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R E V I E W C O P Y N O T F O R D I S T R I B U T I O N
Optical phonon drag and variable range hopping mechanisms of thermoelectric powergeneration in charge density wave system o-TaS
3
D. Stareˇsini´c,
∗
M. Oˇcko, K. Biljakovi´c, and D. Dominko
Institute of physics, Bijeniˇcka cesta 46, P.O.B. 304, HR-10001 Zagreb, Croatia
(Dated: January 15, 2014)We have measured the thermoelectric power of charge density wave system
o
-TaS
3
in a widetemperature range between 10 K and 300 K. All features characteristic for charge density wavesystems have been observed; strong increase below charge density transition temperature
T
P
, peakat
T
M
∼
1/2-1/3
T
P
and change of sign at lower temperatures. These features can be explained bytwo contributions; free carriers drag by the optical phonons with frequencies close to the densitywave gap value in the peak region and above, and hopping of localized carriers of opposite charge tofree carriers at low temperatures. We argue that these are common mechanisms of thermoelectricpower generation in all charge density wave systems.
PACS numbers: 71.45.Lr, 72.20.Pa, 72.10.Di, 72.20.Ee
Charge density wave (CDW) ground state is the bestknown case of electronic crystals emerging from stronginterplay of charge and lattice degrees of freedom incorrelated low-dimensional electronic system. For manydecades they present a host of unusual phenomena andthe ideas that explain these phenomena span much of thecontemporary condensed matter physics [1–3]. Amongexceptional properties of CDW systems are giant dielec-tric constant, nonlinear transport, pulse-duration mem-ory eﬀects, unusual electro-mechanical and thermoelec-tric properties, all of conceptual importance in under-standing systems with various collective ground states.New technologies require use of materials at smaller andsmaller scales, which brings challenges not only to themanufacturing, but also to the understanding of theirproperties which could change drastically by loweringsample size. Therefore the CDW phenomenology be-comes especially attractive as it srcinates from inher-ent low dimensionality of CDW systems and concomitantvery strong electron-phonon (e-ph) coupling.The strong increase of the thermoelectric power (TEP)below the transition temperature
T
P
in CDW systems[4–18], with maximum values entering into the rangeof mV/K, is one peculiarity more which remains un-explained. It was only colloquially referred to as thepossible phonon drag contribution [5, 7, 16, 18]. Thesoftening (Kohn) phonons involved in CDW formationare expected to drag the free carriers [16], however, itwas clearly demonstrated that, once condensed in CDWstate, these phonons do not carry heat [10] and cannotcontribute to TEP. As phonon drag may produce a con-siderable contribution to TEP in general, the deeper un-derstanding how it works in systems with strong e-phinteraction which ﬁnally causes various collective statesis of great importance.Phonon drag was deﬁnitely shown to be important inthe TEP of high-temperature superconductors (HTSC).An anomalous, positive component to TEP, observed inYBaCuO for
T <
160 K, was attributed to freeze-out of carrier-phonon Umklapp processes involving holes in theCuO
2
planes and optical-mode phonons [19]. An uni-versal dependence of the TEP on hole concentration of HTSC cuprates was also found [20]. The direct phonon-drag TEP correlation to
T
c
in HTSC was seen as an in-crease of the phonon-dragcontribution with
T
c
consistentwith strong-coupling theory [21]. Finally, TEP of ZrB12superconductor has been explained by the contributionfrom the optical phonon associated with the vibrationsof the weakly bound Zr atoms in boron cages [22].In this Letter we argue that thermoelectric propertiesof CDW systems show generic temperature dependencefeaturing the
optical phonon drag
peak at temperature
T
M
< T
P
/
2 and change of sign on further cooling indi-cating a new channel of
hopping conductivity
. Our con-sistent interpretation of the integral temperature behav-ior of TEP in CDW systems is demonstrated on thor-oughly investigated system
o
-TaS
3
[7–13] with new mea-surements in a wide temperature range.We used the standard diﬀerential method to investi-gate several samples of 6 mm long ﬁbers of 10-20
µ
m di-ameter synthesized by H. Berger at Ecole PolytechniqueFdrale in Lausanne. Measured values of TEP are morethan two times larger than the values reported in earlierinvestigations indicating higher quality of our samples.Fig. 1 shows the temperature dependence of TEP
α
and resistivity
ρ
of
o
-TaS
3
. At high temperatures TEP ispositive, consistent with the predominantly hole-like car-riers [23]. Typical features of TEP in CDW systems areobserved, which include strong increase below
T
P
=217 Kpeaked at
T
M
∼
90 K (
T
P
/2-
T
P
/3) and change of sign at
T
∗
∼
60 K with almost equally deep minimum at
T
m
∼
30K. As the sample resistance exceeds GΩ, measurementsbelow 20 K were unstable, so there is uncertainty in theobserved decrease of the absolute value of TEP and fur-ther change of sign.We consider ﬁrst the high temperature range with posi-tive TEP. Increase of TEP below
T
P
is usually attributedto the semiconducting nature of CDW. The carrier diﬀu-
2
FIG. 1. (color online) Thermoelectric power
α
and resistivity
ρ
of
o
-TaS
3
measured along the conducting chains in cool-ing (full symbols) and heating regime (empty symbols). Thedashed red line represents the diﬀusion contribution to TEPobtained from Eq. (1) and the full green line represents thecombined diﬀusion contribution and phonon-drag peak ob-tained from Eq. (2).
