7 K this material becomes superconducting, and at 2 K we observe additional Raman peaks near 20 cm-l, one each of A and E symmetry. These are close in energy to the energy gap 2A, as measured by far infrared transmission. The Raman activity of the gap excitations is believed to be caused by coupling to the Raman active CDW phonon modes. Such coupling is shown directly by the changes in the Raman spectra at 2 K due to an applied magnetic field. The zero field results can be explained by a model in which the optical CDW phonon modes couple to superconducting electrons, and the resulting phonon self-energy is calculated using the BCS theory for the electrons.
1. Introduction Superconductivity results from the instability of a normal "nearly free electron" metal to pairing of electrons of wave vector k and spin up with those with - k and spin down, a pairing due to an attractive effective electron-electron interaction mediated by phonons [1]. The energy gap A plays a key role in the superconducting state. At zero temperature an energy of at least 2d is necessary to create an excitation, in this case a pair of quasi-particles. The probe most used to study these excitations is far infrared absorption, which shows a continuous rise from zero at a frequency to equal to 2A/h. Raman scattering should in principle be sensitive to these excitations. Depending on the assumed details of the direct coupling of the electrons to light, the predicted scattering intensity 'should rise from zero at a frequency shift to equal to 2A/h either discontinuously [2-5] or continuously [6]. It should then approach asymptotically the result for free "normal" electrons, which is proportional to to. An experimental result of this type was reported on NbaSn [7] about ten years ago, but to our knowledge it has not been repeated. Such experiments are extremely
difficult, and the results were probably in error. A charge density wave (CDW) is another form of electronic instability in which electrons of the same spin at k and k + Q are correlated, giving a finite value to the electronic charge density - e ( C L o ~ C ~ , ) at wave vector Q [8]. The attraction leading to C D W pairing is again due to phonons, but here they have a single wave vector Q (and - Q ) , whereas essentially all phonons contribute to the pairing responsible for superconductivity. The C D W instability leads to a displacive phase transition, wherein the phonons with wave vectors near Q soften for temperatures above the distortion temperature Td. Below Td phonons of wave vector -+Q "condense" into a static distortion, about which new quasi-harmonic vibrations are possible ( " C D W phonons"). These C D W phase transitions are quite c o m m o n in the layered transition-metal dichalcogenides with their narrow d-bands [9]. T h e C D W phonon modes may be studied by R a m a n scattering. For instance, 2H-TaSe2 has a commensurate 3Q C D W at low temperatures, and amplitude modes and phase modes can be observed [10] and their temperature dependence revealed [11, 12]. In 2H-NbSe2 the C D W is incommensurate, and only amplitude modes are
0378-4363/81/0000-0000/$2.50 O North-Holland Publishing Company and Yamada Science Foundation
R. Sooryakumar and M.V. Klein~Superconducting energy gap in 2H-NbSe2
observable by Raman scattering [12]. While in this state 2H-NbSe2 becomes superconducting with a high Tc. We observe Raman peaks at the gap frequency 2A/h [12, 13]. They are coupled to the CDW phonons, and their Raman activity is believed to result from this coupling [14]. Static coupling between CDWs and superconductivity can be understood using mean-field theories [15]. Dynamic coupling of the type we are concerned with here is a special case of coupling of optical phonons to superconducting electrons [14]. 2. Charge density waves in 2H-NbSe2 Neutron diffraction studies by Moncton et al. [16] show that 2H-NbSe2 undergoes a phase transition from a normal lattice to one with an incommensurate charge-density-wave (CDW) at the onset temperature Td of 33 K. Three wave vectors Qj are simultaneously present in the CDW, lying 120° apart in the basal plane along the FM symmetry directions. Their length is about 0.98 of the value [(2/3)(FM)] that would give a commensurate 3a0x3a0 superlattice.
