Temperature and frequency dependence of the nuclear relaxation rate in Qn(TCNQ)2

Solid State Communications, Vol. 24, PP. 29—32, 1977.

Pergamon Press.

Printed in Great Britain

TEMPERATURE AND FREQUENCY DEPENDENCE OF THE NUCLEAR RELAXATION RATE IN Qn(TCNQ)2 * E. Ehrenfreundl’ and AJ. Heeger Department of Physics and 1.aboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, PA 19104, U.S.A. (Received 11 March 1977; in revised form 16 May 1977 by A.A. Maradudin) The magnitude, frequency dependence and temperature dependence of the proton relaxation rates in Qn(TCNQ)2 are explained in terms of a theory which treats the short wavelength response as coherent and one-dimensional, while the long wavelength response is dominated by anisotropic (intrachain vs interchain) diffusion. The diffusive character of the response near q = 0 is consistent with recent attempts to understand the electronic properties of Qn(TCNQ)2 in terms of weak localization of states in one-dimension. Analysis of the data leads to the conclusion that the effective screened Coulomb interaction is relatively weak. IN RECENT YEARS considerable attention has been devoted to experimental and theoretical investigations of nuclear relaxation rates in the one-dimensional (ld) organic quinolinium—tetracyanoquinodimethaneconductor Qn(TCNQ) 2 [1—5].The temperature dependence of the proton relaxation rates (Ti’) of both Qn(TCNQ)2 [1] and its deuterated analog Qn(D8) (TCNQ)2 [5] is Korringa-like (Tj’ cx T) above 130 K, characteristic metalsstrong with extended wavefunctions and of without exchangeelectronic enhancement, On the other hand, the room temperature frequency dependence of Tj’ showed an W”2 d~pendence[3] characteristic of ld diffusion processes. Devreux [31 attributed this frequency dependence to a ld diffusion process caused by the scattering of the conduction electrons by strong Coulomb repulsion U at long wavelengths, q ~ U/W(where W is the bandwidth). Using the same data [3] Butler, Walker and Zoos [4] fitted the Tj’ data to a In w1 law and concluded that the relaxation process in Qn(TCNQ) 2 results from charge carriers in a 2d random walk process. Both of these diffusion mechanisms are in apparent disagreement with the observed temperature dependence of Tj’. In the strong U limit only a weak temperature dependence of T1’ is expected [3]. Moreover, the proposed [4] random walk motion of charge carriers in Qn(TCNQ)2 is not expected to yield a Korringa-like relaxation rate. Thus, it appears that a more generalized point of view is needed to understand both the temperature and frequency dependence of T1’ in the conducting state of Qn(TCNQ)2. —

*

For nuclear spin relaxation caused by the hyperfine interaction, H = A1 s(r), the rate can be written as [61 k T 2 ~“(q, ~ e ) (1) T1’ = 2 ~ IAql We

q

where ~“(q, We) is the imaginary part of the wavevector (q) and frequency (w) dependent susceptibility (at zero field) evaluated at the Larmor frequencyof We 2 lU~ (0)12electronic with Uq (0) the amplitude and Bloch IAq 12 =wave-function A the at the nucleus. In pseudo-ld metaffic systems where metallic conductivity is observed only along one of the crystalline axes, the major contributions to ~“(q, w) arise from IqI 0 and Iqi 2kF as long as kB T, hWe ~ E~.(the Fermi energy). The IqI 2kF contribution was considered in detail in an earlier publication [7]. Because the divergence of the Lindhard function in id (at T = 0) the associated contribution to the relaxation rate is quite sensitive to the strength of Coulomb interactions. On the other hand, it has been recently recognized [8, 9] that near IqI = 0 (i.e. qi ~ 1 where 1 is the mean free path or id localization length) x(q, W) shows a diffusive character in such pseudo- 1 d systems and can be written [10] 2

x~’w)

=

-

(2)

Dq —1w with the exchange enhanced spin susceptibility X8

=

2j.t~N(0)/(1 cx) —

where N(0) is the density of states at the Fermi energy and a = UN(0). For isotropic 3d metals the “effective” diffusion constant D in equation (2) is related to the

Supported by the Advanced Research Projects Agency through DAHC4 572C~0174.

t Permanent address: Department of Physics,

electronic scatteringtime0Fr via is the theFermi relation velocity. [10] D = ~rv~.(l a) where

Technion, Haifa, Israel.

