PHYSICAL REVIEW B VOLUME 36, NUMBER 8 15 SEPTEMBER 1987-I

Phonon-drag effect in TiSez „S„mixed compounds

J. -M. Lopez-Castillo

Quebec, Canada J1HSX4 and Centre d'Etudes Nucleaires de Grenoble, Departement de Recherche Fondamentale,Service des Basses Temperatures, 85X, 38041 Grenoble Cedex, France

A. Amara and S. JandlGroupe de Recherche sur les Semiconducteurs et les Dielectriques et Centre de Recherche en Physique du Solide,

Uniuersite de Sherbrooke, Sherbrooke, Quebec, Canada J1K2R1

J.-P. Jay-Gerin

Quebec, Canada J1H 5%4 and Centre de Recherche en Physique du Solide, Uniuersite de Sherbrooke,Sherbrooke, Quebec, Canada JIK2R1

C. Ayache

85X, 38041 Grenoble Cedex, France

M. J. AubinGroupe de Recherche sur les Semiconducteurs et les Dielectriques et Centre de Recherche en Physique du Solide,

Uniuersite de Sherbrooke, Sherbrooke, Quebec, Canada J1K2R1(Received 3 April 1987)

Thermoelectric power (TEP) measurements of TiSe& „S solid solutions have been performed inthe temperature range 10—315 K for 05x &2. A phonon-drag eA'ect is identified around 25 K. Itscontribution to the TEP has been analyzed as a function of x. An estimate of the electron-phononrelaxation time 7 ph is derived from this analysis. This estimate is found to be in accordance withthe value of ~, „h obtained from theoretical predictions.

I. INTRODUCTION

In a previous work, ' thermoelectric power (TEP) mea-surements of TiSez S mixed crystals were reported.The observed dip around 150 K in the alloys where thecharge-density-wave phase transition is present was at-tributed to a phonon-drag effect due to a strong I -L inter-pocket electron-phonon interaction.

In the present study, additional measurements of theTEP of these materials are reported. A larger range of xvalues is investigated with an improved accuracy in theexperimental measurements. Moreover, a better crystal-growing process is used, leading to more-stoichiometricmaterials. Our analysis is mainly focused on the low-temperature region below 100 K where we establish theevidence for an intrapocket phonon-drag effect near 25 K.Such a possibility was previously suggested in TiSe2 byDiSalvo et al. , and in V Ti

& „Sez and Zr, Ti& Sez

solid solutions by Gaby et al.

II. EXPERIMENTAL RESULTS

We used iodine-vapor transport to grow homogeneoussingle-crystal samples of TiSe2 „S, the high and lowtemperatures of the gradient being 740 and 640 C, respec-tively. The lattice parameters of the obtained crystals

were checked by x-ray diffraction. In the case of theTiSe2 samples, the characteristic electrical resistivity ratiowas p(160 K)/p(300 K) —3.2. Typically, the dimensionsof our samples were 5 mm long, 1 mm wide, and 0.05mm thick. Electrical as well as thermal contacts were ob-tained with the aid of silver paint. The samples weremounted across the width on gold wires of 0.25 mm di-ameter. The gold wires were in good thermal contactwith two copper blocks. Each of the blocks had awound-wire heater which was used to establish a tempera-ture gradient of less than 1 K along the sample. At eachtemperature, plots of thermoelectric voltage versus thetemperature difference as measured by a differentialChromel —gold —0.07 at. % Fe thermocouple were made.The TEP was deduced from the slope of such plots, henceeliminating any contribution from stray voltages. Simul-taneously, resistivity p and Hall coe%cient RH measure-ments were also carried out. The whole experimentalprocess was computer controlled.

Figure 1 shows the measured electrical resistivities ofseveral TiSe2 S samples. The large resistivity anomalyobserved in pure TiSe2 decreases gradually, as well as thetransition temperature, when the sulfur concentration isincreased. This resistivity behavior confirms the previous-ly reported measurements of DiSalvo et al. , and is quali-tatively similar to the one observed in Ta Ti& Se2,

36 4249 1987 The American Physical Society

&25O 36

~(T)9(3ppK)

J.-M. Z.OP -CASTILE O

invarian t magnitude ana~ound ]50 K

' e and position

strong ev dransforminin al] tr

n of the di

i ence theg crystals s

ip observed

Second thprevious find'

support with

e good ual'n ings of Lak

t 'f'u'TSc t at the TE

Fi 11 hlf d 11

served in TiSeua y reduces

b 1 100 K In thisth

5

n t e TEPn t P ata in this

p0

s

50s

Ipos

I 50s

200 250 300s

350T(K)