sion TEP in semiconductors [24] cannot exceed the value
α
diﬀ
=
k
B
e
·
∆
k
B
T
+
α
0
(1)where ∆ is the eﬀective activation energy for the con-ductivity and
α
0
is a constant of the order of
k
B
/e
≈
86mV/K, with
k
B
the Boltzmann constant and
e
elemen-tary charge. With ∆(
T
) appropriately estimated from
ρ
(
T
) data [25] and
α
0
≈
-1.5
·
k
B
/e
set to match the
T > T
P
value, this contribution, represented by dashedred line in Fig. 1, cannot account for the measured in-crease of TEP below
T
P
.We have therefore to consider other heat transportmechanisms, such as phonons, which can drag the car-riers and produce increased TEP. We use a simpliﬁedexpression for the phonon drag TEP
α
pd
which assumesenergy independent scattering rates as following [26, 27]:
α
pd
=
C
ph
n
fc
·
e
Γ
e
−
ph
Γ
e
−
ph
+ Γ
other
,
(2)where
C
ph
is the phonon speciﬁc heat,
n
fc
is the den-sity of free charge carriers and Γ
e
−
ph
and Γ
other
are thescattering rates of phonon subsystem by charge carriersand by other scattering channels respectively.Both acoustic and optical phonons can contribute toTEP [19, 22], as real optical phonons, unlike the idealEinstein modes, are not dispersionless and they can con-tribute signiﬁcantly to the heat transport [28]. However,the contribution of a particular phonon mode to TEPwill be negligible unless scattering by the free carriers isimportant, i.e. Γ
e
−
ph
/
(Γ
e
−
ph
+ Γ
other
)
≡
Γ
rel
≈
1.Raman measurements [29–31] in
o
-TaS
3
detected threegroups of optical phonons near 280 cm
−
1
, 380 cm
−
1
and500 cm
−
1
that are strongly coupled to the free carriers,as their width (Γ) increases several times at 175 K, 160K and 80 K respectively, temperatures where their fre-quency matches the value of the temperature dependenthalf-gap ∆(
T
) [29]. At these temperatures the free car-rier scattering becomes dominant and Γ
e
−
ph
>
Γ
other
.The mechanism of coupling with electrons was not elu-cidated, however the bolometric measurements indicatedstrong coupling of free carriers with these modes [32],presumably due to the large transverse dipole moment of conduction band states.At suﬃciently high temperatures, where
C
ph
and Γ
rel
are nearly constant,
α
pd
will be inversely proportionalto the free carrier density, and therefore roughly propor-tional to
ρ
.
α
pd
(
T
) is then expected to follow the complextemperature dependence of
ρ
(
T
) around
T
P
and increaseat lower temperatures as evidenced in Fig. 1.On the other hand, at suﬃciently low temperatures
C
ph
of the optical phonon will decrease exponentiallyand, if its frequency is higher than ∆,
α
pd
will decrease asthe temperature is decreased, leading to the maximum inTEP. Experimentally observed Γ
e
−
ph
decreases as well,reducing Γ
rel
and the phonon drag contribution to TEPeven further.In order to model the phonon dragcontribution accord-ing to Eq. (2) we assume that Γ
rel
is constant and close to1, which would be acceptable approximation for the ex-perimentally observed Raman mode near 500cm
−
1
above80 K. Based on the simpliﬁed expression for the free car-rier conductivity
σ
=
n
fc
eµ
, where
µ
represents mobility,
n
fc
(
T
) is estimated as
n
fc
(
T
) =
n
0
·
σ
(
T
)
/σ
(300
K
) fromthe measured temperature dependence of the conductiv-ity
σ
(
T
) = 1
/ρ
(
T
), assuming that
µ
varies slowly withtemperature.
n
0
≈
10
22
e/cm
3
is the free carrier densityin the high temperature metallic state [33].