423
Elastic modulus measurements show that the incommensurate phase persists at least as low as 1.3K [17]. Above Td longitudinal acoustic phonons of wave vector Qj are observed to soften. Below Td these phonons "condense" into a static distortion, about which new quasi-harmonic vibrations are possible. The Raman spectrum at 9 K reveals these vibrations, as shown in fig. 1. The peaks near 40cm -1 are "amplitude modes" of the CDW wherein the amplitude of the components of the distortion at each of the three Qj varies either symmetrically in all of the Qj (A~g) or antisymmetrically in, say, Q1 and Q2 (E~). Modulations in phase ("phasons") are also possible. In the harmonic limit, they have zero frequency at zero wave vector for an incommensurate distortion. In higher order this is no longer true [18]. In the extreme anharmonic (soliton) limit there is both a zero frequency phason branch and one at finite frequency. We are apparently not observing any phasons in 2H-NbSe2, and will ignore them in the following discussion. 3. Gap excitations in superconducting NbSe2
120 I00 8O 6O
2H NbSe2 T-gK
4o ~ .~ 2o z o
g
3C . . : , : , ~ 20
"
10 ,
I 40
,
I
80 RAMAN SHIFT (cm-I) 0
Fig. 1. Raman spectrum of a high quality sample of 2HNbSe2 at 9 K . The 28cm -t EEs peak is the "interlayer" mode in which adjacent S e - N b - S e layers vibrate against each other in the basal plane. The E2s spectrum is depolarized (xy) in the basal plane. The A~8 spectrum has been obtained by subtracting the (xy) spectrum from the polarized (xx) spectrum.
Below 7.2K 2H-NbSe2 becomes a highly anisotropic type II superconductor [19]. Raman spectra when the sample used for fig. 1 is immersed in superfluid He are shown in fig. 2 [12, 13]. Two new peaks are seen: one of Alg symmetry at 19cm -1 and a n E2s peak at 15.5cm-L Their weighted average agrees with the position of the peak in far infrared transmission observed by Clayman and Frindt [20]. We have studied approximately 10 samples of 2H-NbSe2. The one used for figs. 1 and 2 gave the strongest and narrowest CDW Raman peaks. Other samples gave broader CDW peaks, particularly in the E2s spectrum, but the peaks at the gap were only slightly affected [13]. Changes in the technique of crystal growth can vary the presence of non-magnetic impurities, which tend to inhibit the formation of CDWs [21], whereas they have a small effect on superconductivity [22]. Raman measurements have been made on several samples studied by Huntley and Frindt [21] which have no CDWs but which still are
424
R. Sooryakumar and M. V. Klein~Superconducting energy gap in 2H-NbSe2
35!
presented in fig. 3 [13], which shows that a magnetic field completely suppresses the gap peak and transfers its strength to the CDW phonon. The upper critical field He2 for H perpendicular to the layers is about 40 kG at 2 K [19]. It is seen that suppression is complete for H ~Hc2/3. This result holds for both Raman symmetries and for H parallel and perpendicular to the layers.
2H NbSe2 T:2K
AIg
E2g
.~ 25 __m
g ~)- 2o
1:
F-
m
4. Theory of the Raman-active gap excitations
\
I0
1
0
20
I
I
I
I
40 60 0 20 40 RAMAN SHIFT (cm-I)
I
60
Fig. 2. Raman spectrum of the sample in fig. 1 when immersed in superfluid He.
superconductors below 7 K [23]. The spectra below 60 cm -1 show no CDW phonon peaks and no superconducting gap peaks. This last and very recent result had been predicted theoretically by Balseiro and Falicov [14] before the measurements were made. Somewhat older results proving coupling between CDW phonons and gap excitations are A Spectrum
HI
03 I--
I// 7:.~.. \ \ ";: ...... ?5:',,,
A
4-
n~
~3
tn a~
<
f
>-2 I.-
.
:r ~,\'
.?
0.0
H'=
+
+
,•Ggk(bq+ b-q)Ck+q.~,Ck,,,.
(1)
Here gk is the electron-phonon coupling constant, q is the phonon wave vector, tr is the spin index, and bq and Ck,~ are phonon and electron destruction operators, respectively. Assumption 3: The renormalized phonon spectrum is calculated using the electron "vacuum polarization" diagram for BCS electrons. This leads to a phonon self-energy for frequency to at zero temperature proportional to
Layers
A5
>-
The shape of the observed gap peaks and their coupling to CDW phonons suggests that previous theories [2-6] of Raman scattering by gap excitations do not apply in 2H-NbSe2. The direct scattering of light by the electrons is too weak to be observed in these experiments. Indirect coupling is assumed in the theory of Balseiro and Falicov [14]. There are four implicit or explicit assumptions in their theory. Assumption 1: A Raman-active phonon lies close in energy to the superconducting gap 2A. Assumption 2: The phonon couples to the electrons via the usual type of electron-phonon coupling
i
]~ g2 (1
Z bJ Z
..... I
I0
×
13.8 kG I
20 30 40 RAMAN SHIFT (cm -;)
1
1
(2)
I
50
60
Fig. 3. The Al= Raman spectrum of a NbSe2 sample (different from that in figs. 1 and 2) at 2 K as a function of an applied magnetic field perpendicular to the layers.
where
to + = to + i0 +, E = (e2+ A2)1/2, E' = ( e ' 2 + A 2 ) 1/2, e = $k, and e ' = ek+q, the e's being normal electron excitation energies relative to the Fermi surface.