—

29

30

DEPENDENCE OF THE NUCLEAR RELAXATION RATE IN Qn(TCNQ)2 Vol. 24, No. 1 2P(We) (2we)”2 which yields the expected We112 Soda eta!. [8] first applied (2)coupling. for the case r~ pseudo id metals with finiteequation interchain In of dependence of Tj~.At very low frequencies (WeTc ~ 1), their treatment the interchain hopping of the electrons is F(We) = 2 and T~1becomes frequency independent, as modeled by a cutoff exp [— t/r~]to the pure Id difexpected for a 3d system. Both these limits have been fusion spin auto-correlation function. We present here a observed for the id metallic system TTF—TCNQ at somewhat different approach in which we consider room temperature [13, 14]. anisotropic diffusion with two different diffusion conWhen the Coulomb interaction is relatively large stants D 11 and D1 for intra- and interchain processes, (U ~ W), the expression (2) for x(q, w) is inadequate, respectively. Consequently, we write the imaginary part since it treats the exchange enhancement in RPA. In of the diffusive susceptibility as the strong coupling limit, we expect a weak temperature x “( q, w ) w

=

Xs(DD 2+2 +DD 2\22 + ,~ ~q~1 1q1,

(3)

2 W

The metallic conductivity along the chain suggests [lO—12] D11 rv~(l— a), whereas the transverse diffusion constant and the interchain hopping rate T~’ can 2with a being the interchain disbe related by D1 r~’a tance. The contribution of the susceptibility given by equation (3) to Tj’ is obtained after substitution into equation (1) and integration over the small momentum region near q 0. In the transverse directions, q 1 is(3) is 1 [equation taken arbitrarily to be smaller than a valid only for q/ < 1]. The aclual limit of q 11, on the other hand, does not significantly affect the value of the integral and may be chosen as infinity. The final 2 = (A2> is independent of result q may for T~’ assuming Aq1 be written as irkB Tx~(A>

=

2h.zB

(gji~)

(~iI2 \T/

(1

— a)~2 F (We)

+ K(cx) (4)

where ___________________

F(w)

=

~J2(l +‘~/l+ w2r~)_~/~~ -2

(1 —a2)([l —o’F(q)] >q 2k~ with F(q) being the ld Lindhard function [7]. The first term in equation (4) represents the contribution of the K(cx)

=

IqI 0 diffusive mode and the second term contri2kF [7]. Wethenote bution from the region near q = again here that similar results have been obtained by Soda eta!. [8] who introduced a cut-off ethTc to the ld auto-correlation function to account for the interchain hopping. The anisotropic diffusion treatment introduced in the present paper is somewhat more general and allows a natural extension to 3d systems as the interchain interactions are increased. Equation (4) represents a modified Korringa relation, appropriate to pseudo-ld metals, which includes both the effects of the exchange enhancement and the diffusive character of the long wavelength susceptibility. The frequency (or field) dependence of Tj1 is contained in F(w). For purely ld system (r~ 0), -~

dependence ofband T1’.Hubbard Coil [15]model, has shown that for quarter-filled the orbital andthe spin decoupling leads to a well-defined spinwave spectrum J) in addition to the single particle band excitations (Cm~ W). Based on our understanding of the in half-filled band case[16], where and theory are good agreement theexperiment spin-wave excitations should give a contribution to Tj’ of order A 2/J possibly reduced by the lifetime of a given electron on a given site. A contribution from the single particle excitations may add an additional linear term. The

(Cmax

actual of the two terms would be determined magnitudes by the spectral weight function associated with each part of the spectrum. These spectral weight functions are presently unknown. It is likely however that for W the will dominate whereas for U~’ U~W the spin spin wave wave process contribution will be negligible and the above approach [equation (4)] should be a good approximation. In any case we expect the spinwave contribution to be typical for the s = ld, antiferromagnet, i.e. dominated by diffusion processes at high temperatures and approaching a finite value (of order A 2/J where J is the intrachain exchange) at low temperatures. The band contribution [17] is perhaps given by an expression similar to equation (4). These physical arguments based on the Coll [151 theory of the quarter-filled band Hubbard model therefore suggest a large temperature independent contribution to T 1’ which is not observed in Qn(TCNQ)2. Moreover, inclusion of nearest neighbor a generalized Hubbard model would put interactions a gap in the in single particle spectrum leaving only the constant term at kB T W is given by Devreux [3] and can be written (for the case of isotropic hyperfine interaction) as 1 /a\2 irk) ~(We) 1 \ J where a2 = A 2p and p is the fractional charge per TCNQ TCNQ. For a Id system ~,