C
ph
(
T
) is estimated from the Einstein formula for theheat capacity which is a fair approximation for opticalphonons of frequency
ω
E
. As the unit cell of
o
-TaS
3
consists of 24 chains (or formula units) and the mainoptical modes come from the prismatic structure of singleTaS
3
chain, the basic degeneracy of the single mode is 24[30].We have varied only the optical phonon frequency
ω
E
and obtained quite satisfactory ﬁt for the peak in TEPwith
α
pd
+
α
diﬀ
, as shown in Fig. 1 (solid green line),for
ω
E
≈
480 cm
−
1
. This frequency is close to the 500cm
−
1
optical phonons for which a strong increase of theline width has been observed at 80 K [30], i.e. below theposition of the peak in TEP. The corresponding energy¯
hω
E
/k
B
=690 K is also close to the eﬀective activationenergy ∆
≈
750 K for the conductivity. This justiﬁessome of the most important assumptions we have madein the calculation.We consider now the low temperature regime below
3
FIG. 2. (color online) a) electrical conductivity of
o
-TaS
3
measured (symbols) and decomposed in two contributionsfrom free (red dashed line) and hopping (blue dash-dottedline) carriers as described in the text. The green solid line isthe ﬁt to Eq. (4. b) TEP of
o
-TaS
3
measured (symbols) andestimated from Eq. (3) (green solid line) with partial con-tributions (Eq. (5)) from the free carriers (red dashed line),including carrier diﬀusion and phonon drag, and hopping car-riers (blue dash-dotted line).
the maximum of TEP, which is dominated by the changeof sign at
T
∗
. Apparently, it should be attributed to thenew transport channel with the opposite carrier charge.This low temperature behavior can be understood inthe frame of new generic phase diagram of DW systemsin general (including also SDW ground state) with new”density wave glass” ground state established below aﬁnite temperature
T
g
∼
T
P
/
4
−
T
P
/
5 [34, 35]. Belowthis temperature the free carrier contribution to the con-ductivity continues to decrease with the same activationenergy [23, 36], but another channel with lower appar-ent activation energy
E
a
= 200
−
400 K (
∼
T
P
) becomesdominant in electrical transport [37], as demonstrated inFig. 2a.Several microscopic models of this low temperatureCDW state predict localized carriers of opposite charge[38, 39] (i.e. the electrons in the case of
o
-TaS
3
) whichcould explain the observed change of sign in TEP. More-over, the hopping mechanism of the corresponding con-ductivity can lead to the temperature independent TEP,as shown in [40] for the case of variable range hopping(VRH) in 1D Mott and Coulomb gap (Efros-Shklovskii- ES) models, which is a reasonable approximation forthe nearly constant negative TEP of -20
k
B
/e
observedbelow 50 K.With two transport channels the overall TEP can beobtained from
α
th
=
α
1
σ
1
+
α
2
σ
2
σ
1
+
σ
2
(3)where
α
1
and
σ
1
represent the free carrier contributionto TEP and conductivity, and
α
2
and
σ
2
the hoppingcontribution.Taking into account that
σ
1
shoud decrease in the acti-vated manner in the entire temperature range below
T
P
[23, 36],
σ
2
can be obtained numerically. However, inorder to simplify further calculations we decompose thetemperature dependence of conductivity
σ
(
T
) below 150K as follows:
σ
th
(
T
) =
σ
1
+
σ
2
=
σ
01
e
−
∆
kBT
+
σ
02
1 +
be
(
T
0
T
)
12
(4)represented by green line in Fig. 2a. Above 150 K
σ
1
is taken directly from the experiment. Resulting free car-rier and hopping carrier conductivities are represented bydashed red (
σ
1
) and dash-dotted blue (
σ
2
) lines respec-tively in Fig. 2a.The value of ∆=750 K is typical for
o
-TaS
3
. Theanalytical expression for
σ
2
conforms to the 1D Mottor the ES VRH model below 50 K. At higher tempera-tures it mimics the numerical VRH results of [41] where aconstant conductivity is obtained above suﬃciently hightemperature. The parameter
b
= 2
.
2
·
10
−
8
determinesthe crossover temperature of 60 K. VRH model has notbeen previously applied to the in-chain conductivity of
o
-TaS
3
above 20 K, however, the ﬁt is reasonable and thevalue of
T
0
= 18
.