R. Sooryakumar and M.V. Klein~Superconducting energy gap in 2l.-I-NbSe2
Assumption 4: The small q limit holds, i.e. hq • Vk '~ 2A,
(3)
where hvk = 8e/Ok. Then e = e' and E = E', and the phonon self-energy becomes 2f ~
de
1
to2
A: - e 2]
(4)
Here (g2p) is the average of g~, times the local density of states averaged over the Fermi surface. If gk is assumed to be a constant g we find
(g2p) = g:Po,
(5)
where p0 is the usual density of normal electrons per spin at the Fermi level. This is the form assumed by Balseiro and Falicov. The real part of (4) has an inverse square root singularity when hto approaches 2A from below. If the bare phonon is assumed to be undamped and have the frequency tOo, this leads to a pole in the phonon Green's function at a frequency Ep/h < 2A/h that is the solution of [14] 16htooA 2
(6) A solution exists for all values of the coupling constant g. The self-energy (4) also leads to phonon damping when t o - too > 2A and renormalization of the phonon peak near too to a higher value. An example of the resulting phonon spectral density is shown in fig. 4 [14] for a coupling constant
425
The C D W phonon will have an intrinsic width in the absence of electron-phonon interactions due to anharmonicity. It may also be inhomogeneously broadened due to interactions of impurities with the CDW. Balseiro and Falicov chose an inhomogeneous distribution of to0's to fit the Alg spectrum of fig. L The fit is shown in fig. 5(a). They then calculated the resulting phonon spectral function in the superconducting state. The result for (g:p)= 0.08hto0 is shown in fig. 5(b). The experimental Alg spectrum of fig. 2 is reproduced in fig. 5(c). Qualitatively there is good agreement. A similar qualitative result is obtained if lifetime broadening is assumed for the bare phonon. It is not obvious that the small q limit (3) should apply. In the pseudo-backscattering geometry used for the experiments q will have two components: a fixed qll-~ 1.2 x 105 cm -1 in the basal plane equal to the parallel component of the incident photon wave vector and a variable q, of order of the inverse skin depth 8 - 1 ~ 5 x 105 cm -I [24]. Calculations of the conduction band give it a width of about l e V [25, 26], leading to an estimate of vii, the Fermi velocity in the basal plane of 1.6 x 107 cm/s and an estimate of qllVll of about 2 x 1012~ 10 cm -~. In the other direction one might estimate q,v,=6Ocm -~. J
(b)
(c)
¢ \
24 = 0.5h~i
i 40
80 0
L
,40 80 0 Raman shift (cm -1)
I
I
40
I
80
Fig. 5. P h o n o n spectral density for the coupled modes when the phonon is inhomogeneously broadened to fit the data of 0.5
1.0
1.5
Fig. 4. P h o n o n spectral density for the coupled m o d e s [14].
fig. 2 [14]. (a) Fit to Als Raman data of fig. 1. Co) Calculated spectrum of coupled modes. (c) Experimental At s Raman spectrum from fig. 2.