=

Vol. 24, No. 1 10.0

DEPENDENCE OF THE NUCLEAR RELAXATION RATE IN Qn(TCNQ)2 2and B I with B, = 15.6 sec’ MHz~ 2 Qn (TCNQ)2

•

•

7 5

—

—

5.0

—

—

2.5

—

—

—

U

0 0

I 0.2 12( MHI”2)0.3

0.1

0.4

w~’ Fig. 1. The frequency dependence of Tj~at room temperature for Qn(TCNQ)2. Data points were taken from Devreux [3]. The solid line is a linear fit to the data. 2 + ~2 ~ ir (D 2)2 ~ v’ GSq~Sq> 11q D11q In equation (5), D 11 ~W for the quarter-filled band case, and 2 D / w 11 ,j ~(w)

— —

hA ~

for the nearly half-filled case. If we again assume that for pseudo-ld systems the interchain hopping can be treated as transverse diffusion, we immediately obtain /a\2 2r~Y2 F(We) (6) T1’ = c~ (ntnt>D~” ~

“ “

where n~is the population operator for an electron of spin o(= I~or f) at a given site, and c (of order unity) is defined in reference [3]. The frequency dependence predicted by equation (6) is similar to that of equation (4); however, the magnitude and temperature areof significantly different. Compared with thedependence Korringa-law equation (4), equation (6) yields nearly no temperature dependence of T~’[3]. Thus, we emphasize that caution has to be exercised before reaching a conclusion based upon the frequency dependence alone, The proton relaxation rates in Qn(TCNQ)2 can now be understood [14]. At fixed temperature (T = 300 K) and in the NMR frequency range, i’N = 7—100 MHz, the data as shown in Fig. I (see reference [3]) can be fit to the expression 1 = B 2+ B Tj 1v~/’ 2 (6)

31 =

3.1 sec~.This is

the expected frequency dependence according equation (4) when the interchain hopping timeto is longer than the electronic Larmor period (Were > I). At constant frequency [1, 5] r1’ cx Tat 130 ~ T~300 for both Qn(TCNQ)2 and Qn(D8)(TCNQ)2. Since the susceptibility in this temperature range is temperature independent [18, 19] this temperature variation is in accordance with equation (4) when the Coulomb exchange enhancement factor K(a) is temperature independent. Although the dipolar contribution is relatively small [14] and probably frequency independent in the above frequency regime, its contribution to the small constant term B2 [equation (6)] might be significant; If we assume that the other ~Bits contribution is about ~B2 and associate 2 with the second term in equation (4) we Qn(TCNQ) obtain a U/W 3 per 0.4 mole for theofmeasured susceptibility = 5 x l0~ cm 2 units [18 19]. This relatively small value of a is consistent .

-1

‘

with the observed temperature dependence, T, cx T. On comparison of the slope B1 in equation (6) with the firsttime, termrin= equation (4), we obtain an effective scattering 3 x 10~4sec. This suggests a ld localization length, I

VFT

of order 10—20 lattice coimtants.

However, because of the complexity of phonon assisted diffusion [11, 12] through the wealdy localized states in Id, direct comparison of r with transport results requires a more detailed theory. The nuclear relaxation rate of Qn(TCNQ)2 is often discussed [1, 3, 4] together with that of (NMP)(TCNQ) [NMP= N-methylphenazinium]. However, (NMP) (TCNQ) is a ld conductor with much stronger Coulomb interaction (U/W 1—2 [1, 20] than the above estimate for Qn(TCNQ) 2. It was indeed shown by Devreux [3] that the magnitude and weak temperature dependence of T~ in (NMP)(TCNQ) are better described by equation (6), appropriate in the strong Coulomb limit, rather than equation (4), which is appropriate in the weak Coulomb limit. The frequency dependence observed for NMP—TCNQ can also bewhich obtained 2sec givesfrom a equation (6) with r~, 6 x l0~ reasonable value of iO~for the ratio of the intra- to interchain transfer integrals [3, 21]. In summary, we have shown that the temperature dependence, frequency dependence, and the magnitude of the proton relaxation rates in Qn(TCNQ) 2 can be understood by a theory which treats the short wavelength response function as coherent and Id while the long wavelength response is assumed to be diffusive [equation (4)]. The analysis of the data as presented here leads to the conclusion that the effective screened on-site Coulomb interaction on the TCNQ-chain relatively weak (U/W 0.4).

32

DEPENDENCE OF THE NUCLEAR RELAXATION RATE IN Qn(TCNQ)2

The diffusive character of the response near q = 0 is consistent with recent attempts [11, 12] to understand the electronic properties of Qn(TCNQ)2 in terms of ld localization. The positive low frequency dielectric constant and temperature dependent conductivity have

1.