FIG. 1. Ternernperature de par to the c

e electri

urve 7, 0.8

S(IQ V K )

0—0+

-oo 4-5 —o xo

x00

00—IQ

—15

~oo0+ 0

o +Oo~ 0oo + (&

p 0x&O&0go 0+0

o0x+O

ooo

0000

00 0oooo 0

o O+

oo

0 oo o

p

p 0, + o+ o

QO xxx

x"

X=O

X=0.I

X =0.2

X = 0,3

X =Q.4

Hf~ Ti& „Sez, anin c ese latter solid ol tsolutions where the

0.7.

va ues, inserved for x a

In Fig. 2, Hall

se d x as large as

measurementses with x=0, 0.2 . .. 0 '

e

reases with theesi ual electron

becomes les d d t dI 1 R

temperatP mea

res.constant

easurements o a e samg1 k hs s ould be mad .e made. First, the

-200

0

—IQ

-200

0

I s s s s

I 00 I 50s s s I s s s s I s I 1 I l s 1 1

200 250s I

300 350T(K)

I s «s I s s s s I s s s

50 IOO l5Q

s s s s I s s s s I

200 250I s, I

300 350T(K)

(c)

4+p +0

p +0

o +0 + ++++ X=Q.5

0++

o$ go+ +O OOOo oo oooooX=

0 0.6

Ooo 0 op oooo X =Q.7

RH(IO m C-7 3

)

S( IO V 0000

00

—0.5—

—I.Q—

5 I s

0 50 IQQ

s s s s I s I s

I50 200 250s s s s I s I s s I

300 350T(K)

FIG. 2. TTemperature dS single crystals.

l coefficient RH

0 xx X 0

x)p„o x0xgo "X0

00

ooo 00 00~

s s s s s s s I s

0s s I s s

IOO l50s s s s I s s s s I s s I

300

—IO

X =1.0—15X =0.8X =0.7

—200 50 350

FIG. 3. ectri

T(K)

Therm oelecforming [l )a and (b)

ectric power S p ratvs temperat

ns. ote that(

tween th e two cases.h 1 =0 i s t e boundary be-

36 PHONON-DRAG EFFECT IN TiSe2 S„MIXED COMPOUNDS 4251

S(~O V K )

+++++++

+

lOI 1 l

T(K)

+

S(IO V K

—10

—l5

oooo

oooo

00

00000

000000

0 00

0 0o0 0o

oooo

FIG. 4. Low-temperature TEP measurements of TiSeq downto 1.8 K, showing the linear drift-diffusion regime. The line sim-

ply serves as an eye guide.

temperature region in order to identify the origin of thisplateau. As the linear behavior of the TEP data below 20K does not extrapolate to zero, we undertook carefulmeasurements down to 1.8 K in TiSez using the standardtwo-Aux method. The results, presented in Fig. 4, clearlyshow how the TEP recovers its 0-K limit with a shoulderaround 3.5 K.

III. ANALYSIS AND DISCUSSION

As seen in Fig. 4, the TEP slope increases rapidly withT above 3.5 K. If the TEP were only controlled by drift-diffusion mechanisms, one would expect the opposite be-havior. At low temperatures, the electron relaxation timeis controlled by impurities (either ionized or neutral) andthe TEP slope would be higher (by a factor of 3 or —,', re-

spectively) than that related to the electron-phonon in-teraction that prevails at higher temperatures. Our resultsthus confirm that the TEP of TiSe2 at low temperatures isnot only controlled by drift-diffusion mechanisms but alsoby a significant phonon-drag effect which is responsiblefor the plateau observed below 100 K. Consequently, thetotal TEP can be written as S =Sd +Sg, where Sd and S~are the drift-diffusion and the phonon-drag contributions,respectively.

To determine S~, the knowledge of Sd is required.Since Sd of TiSe2 is observed to be linear with tempera-ture below 3 K (see Fig. 4) and expected to remain so un-til around 100 K (our estimate of the Fermi energy EF isabout 900 K, as shown below), we simply extrapolatelinearly our low-temperature data to the plateau high-temperature limit. This is shown in Fig. 5. It is interest-ing to observe that the measured TEP around 100 K coin-cides with the extrapolated Sd values, thus identifying thephonon-drag contribution. In order to also obtain anevaluation of Sd for the mixed compounds, we generalizethis approach by plotting the tangent to the TEP curvemeasured around 100 K which passes through the origin.Figure 6 shows the deduced S~-versus -T values for vari-ous x. As can be seen, a minimum of Sg is clearly presentaround 25 K for small x values. The magnitude of thisminimum decreases gradually with increasing x. We alsonote that Sg is strongly dependent on the sample quality

as illustrated by two measured TiSe2 compounds. Non-stoichiometry acts similarly to sulfur addition in reducingthe absolute Sz value at a given T. The disappearance ofS~ occurs when x approaches 0.7. It is remarkable tonote that this value corresponds to the same sulfur con-centration that destroys both the phase transition and thelarge TEP dip around 150 K.