426
R. Sooryakumar and M. V. Klein~Superconducting energy gap in 2H-NbSe2
These represent overall averages. From (2) it can be seen that one needs to know the Fermi surface properties where the coupling constant gk is large. For charge-density-wave phonons this will be near the intersections of the high temperature Fermi surface with itself shifted by the wave vectors Qj. These regions depend on the extent of "nesting" and are not really well known. For instance, the calculated band structure in [25, 26] is unable to account for the observed Landau quantum oscillations in 2HNbSe2 [27]. As with the latter measurements, CDW effects are sensitive to subtle changes in the Fermi surface. We conclude that not enough is known to prove the inequality (3), but it is probably not grossly violated. A calculation of the phonon spectral density has been made by Schuster in the other limit, namely hqvf ~, 2A [28]. In this limit and with a simple Fermi surface the inverse square root singularity of the real part of the phonon selfenergy in (6) becomes merely a logarithmic singularity. Strong modifications of the spectrum are obtained only when 2A is close to hto0; for 2A ~ hwo/2 they are essentially negligible (1% effects). It appears that the experiments require the small q limit if this model is to work. An interesting situation should occur for optical phonons in metals with a simple Fermi surface. "Anomalous" dispersion is obtained for qvf
The superconducting gap has been seen via its coupling to CDW phonons. Other low-lying Raman-active phonons should show the same effect if the coupling is strong enough [14] and if the "small q" limit (3) holds. (/'he "interlayer"
mode at 28 cm -1 in fig. 1 shows no evidence of such coupling and thus presumably violates one or both of these conditions.) The spectra in a magnetic field (fig. 3) contain as yet unanalyzed information about electronphonon coupling and hence about electron "vacuum polarization" in the vortex state of this anisotropic superconductor. Qualitatively, the observed suppression of the gap peak is probably due to the "gapless" nature of the vortex state, i.e. the "normal" electrons in the vortex cores [31]. Acknowledgments
We thank S.F. Meyer, F.J. DiSalvo, R.V. Coleman and R.F. Frindt for providing crystals and J.P. Wolfe for the use of a cryostat for the magnetic field-dependent studies. Thanks go to W.L. McMillan and John Bardeen for theoretical discussions, and especially to L.M. Falicov for many conversations for his patient explanation of the theory of ref. 14, and for permission to present it here in figs. 4 and 5. This work was supported by the National Science Foundation under the MRL Grant DMR-77-23999. References [1] J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175. [2] A.A. Abrikosov and L.A. Falkovskii, Zh. Eksp. Teor. Fiz. 40 (1961) 262 [Soy. Phys. JETP 13 (1961) 179]. [3] S.Y. Tong and A.A. Maradudin, Mater. Res. Bull. 4 (1969) 563. [4] D.R. Tilley, Z. Phys. 254 (1972) 71; J. Phys. F 38 (1973) 417. [5] A.A. Abrikosov and V.M. Genkin, Zh. Eksp. Teor. Fiz. 65 (1973) 842 [Sov. Phys. JETP 38 (1974) 417]. [6] C.B. Cuden, Phys. Rev. B13 (1976) 1993; Phys. Rev.
B18 (1978)3150. [7] L.M. Fraas, P.F. Williams and S.P.S. Porto, Solid State Commun. 8 (1970) 2133. [8] S.K. Chan and V. Heine, J. Phys. F 3 (1973) 795. [9] J.A. Wilson, F.J. DiSalvo and S. Mahajan, Advan. Phys. 24 (1975) 117. [10] J.A. Holy, M.V. Klein, W.L. McMillan and S.F. Meyer, Phys. Rev. Lett. 37 (1976) 1145. [11] E.F. Steigmeyer, G. Harbeke, H. Auderset and F.J. DiSalvo, Solid State Commun. 20 (1976) 667. [12] R. Sooryakumar, D.G. Bruns and M.V. Klein, in: Light Scattering in Solids, J.L. Birman, H.Z. Cummins and K.K. Rebane, eds. (Plenum, New York, 1980), p. 347.
R. Sooryakumar and M. V. Klein/Superconducting energy gap in 2H-NbSe2 [13] R. Sooryakumar and M.V. Klein, Phys. Rev. Lett. 45 (1980) 660. [14] C.A. Balseiro and L.M. Falicov, Phys. Rev. Lett. 45 (1980) 662. [15] G. Bilbro and W.L. McMillan,' Phys. Rev. B14 (1976) 1887. C.A. Balseiro and L.M. Falicov, Phys. Rev. B20 (1979) 4457. [16] D.E. Moncton, J.D. Axe and F.J. DiSalvo, phys. Rev. B16 (1977) 801. [17] M. Barmatz, L.R. Testardi and F.J. DiSalvo, Phys. Rev. B12 (1975) 4367. [18] W.L. McMillan, Phys. Rev. B16 (1977) 4655. A.D. Bruce and R.A. Cowley, J. Phys. C l l (1978) 3609. [19] P. de Trey, Suso Gygax and J.P. Jan, J. Low Temp. Phys. 11 (1973) 421. [20] B.P. Clayman and R.F. Frindt, Solid State Commun. 9 (1971) 1881. [21] D.J. Huntley and R.F. Frindt, Can. J. Phys. 52 (1974) 861. [22] D.J. Huntley, Phys. Rev. Lett. 36 (1976) 490.