4
·
10
3
K, is very close to
T
0
obtainedfor VRH transversal conductivity of
o
-TaS
3
[42].The free carrier contribution to TEP
α
1
includes thediﬀusion and phonon drag, however, with ∆(
T
) in Eq.(1) and
n
fc
(
T
) in Eq. (2) estimated from
σ
1
. VRH con-tribution to TEP,
α
2
=20
·
k
B
/e
, is taken directly from ex-perimental results.Two adjustable parameters that we have varied in or-der to obtain a good ﬁt to experimental data are the op-tical phonon frequency
ω
E
and the scattering term Γ
rel
inEq. (2). TEP evaluated for Γ
rel
≈
0.6 and
ω
E
≈
380 cm
−
1
is presented in Fig. 2b (green solid line), together withpartial contributions of free carriers (
α
1p
) and hopping(
α
2p
):
α
1p
=
α
1
σ
1
σ
1
+
σ
2
α
2p
=
α
2
σ
2
σ
1
+
σ
2
(5)The frequency
ω
E
is within the range of the opti-cal phonons observed by Raman spectroscopy [30] to be
4
TABLE I. Comparative values for the CDW gap ∆, CDWtransition temperature
T
P
and the temperature of the max-imum in TEP
T
M
and their ratios for various CDW sys-tems,
o
-TaS
3
(this work), K
0
.
3
MoO
3
[15], TTF-TCNQ [5],HMTTF-TCNQ [6], BDT-TTF [18], KCP [4], (TaSe
4
)
2
I [14],(NbSe
4
)
10
/
3
I[17].∆ (K)
T
P
(K)
T
M
(K) ∆
/T
P
T
P
/T
M
∆
/T
M
o
-TaS
3
750 210 90 3.6 2.3 8.3K
0
.
3
MoO
3
600 175 60 3.4 2.9 10TTF-TCNQ 220 53 20 4.2 2.7 11HMTTF-TCNQ 250 50 25 5 2.0 10BDT-TTF 500 150 40 3.3 3.8 12.5KCP 500 130 50 3.8 2.6 10(TaSe
4
)
2
I 1600 260 130 6.2 2.0 12.3(NbSe
4
)
10
/
3
I 1850 280 190 6.6 1.5 9.7
strongly scattered by the free carriers. For particular 380cm
−
1
phonon the free carrier scattering actually becomesdominant at temperatures higher than
T
M
, which is re-ﬂected in the reduced value of Γ
rel
. However, these quan-titative considerations should not be taken too strictly, aswithin our simple model optical phonon parameters rep-resent the average contribution of all aﬀected phonons.Anomalous increase of TEP below
T
P
and subsequentmaximum are common features of CDW systems. In pro-posed optical phonon drag mechanism of TEP, these fea-tures are related to the phonons with frequencies crossingthe CDW gap energy. In Table 1. we compare the ∆,
T
P
and
T
M
values of several CDW systems for whichTEP has been measured in suﬃciently broad tempera-ture range. In respect to the wide variation of
T
P
and ∆,as well as 2-fold variation of the ∆
/T
P
ratio, the mea-sure of the coupling strength,
T
M
scales reasonably wellwith the value of ∆. It is a good indication that the sameuniversal mechanism with ∆ as the relevant energy scale,such as proposed optical phonon drag, is responsible forTEP in CDW systems below
T
P
.The change of sign in TEP at some temperature
T
∗
below
T
M
is also a common feature of CDW systems.However, measurements presented in literature have notbeen performed at suﬃciently low temperatures to per-mit the comparison with our model of VRH contributionto conductivity and TEP.Our interpretation of TEP of
o
-TaS
3
in a wide temper-ature range relies of two important presumptions, the freecarrier drag by optical phonons and hopping contributionto the conductivity at low temperatures, formulated ac-cording to the experimental results. The resulting ﬁt of TEP, based on simple theories, uses parameters whichcan be independently obtained from experiments. Theconsistency of our approach appeals for further theoreti-cal consideration, particularlyas the increase of the ﬁgureof merit in some systems has been recently attributed tothe CDW instablility [43–45]. It should also address thesrcin of the strong coupling of optical phonons with freecarriers seen in the Raman scattering of CDW systemswhich occurs at the phonon energies close to ∆, whichis only half of the optical gap, and therefore does notcorrespond to the simple creation of electron-hole pairs.In summary, we have successfully described the ther-moelectric power of CDW system
o
-TaS
3
in a wide tem-perature range with two contributions corresponding tothe optical phonon drag of free carriers excited over thegap and the hopping of localized carriers of charge oppo-site to free carriers. Comparison with other CDW sys-tems suggeststhat these arecommon mechanisms of TEPin all of them.The authors thank Pierre Monceau and Ivan Kupˇci´cfor useful discussions and suggestions. This study wassupported by Croatian MSES projects 035-0352827-2841and 035- 0352827-2842.
∗
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