The Hall effect in the organic conductor TTF–TCNQ: choice of geometry for accurate measurements of a highly anisotropic system

As the first realization of an organic metal, the quasi-one-dimensional molecular crystal TTF–TCNQ (tetrathiafulvalene–tetracyanoquinodimethane) has been studied thoroughly for more than 30 years [1–4]. The planar molecules of TTF–TCNQ form segregated stacks in a plane-to-plane manner and the molecular π-orbitals interact preferably along the stacking direction (crystallographic b direction of the monoclinic structure), leaving only weak interactions in the perpendicular a and c crystallographic directions. Due to the formation of linear molecule stacks in the crystal structure and an electronic charge transfer from cationic (TTF) to anionic (TCNQ) complexes, both types of stack are metallic: the TTF column is hole-conducting whereas the TCNQ column is electron-conducting. As a consequence, TTF–TCNQ displays strongly anisotropic metallic conductivity with σ_b > σ_c > σ_a in a wide temperature range down to about 60 K, below which a cascade of phase transitions starts and progressively destroys the metallic character.

Renewed interest in TTF–TCNQ started recently as it was the first material for which angle-resolved photoemission spectroscopy (ARPES) positively identified 1D single-band Hubbard model spectral features [5, 6]. Based on these findings the spectral behaviour of TTF–TCNQ was interpreted as evidence for spin–charge separation, signalling a breakdown of the Fermi liquid quasi-particle picture and leading to the appearance of a new state—a one-dimensional quantum many-body system known as a Luttinger liquid. However, some quantitative values for different parameters are not yet fully available [7–9].

It is worth pointing out that although the electrical properties of TTF–TCNQ have been studied intensively, investigations of one basic magnetotransport property, the Hall effect, have not been unambiguously completed. The only published Hall effect measurements for TTF–TCNQ date back to 1977. They were performed in the high temperature metallic region: dc Hall effect measurements in the metallic phase above the phase transitions (for j || b, B || a) [10], and microwave measurements of the Hall mobility at room temperature (for j || b, B || c) [11]. However, the results obtained vary in their sign and values. The average value of the Hall coefficient, obtained at room temperature from the dc Hall effect measurements, is negative and approximately consistent with estimates of the electron density, implying that the Hall effect and conductivity are dominated by the TCNQ chains. On the other hand, the positive Hall mobility at room temperature, obtained from microwave Hall effect measurements, was interpreted as an indication that the carriers in the TCNQ chain relax more rapidly. A direct comparison between those results could not be done, not only because the quality of the samples used was most probably not the same, but also because the magnetic field orientations were different in the two experiments. Measurements in the same geometry (to compare the data and to resolve the question of the importance of the geometry conditions) were not performed. Our aim has been to perform a new and systematic Hall effect study on TTF–TCNQ  extending measurements to higher magnetic fields and lower temperatures (i.e. below the phase transitions) and using different geometries to investigate the possibility of an influence of the geometry on the value and sign of the Hall coefficient and to try to detect phase transitions, thereby gaining new information.

Measurements of the conductivity and Hall effect were done in the temperature region 30–300 K. The Hall effect was measured in magnetic fields of 5 and 9 T. All the single crystals used came from the same batch. Their typical configuration is shown in figure 1: the b direction is the highest conductivity direction, the a direction, with the lowest conductivity, is perpendicular to b in the a–b plane, and the c direction, with intermediate conductivity, is perpendicular to the a–b plane. Our average room temperature conductivity values for σ_b (400 Ω⁻¹ cm⁻¹), σ_c (5 Ω⁻¹ cm⁻¹) and σ_a (0.5 Ω⁻¹ cm⁻¹) are in good agreement with previously published data [3, 4]. With respect to the samples used in previous measurements, we can state that our recently synthesized samples have much larger dimensions and very good defined geometry, show homogeneous current flow, and are of high quality (concerning high R (300 K)/R (min) ratios).

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The samples were cooled slowly ( ∼ 3 Kh⁻¹) in order to avoid irreversible jumps in resistance that are caused by microcracks well known to appear in all organic conductors. Two (or sometimes three, depending on the size of the crystal) pairs of Hall contacts and one pair of current contacts were made on the sides of the crystal by evaporating gold pads to which the 30 µm gold wires were attached with silver paint. An ac current (10 µA to 1 mA, 22 Hz) was used. For temperatures around and below the phase transitions, a dc technique was also used because of the large resistance increment. Particular care was taken to ensure the temperature stabilization. The Hall voltage was measured at fixed temperatures and in field sweeps from  − B_max to  + B_max in order to eliminate the possible mixing of the magnetoresistance component. At each temperature the Hall voltage was measured for each pair of Hall contacts to test and/or control the homogeneous current distribution through the sample. However, due to the very steep slope of resistivity in the low temperature region, much better and accurate results were obtained in temperature sweeps in fixed magnetic fields ( − B_max and  + B_max). The Hall voltage V_H was determined as [V_xy(B) − V_xy(−B)]/2 and the Hall coefficient R_H was obtained as R_H = (V_H/IB)t (I is the current through the crystal and t is the sample thickness). The linearity of the Hall signal with magnetic field was checked in the whole temperature region investigated.