[23] [24] [25] [26]
[27] [28] [29]
[30]
[31]
427
J.R. Long, S.P. Bowen and H.E. Lewis, Solid State Commun. 22 (1977) 363. R. Sooryakumar, M.V. Klein and R.F. Frindt, unpublished work. R.T. Harley and P.A. Fleury, J. Phys. C12 (1979) L863. L.F. Mattheis, Phys. Rev. B8 (1973) 3719. G. Wexler and A.M. Wolley, J. Phys. C9 (1976) 1185. N.J. Doran, B. Ricco, D.J. Titterington and G. Wexler, J. Phys. C l l (1978) 685. J.E. Crraebner and M. Robbins, Phys. Rev. Lett. 36 (1976) 422. H.G. Schuster, Solid State Commun. 13 (1973) 1559. I.P. Ipatova, A.V. Subashiev and A.A. Maradudin, in: Light Scattering in Solids, M. Balkanski, ed. (Flammarian Sciences, Pads, 1971) p. 86. I.P. Ipatova and A.V. Subashiev, Zh. Eksp. Teor. Fiz. 66 (1974) 722 [Sov. Phys. JETP 39 (1970 349]. M.V. Klein, in: Proc. Sergio Porto Memorial Conf. on Lasers and Applications, Rio di Janiro, Brazil (30 June3 July, 1980) (Springer, Berlin, 1981) to be published. K. Maki, Phys. Rev. 156 (1967) 437.
THE S U P E R C O N D U C T I N G E N E R G Y GAP IN 2H-NbSe2 AND ITS C O U P L I N G TO C H A R G E DENSITY WAVES R. S O O R Y A K U M A R and M.V. K L E I N Department of Physics and Materials Research Laboratory, University of Illinois at Urbana-Champaign, Urbana IL. 61801, USA Invited paper At 33 K 2H-NbSe2 distorts into an incommensurate charge density wave (CDW) state. The Raman spectrum shows amplitude modes of A and E symmetrynear 40 cm-I induced by the CDW. At 7 K this material becomes superconducting, and at 2 K we observe additional Raman peaks near 20 cm-l, one each of A and E symmetry. These are close in energy to the energy gap 2A, as measured by far infrared transmission. The Raman activity of the gap excitations is believed to be caused by coupling to the Raman active CDW phonon modes. Such coupling is shown directly by the changes in the Raman spectra at 2 K due to an applied magnetic field. The zero field results can be explained by a model in which the optical CDW phonon modes couple to superconducting electrons, and the resulting phonon self-energy is calculated using the BCS theory for the electrons.
1. Introduction Superconductivity results from the instability of a normal "nearly free electron" metal to pairing of electrons of wave vector k and spin up with those with - k and spin down, a pairing due to an attractive effective electron-electron interaction mediated by phonons [1]. The energy gap A plays a key role in the superconducting state. At zero temperature an energy of at least 2d is necessary to create an excitation, in this case a pair of quasi-particles. The probe most used to study these excitations is far infrared absorption, which shows a continuous rise from zero at a frequency to equal to 2A/h. Raman scattering should in principle be sensitive to these excitations. Depending on the assumed details of the direct coupling of the electrons to light, the predicted scattering intensity 'should rise from zero at a frequency shift to equal to 2A/h either discontinuously [2-5] or continuously [6]. It should then approach asymptotically the result for free "normal" electrons, which is proportional to to. An experimental result of this type was reported on NbaSn [7] about ten years ago, but to our knowledge it has not been repeated. Such experiments are extremely
difficult, and the results were probably in error. A charge density wave (CDW) is another form of electronic instability in which electrons of the same spin at k and k + Q are correlated, giving a finite value to the electronic charge density - e ( C L o ~ C ~ , ) at wave vector Q [8]. The attraction leading to C D W pairing is again due to phonons, but here they have a single wave vector Q (and - Q ) , whereas essentially all phonons contribute to the pairing responsible for superconductivity. The C D W instability leads to a displacive phase transition, wherein the phonons with wave vectors near Q soften for temperatures above the distortion temperature Td. Below Td phonons of wave vector -+Q "condense" into a static distortion, about which new quasi-harmonic vibrations are possible ( " C D W phonons"). These C D W phase transitions are quite c o m m o n in the layered transition-metal dichalcogenides with their narrow d-bands [9]. T h e C D W phonon modes may be studied by R a m a n scattering. For instance, 2H-TaSe2 has a commensurate 3Q C D W at low temperatures, and amplitude modes and phase modes can be observed [10] and their temperature dependence revealed [11, 12]. In 2H-NbSe2 the C D W is incommensurate, and only amplitude modes are
0378-4363/81/0000-0000/$2.50 O North-Holland Publishing Company and Yamada Science Foundation
R. Sooryakumar and M.V. Klein~Superconducting energy gap in 2H-NbSe2
observable by Raman scattering [12]. While in this state 2H-NbSe2 becomes superconducting with a high Tc. We observe Raman peaks at the gap frequency 2A/h [12, 13]. They are coupled to the CDW phonons, and their Raman activity is believed to result from this coupling [14]. Static coupling between CDWs and superconductivity can be understood using mean-field theories [15]. Dynamic coupling of the type we are concerned with here is a special case of coupling of optical phonons to superconducting electrons [14]. 2. Charge density waves in 2H-NbSe2 Neutron diffraction studies by Moncton et al. [16] show that 2H-NbSe2 undergoes a phase transition from a normal lattice to one with an incommensurate charge-density-wave (CDW) at the onset temperature Td of 33 K. Three wave vectors Qj are simultaneously present in the CDW, lying 120° apart in the basal plane along the FM symmetry directions. Their length is about 0.98 of the value [(2/3)(FM)] that would give a commensurate 3a0x3a0 superlattice.