Typically, S~ is given by

1/r, phSg ——So 1~r - h+ 1~r h- h+ 1~rd- h

where So~n, ', n, is the electron concentration, and

ph rph ph and rd ph represent the relaxation times ofphonons interacting with electrons, other phonons, anddefects, respectively. S~ tends to zero at both low andhigh temperatures and reaches its maximum at an inter-mediate value of T where the electron-phonon interac-tion becomes dominant with respect to the other phononinteractions. Our above evaluations of S~ all show this

0

Sg(IO V K )X=OX=OX=O

X=O

X=0

X=0X=O

I I I I I I I I I I ~ ~ I I I I I I j I I I I I I I I I I

IO 20 30 40 50 60

FIG. 6. Plot of Sg versus T for various transformingTiSe2 „S„compounds, including two TiSe2 samples of diFerentstoichiometry.

2Q I I I I I I I I I I I I I I I I I i I I I I i I I i I I I I I I I I

0 50 IOO I50 200 250 300 350T(K)

FIG. 5. Linear extrapolation of our low-temperature Sd data,showing its coincidence with the measured TEP around 100 K.

4252 J.-M. LOPEZ-CASTILLO et al. 36

2n„a co

4~U,'(2)

behavior qualitatively but the latter becomes less pro-nounced as x is increased. However, a quantitativetreatment of S~ as a function of x is difficult because ofthe simultaneous dependences of So 7 d ph and 7 ph onx. The inAuence of So is readily seen. This term variesas the inverse of n„ that is, proportionally to R&.Indeed, the reduction of S~ is consistent with the ob-served evolution of RH when x increases. On the otherhand, in the temperature range where S~ is the strong-est, rd ~g is mainly limited by mass defects (bM/M). '

For small x values, ~d ph can be written as'

The phonon-phonon relaxation time is not expected tovary strongly with x. Moreover, as r, ph((~pQ pQ around25 K, ' we ignore the variation of ~, ph/mph ph with x inEq. (1) which can be rewritten for low x values in the fol-lowing form:

Sp(x) = 2 +Cr, ~q(x)x,Sg x (7)

where 3 =1+(r, „z/r„z ~b) is close to unity. Consideringthe proportionality of So with R& and the dependence of7 ph on x through n, and RH, it is possible to evaluateCr, „&(0) for TiSez. For this purpose, we use the experi-mental RH and the estimated Sg for the two concentra-tions x =0 and x =0.2. The result is

where nd ——x/3 is the ratio of the number of point defectsto the number of lattice sites, a'= V/3 is the volume oc-cupied by a defect in the lattice, V is the unit-cell volume,U, is the sound velocity, and m is the acoustic-phonon fre-quency. Considering that the substitution of selenium bysulfur introduces negligible variations in the lattice param-eters" and the sound velocities, '

~d ph dePends on x onlythrough nd, so that

+e-ph

2 22c 2D

cx kFJd

(4)

where c is the interlayer distance, d is the mass density, Dis the electron-phonon coupling constant, rnid is the in-plane electron effective mass, kF~ is the transverse Fermiwave number, and a=kF~~/kF~ ——(m~ /m~)' is the anisot-ropy ratio. The phonon-drag contribution reaches itsmaximum value around the temperature'

T = = u, (2kFi) .B B

(5)

Since

kFj—1/3

3w(6)

we see from Eq. (4) that r, ~z varies as n, while we ex-pect from Eq. (5) that T ~n, ' As n, inc. reases with x,the inhuence of 1/~, ph on Sg thus counteracts those of Soand 7d ph This could exPlain why a significant Phonon-drag effect is still observed for values of x as large as 0.5or 0.6. The variation of T with x cannot be resolvedfrom our experimental data. This can be due to our eval-uation procedure of Sg or, when too many defects arepresent, to the shortcoming of Eq. (5).

+d-ph

where C is a constant which can be deduced from Eq. (2).Thus, 1/~d ph acts similarly to So to reduce Sg as x in-creases. Turning our attention to ~„ph, we assume thatTiSe2 around 25 K can be described by a single ellipsoidalelectronic Fermi pocket, ' a two-dimensional phonon sys-tem, ' and an intrapocket electron-phonon interaction.Under these conditions, an estimate of ~, ph can be givenb 15

1 =4.2& 10 sr, pg(0)

Evaluating C from Eqs. (2) and (3) for co=co, we getC =2.4 & 10 s '. Consequently,

Cr, g(0) =6, (9)

which compares well with Eq. (8). However, consideringthe various approximations used in the present work, thisagreement should be regarded as essentially qualitative.