Vol. 24, No. 1

been explained in these terms. The nuclear relaxation results and their interpretation as described above furthermore imply that at least near kF, the localized states extend over many lattice constants along the onedimensional chains.

REFERENCES EHRENFREUND E., ETEMAD S., COLEMAN L.B., RYBACZEWSKI E.F., GARITO A.F. & HEEGER A.J., Phys. Rev. Lett. 29, 269 (1972).

2.

AZVEDO L.J., CLARK W.G., McLEAN E.O. & SELIGMAN P.F., Solid State Commun. 16, 1267 (1975).

3. 4. 5.

DEVREUX F., IVth International Symposium on the Organic Solid State, p. 74. Bordeaux, France (1975): DEVREUX F.,Phys. Rev. B13,465l (1976). BUTLER M.A., WALKER L.R. & SOOS Z.G., J. Chem. Phys. 64, 3592 (1976). EHRENFREUND E. & GARITO A.F., Solid State Commun. 19, 815 (1976).

6.

MORIYAT.,J. Phys. Soc. Japan 18, 516 (1963).

7.

EHRENFREUND E., RYBACZEWSKI E.F., GARITO A.F. & HEEGER A.J., Phys. Rev. Lett. 28, 873 (1972). In this paper the q = 0 contribution was neglected. SODA G., JEROME D., WEGER M., FABRE J.M., GIRAL L. & BECHGAARD K.,&oc. of the Siofok Conference, Siofok, Hungary (1976). LEE P.A., RICE T.M. & KLEMM R.A. (preprint). FULDE P. & LUTHER A.,Phys. Rev. 170, 570 (1968).

8. 9. 10. 11. 12. 13.

GOGOLIN A.A., ZOLOTUKHIM S.P., MELNIKOV V.1., RASHBA E.I. & SHCHEGOLEV 1.F., Phys. Zh. Eksp. Teor. Fiz. 22,564 (1975), [Soy. Phys.—JETPLett. 22,278 (1975)1. GOGOLIN A.A., MELNIKOV V.1. & RASHBA E.I., Zh. Eksp. Teor. Fiz. 65, 1251 (1973), [Soy. Phys.---JETP 38,620 (1974)]~ SODA G., JEROME D., WEGER M., FABRE J.M. & GIRAL L., So/id State Commun. 18, 1417 (1976).

14.

The dipolar part of the hyperfine coupling also contributes to the nuclear relaxation. The scalar part (proportional to I S) may be incorporated with the contact term to give an effective A in equation (l).The non-scalar part (proportional to 1z5z) can be written in a way similar to (1) but with the susceptibility evaluated at WN — the nuclear resonance frequency [3]. This would 2>, where give(cj2> a contribution is the mean to square T~ similar of the to equation dipolar part (4) of with the F(WN) hyperfine replacing coupling. F(We) Forand the (j2> anisotropies replacing appropriate (A to TCNQ salts (~102_l0~)one would expect r~~ WN (for WN < iO~sec’) and thus no frequency dependence is expected from the nonscalar interaction. Moreover, in TCNQ salts, the scalar part should be dominant, since a strong positive Overhauser effect has been observed; see the following: MARTICORENA B. & NECHTSCHEIN M.,P~oc.XVIIth Colloque Ampere p. 505. (Edited by HOVI V.), North Holland, Amsterdam (1973); ALIZON J., BERTHET G., BLANC J.P., GALLICE J. & ROBERT H. (preprint). 15. COLL C.F., Phys. Rev. B9, 2150 (1974). 16.

17. 18. 19.

EHRENFREUND E., RYBACZEWSKI E.F., GARITO A.F., HEEGER A.J. & PINCUS P., Phys. Rev. B7, 421 (1973); SMITH L.S., EHRENFREUND E., HEEGER A.J., INTERRANTE L.V., BRAY J.W., HART H.R., Jr., & JACOBS I.S.,Solid State Commun. 19, 377 (1976). PINCUS P. (private communication). KEPLER R.G., J. Chem. Phys. 39, 3528 (1963).

20.

BULAEVSKII L.N., LYUBOVSKII R.B. & SHCHEGOLEV I.F., Pis. ZhETF 16, 42 (1972), [JETPLett. 16 29(1972)]. EPSTEIN A.J., ETEMAD S., GARITO A.F. & HEEGER A.J.,Phys. Rev. B5, 952 (1972).

21.

The diffusion model used in reference [4] would predict an unreasonable anisotropy of 106 for (NMP)(TCNQ).