423
Elastic modulus measurements show that the incommensurate phase persists at least as low as 1.3K [17]. Above Td longitudinal acoustic phonons of wave vector Qj are observed to soften. Below Td these phonons "condense" into a static distortion, about which new quasi-harmonic vibrations are possible. The Raman spectrum at 9 K reveals these vibrations, as shown in fig. 1. The peaks near 40cm -1 are "amplitude modes" of the CDW wherein the amplitude of the components of the distortion at each of the three Qj varies either symmetrically in all of the Qj (A~g) or antisymmetrically in, say, Q1 and Q2 (E~). Modulations in phase ("phasons") are also possible. In the harmonic limit, they have zero frequency at zero wave vector for an incommensurate distortion. In higher order this is no longer true [18]. In the extreme anharmonic (soliton) limit there is both a zero frequency phason branch and one at finite frequency. We are apparently not observing any phasons in 2H-NbSe2, and will ignore them in the following discussion. 3. Gap excitations in superconducting NbSe2
120 I00 8O 6O
2H NbSe2 T-gK
4o ~ .~ 2o z o
g
3C . . : , : , ~ 20
"
10 ,
I 40
,
I
80 RAMAN SHIFT (cm-I) 0
Fig. 1. Raman spectrum of a high quality sample of 2HNbSe2 at 9 K . The 28cm -t EEs peak is the "interlayer" mode in which adjacent S e - N b - S e layers vibrate against each other in the basal plane. The E2s spectrum is depolarized (xy) in the basal plane. The A~8 spectrum has been obtained by subtracting the (xy) spectrum from the polarized (xx) spectrum.
Below 7.2K 2H-NbSe2 becomes a highly anisotropic type II superconductor [19]. Raman spectra when the sample used for fig. 1 is immersed in superfluid He are shown in fig. 2 [12, 13]. Two new peaks are seen: one of Alg symmetry at 19cm -1 and a n E2s peak at 15.5cm-L Their weighted average agrees with the position of the peak in far infrared transmission observed by Clayman and Frindt [20]. We have studied approximately 10 samples of 2H-NbSe2. The one used for figs. 1 and 2 gave the strongest and narrowest CDW Raman peaks. Other samples gave broader CDW peaks, particularly in the E2s spectrum, but the peaks at the gap were only slightly affected [13]. Changes in the technique of crystal growth can vary the presence of non-magnetic impurities, which tend to inhibit the formation of CDWs [21], whereas they have a small effect on superconductivity [22]. Raman measurements have been made on several samples studied by Huntley and Frindt [21] which have no CDWs but which still are
424
R. Sooryakumar and M. V. Klein~Superconducting energy gap in 2H-NbSe2
35!
presented in fig. 3 [13], which shows that a magnetic field completely suppresses the gap peak and transfers its strength to the CDW phonon. The upper critical field He2 for H perpendicular to the layers is about 40 kG at 2 K [19]. It is seen that suppression is complete for H ~Hc2/3. This result holds for both Raman symmetries and for H parallel and perpendicular to the layers.
2H NbSe2 T:2K
AIg
E2g
.~ 25 __m
g ~)- 2o
1:
F-
m
4. Theory of the Raman-active gap excitations
\
I0
1
0
20
I
I
I
I
40 60 0 20 40 RAMAN SHIFT (cm-I)
I
60
Fig. 2. Raman spectrum of the sample in fig. 1 when immersed in superfluid He.
superconductors below 7 K [23]. The spectra below 60 cm -1 show no CDW phonon peaks and no superconducting gap peaks. This last and very recent result had been predicted theoretically by Balseiro and Falicov [14] before the measurements were made. Somewhat older results proving coupling between CDW phonons and gap excitations are A Spectrum
HI
03 I--
I// 7:.~.. \ \ ";: ...... ?5:',,,
A
4-
n~
~3
tn a~
<
f
>-2 I.-
.