In conclusion, we unambiguously confirm the charac-teristics of the 150-K dip in the TEP of all transformingcrystals of the TiSe2 S family. The evidence of aphonon-drag effect around 25 K, controlled by intrapock-et transitions, is established in TiSe2 and related mixedmaterials with an estimation of the relative magnitude ofthe electron-phonon and mass defect interactions.

ACKNOWLEDGMENTS

We wish to thank Professor K. Motizuki for providingus with distorted-phase band-structure calculations ofTiSe2 prior to publication, and Y. Frongillo for his assidu-ous technical assistance.

Cr, pg(0)=5 .

Alternatively, we can deduce the same productCr, ~„(0) from the well-known physical parameters ofTiSe2, namely, c =6.004)&10 ' m, d =5.2)&10 kg/m,u, =2.15X10 m/s, ' m*=1.7mp (mp is the free-electronmass and m * =m j n is the density-of-states effectivemass) determined from low-temperature specific-heatdata, ' and n, = 5 && 10 m as obtained from Hall-coefficient measurements. Then, taking T =25 K, Eqs.(5) and (6) yield kF&=7 6X10 m ' and a=3.3, andfrom m * we get m z ——0.77m o. It is worth noting herethat TF =(A kF~/2m ~k~ ) =900 K, which justifies our pre-vious assumption regarding the degeneracy of the electrongas in TiSe2 up to 100 K. It should also be noted that kF~and o. agree well with the results of band-structure calcu-lations of Suzuki et al. ' for the electron pocket. An esti-mate of the deformation potential D has been determinedby Caille et al. ' Using our U, and m* values in theircalculation, we find D=0.52 eV. Equation (4) then gives

36 PHONON-DRAG EFFECT IN TiSe2 „S„MIXEDCOMPOUNDS 4253

'A. A. Lakhani, S. Jandl, C. Ayache, and J.-P. Jay-Gerin, Phys.Rev. B 28, 1978 (1983).

2F. J. DiSalvo, D. E. Moncton, and J. V. Waszczak, Phys. Rev.B 14, 4321 (1976).

J. H. Gaby, B. DeLong, F. C. Brown, R. Kirby, and F. Levy,Solid State Commun. 39, 1167 (1981).

A. Amara, Memoire de Maitrise es-Sciences, Universite deSherbrooke, Sherbrooke, 1986;

~F. J. DiSalvo and J. V. Waszczak, Phys. Rev. B 17, 3801(1978).

6I. Taguchi, Solid State Commun. 32, 679 (1979).7R. D. Kirby, R. L. Fagerquist, and W. N. Nieveen, Solid State

Commun. 51, 131 (1984).J.-M. Lopez-Castillo, These de 3e cycle, Universite Scientifique

et Medicale de Grenoble, Grenoble, 1984.See, for example, C. Herring, Phys. Rev. 96, 1163 (1954); J. M.

Ziman, Principles of the Theory of Solids, 2nd ed. (CambridgeUniversity Press, Cambridge, England, 1972).R. Berman, Thermal Conduction in Solids (Clarendon, Oxford,1976).

H. P. B. Rimmington and A. A. Balchin, J. Cryst. Growth 21,

171 (1974).' A. Caille, Y. Lepine, M. H. Jericho, and A. M. Simpson, Phys.

Rev. B 28, 5454 (1983).~3N. Suzuki, A. Yamamoto, and K. Motizuki, Solid State Com-

mun. 49, 1039 (1984); K. Motizuki and N. Suzuki, inStructural Phase Transitions in Layered Transition MetalCompounds, edited by K. Motizuki (Reidel, Dordrecht, 1986),p. 1; K. Motizuki, private communication quoted in Ref. 8.

4N. Wakabayashi, H. G. Smith, K. C. Woo, and F. C. Brown,Solid State Commun. 28, 923 (1978).

'5J.-P. Jay-Gerin and R. Maynard, J. Low Temp. Phys. 3, 377(1970).

Comparison of the thermal conductivities of the semicon-ducting compound SnSe2 and of TiSe2 would give

0. 1r~q ~q [C. Ayache (unpublished)].The sound velocity used here is calculated from the data ofRef. 14 and is consistent with kF~~ that constrains the electronpocket within the Brillouin zone.R. A. Craven, F. J. DiSalvo, and F. S. L. Hsu, Solid StateCommun. 25, 39 (1978).