:r ~,\'
.?
0.0
H'=
+
+
,•Ggk(bq+ b-q)Ck+q.~,Ck,,,.
(1)
Here gk is the electron-phonon coupling constant, q is the phonon wave vector, tr is the spin index, and bq and Ck,~ are phonon and electron destruction operators, respectively. Assumption 3: The renormalized phonon spectrum is calculated using the electron "vacuum polarization" diagram for BCS electrons. This leads to a phonon self-energy for frequency to at zero temperature proportional to
Layers
A5
>-
The shape of the observed gap peaks and their coupling to CDW phonons suggests that previous theories [2-6] of Raman scattering by gap excitations do not apply in 2H-NbSe2. The direct scattering of light by the electrons is too weak to be observed in these experiments. Indirect coupling is assumed in the theory of Balseiro and Falicov [14]. There are four implicit or explicit assumptions in their theory. Assumption 1: A Raman-active phonon lies close in energy to the superconducting gap 2A. Assumption 2: The phonon couples to the electrons via the usual type of electron-phonon coupling
i
]~ g2 (1
Z bJ Z
..... I
I0
×
13.8 kG I
20 30 40 RAMAN SHIFT (cm -;)
1
1
(2)
I
50
60
Fig. 3. The Al= Raman spectrum of a NbSe2 sample (different from that in figs. 1 and 2) at 2 K as a function of an applied magnetic field perpendicular to the layers.
where
to + = to + i0 +, E = (e2+ A2)1/2, E' = ( e ' 2 + A 2 ) 1/2, e = $k, and e ' = ek+q, the e's being normal electron excitation energies relative to the Fermi surface.
R. Sooryakumar and M.V. Klein~Superconducting energy gap in 2l.-I-NbSe2
Assumption 4: The small q limit holds, i.e. hq • Vk '~ 2A,
(3)
where hvk = 8e/Ok. Then e = e' and E = E', and the phonon self-energy becomes 2f ~
de
1
to2
A: - e 2]
(4)
Here (g2p) is the average of g~, times the local density of states averaged over the Fermi surface. If gk is assumed to be a constant g we find
(g2p) = g:Po,
(5)
where p0 is the usual density of normal electrons per spin at the Fermi level. This is the form assumed by Balseiro and Falicov. The real part of (4) has an inverse square root singularity when hto approaches 2A from below. If the bare phonon is assumed to be undamped and have the frequency tOo, this leads to a pole in the phonon Green's function at a frequency Ep/h < 2A/h that is the solution of [14] 16htooA 2
(6) A solution exists for all values of the coupling constant g. The self-energy (4) also leads to phonon damping when t o - too > 2A and renormalization of the phonon peak near too to a higher value. An example of the resulting phonon spectral density is shown in fig. 4 [14] for a coupling constant
425
The C D W phonon will have an intrinsic width in the absence of electron-phonon interactions due to anharmonicity. It may also be inhomogeneously broadened due to interactions of impurities with the CDW. Balseiro and Falicov chose an inhomogeneous distribution of to0's to fit the Alg spectrum of fig. L The fit is shown in fig. 5(a). They then calculated the resulting phonon spectral function in the superconducting state. The result for (g:p)= 0.08hto0 is shown in fig. 5(b). The experimental Alg spectrum of fig. 2 is reproduced in fig. 5(c). Qualitatively there is good agreement. A similar qualitative result is obtained if lifetime broadening is assumed for the bare phonon. It is not obvious that the small q limit (3) should apply. In the pseudo-backscattering geometry used for the experiments q will have two components: a fixed qll-~ 1.2 x 105 cm -1 in the basal plane equal to the parallel component of the incident photon wave vector and a variable q, of order of the inverse skin depth 8 - 1 ~ 5 x 105 cm -I [24]. Calculations of the conduction band give it a width of about l e V [25, 26], leading to an estimate of vii, the Fermi velocity in the basal plane of 1.6 x 107 cm/s and an estimate of qllVll of about 2 x 1012~ 10 cm -~. In the other direction one might estimate q,v,=6Ocm -~. J
(b)
(c)
¢ \
24 = 0.5h~i
i 40
80 0
L
,40 80 0 Raman shift (cm -1)
I
I
40
I
80
Fig. 5. P h o n o n spectral density for the coupled modes when the phonon is inhomogeneously broadened to fit the data of 0.5
1.0
1.5
Fig. 4. P h o n o n spectral density for the coupled m o d e s [14].
fig. 2 [14]. (a) Fit to Als Raman data of fig. 1. Co) Calculated spectrum of coupled modes. (c) Experimental At s Raman spectrum from fig. 2.
426
R. Sooryakumar and M. V. Klein~Superconducting energy gap in 2H-NbSe2
These represent overall averages. From (2) it can be seen that one needs to know the Fermi surface properties where the coupling constant gk is large. For charge-density-wave phonons this will be near the intersections of the high temperature Fermi surface with itself shifted by the wave vectors Qj. These regions depend on the extent of "nesting" and are not really well known. For instance, the calculated band structure in [25, 26] is unable to account for the observed Landau quantum oscillations in 2HNbSe2 [27]. As with the latter measurements, CDW effects are sensitive to subtle changes in the Fermi surface. We conclude that not enough is known to prove the inequality (3), but it is probably not grossly violated. A calculation of the phonon spectral density has been made by Schuster in the other limit, namely hqvf ~, 2A [28]. In this limit and with a simple Fermi surface the inverse square root singularity of the real part of the phonon selfenergy in (6) becomes merely a logarithmic singularity. Strong modifications of the spectrum are obtained only when 2A is close to hto0; for 2A ~ hwo/2 they are essentially negligible (1% effects). It appears that the experiments require the small q limit if this model is to work. An interesting situation should occur for optical phonons in metals with a simple Fermi surface. "Anomalous" dispersion is obtained for qvf
The superconducting gap has been seen via its coupling to CDW phonons. Other low-lying Raman-active phonons should show the same effect if the coupling is strong enough [14] and if the "small q" limit (3) holds. (/'he "interlayer"
mode at 28 cm -1 in fig. 1 shows no evidence of such coupling and thus presumably violates one or both of these conditions.) The spectra in a magnetic field (fig. 3) contain as yet unanalyzed information about electronphonon coupling and hence about electron "vacuum polarization" in the vortex state of this anisotropic superconductor. Qualitatively, the observed suppression of the gap peak is probably due to the "gapless" nature of the vortex state, i.e. the "normal" electrons in the vortex cores [31]. Acknowledgments
We thank S.F. Meyer, F.J. DiSalvo, R.V. Coleman and R.F. Frindt for providing crystals and J.P. Wolfe for the use of a cryostat for the magnetic field-dependent studies. Thanks go to W.L. McMillan and John Bardeen for theoretical discussions, and especially to L.M. Falicov for many conversations for his patient explanation of the theory of ref. 14, and for permission to present it here in figs. 4 and 5. This work was supported by the National Science Foundation under the MRL Grant DMR-77-23999. References [1] J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108 (1957) 1175. [2] A.A. Abrikosov and L.A. Falkovskii, Zh. Eksp. Teor. Fiz. 40 (1961) 262 [Soy. Phys. JETP 13 (1961) 179]. [3] S.Y. Tong and A.A. Maradudin, Mater. Res. Bull. 4 (1969) 563. [4] D.R. Tilley, Z. Phys. 254 (1972) 71; J. Phys. F 38 (1973) 417. [5] A.A. Abrikosov and V.M. Genkin, Zh. Eksp. Teor. Fiz. 65 (1973) 842 [Sov. Phys. JETP 38 (1974) 417]. [6] C.B. Cuden, Phys. Rev. B13 (1976) 1993; Phys. Rev.
B18 (1978)3150. [7] L.M. Fraas, P.F. Williams and S.P.S. Porto, Solid State Commun. 8 (1970) 2133. [8] S.K. Chan and V. Heine, J. Phys. F 3 (1973) 795. [9] J.A. Wilson, F.J. DiSalvo and S. Mahajan, Advan. Phys. 24 (1975) 117. [10] J.A. Holy, M.V. Klein, W.L. McMillan and S.F. Meyer, Phys. Rev. Lett. 37 (1976) 1145. [11] E.F. Steigmeyer, G. Harbeke, H. Auderset and F.J. DiSalvo, Solid State Commun. 20 (1976) 667. [12] R. Sooryakumar, D.G. Bruns and M.V. Klein, in: Light Scattering in Solids, J.L. Birman, H.Z. Cummins and K.K. Rebane, eds. (Plenum, New York, 1980), p. 347.
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