The Hall effect was measured in several different geometries: (i) j || b, B || a; (ii) j || b, B || c; and (iii) j || a, B || c. Moreover, we took care of the concept of the equivalent isotropic sample (EIS) that is crucial for anisotropic conductors [12]. Generally, for Hall effect measurements it is important to avoid either shorting of the Hall voltage by the current contacts or a non-uniform current distribution. For isotropic materials this requires that the length/width (L/W) ratio obeys the condition L/W ≥ 3 (where the current is passed along L, the Hall voltage develops along W and the magnetic field is along the thickness t of the parallelepiped). For highly anisotropic materials this condition has to be replaced with the more feasible one that takes into account the anisotropy of the conductivity. EIS provides a way of thinking about the current distribution in the anisotropic sample and is realized by a coordinate transformation according to the formulae ℓ'_i = ℓ_i(σ/σ_i)^1/2, where σ = (σ₁σ₂σ₃)^1/3, ℓ_i is the actual sample dimension along the ith principal axis of conductivity σ_i and ℓ'_i is EIS dimension along the ith axis. Following this notation, we consider that an acceptable geometry for highly anisotropic materials is the one where L'/W' = L/W(σ_W/σ_L)^1/2 ≥ 3.

Figure 1 shows typical crystal configurations with corresponding contacts as well as a comparison of the size of the real crystals with EIS for the different geometries used in our Hall effect measurements. The typical dimensions of our TTF–TCNQ crystals were 4.7 × 0.7 × 0.08 mm³ for the b  a and c directions, respectively. Considering first the geometry used in [10] (j || b, B || a—cf figure 1) the EIS coordinate transformation gives the ratio L'/W' = ℓ'_b/ℓ'_c = 6.3, which is the acceptable value. However the anisotropy is temperature dependent; the ratio σ_b/σ_c increases from about 10² at room temperature to almost 10³ at 60 K, and therefore the related L'/W' values also change with temperature. For our sample dimensions this gives at 60 K ℓ'_b/ℓ'_c ≈ 1.8, implying that this geometry shows pronounced deterioration with cooling and as a consequence could yield potentially inaccurate results. Indeed, we have obtained very poor results for this geometry. This was also true for the j || b, B || c geometry (that was used in [11]) which, for our samples, already at room temperature had an unsatisfactory value L'/W' = ℓ'_b/ℓ'_a ≈ 0.2, and got even worse with cooling. These obstacles motivated us to search for yet another geometry that implies a current direction other than j || b and that fulfils the L'/W' ≥ 3 condition for Hall effect measurements. However, the choice of the sample geometry is limited given the sample dimensions and available area for contacts. The only choice turned out to be j || a because it is impossible to position the Hall contacts along the tiny c direction. We have also selected B || c  which ensures a remarkable L/W value. Finally, in order to decrease ℓ_b and consequently ℓ'_b we have cut the sample along the b direction to approximately 1/4 of its original dimension. Thus the final dimensions of one of the measured samples which showed the best Hall effect results were 1.01 × 0.88 × 0.07 mm³ for the b, a and c directions, respectively, with L'/W' = ℓ'_a/ℓ'_b = 24 at room temperature and increasing up to 64 at 60 K (the real crystal and corresponding EIS are also shown in figure 1). Summing up this part, we point out that our analysis indicates that the j || a, B || c geometry should be much more favourable for TTF–TCNQ Hall effect measurements.

Figure 2 shows the temperature dependence of the resistivity along the highest conductivity direction ρ_b(T) and along the lowest conductivity direction ρ_a(T) in the temperature range 30 K < T < 300 K, measured on the samples used for Hall effect measurements. The results show good agreement with the previously published data [3] comprising the values of room temperature resistivity, the metallic behaviour for both directions from room temperature down to about 60 K below which the increase of resistivity indicates a cascade of phase transitions (at 54, 49, 38 K) which destroy the metallic character. Further below the phase transitions the resistivity increases exponentially.

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The magnetic field dependence of the Hall resistance (up to B = 5 T) is shown in figure 3 for T = 34 K and for the j || a, B || c geometry, which we consider to be an optimal one. The data show that the Hall resistivity is linear with magnetic field.

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Figure 4 shows the temperature dependence of the Hall coefficient R_H for 30 K < T < 300 K. The results are shown for several samples and for different geometries: (i) j || b, B || a; (ii) j || b, B || c; and (iii) j || a, B || c. At the increased scale (figure 4, inset) it can be clearly seen that R_H is around zero for T > 150 K. Below this temperature, R_H becomes positive down to the phase transition temperature region, within which it changes its sign to negative values. It should be noted that for the first two geometries we have obtained quite scattered data at high temperatures and the signals were not at all detectable at lower temperatures. However, for the j || a, B || c geometry, where the Hall voltage V_H develops along the b axis, we have obtained very good results for T < 150 K. These measurements were performed on two samples, one of which is presented in figure 4. Although both samples were good quality ones, we show the data for the sample for which the Hall contacts were positioned strictly opposite to each other. This is indeed a textbook requirement for a good Hall effect measurement, but due to the extremely small dimensions for this geometry not at all easy to achieve. The results for another sample confirm the obtained data with positive R_H for phase transitions at T < 150 K and above, and negative R_H below.

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The data for T > 150 K were determined as the average of the several measurements taken at fixed temperature, during the cooling and heating cycles, using several samples and Hall contacts. The error bars show the uncertainty in the determination of small R_H values. The scattering in the data also comes from the fact that during field sweeps from  − B_max to  + B_max at fixed temperature a small temperature drift occurs (a common and well known source of error in this kind of measurement). Figure 4 also contains the data from [10] and [11].

As mentioned above, TTF–TCNQ shows metallic behaviour down to about 60 K and undergoes phase transitions at T_H = 54 K, T_I = 49 K and T_L = 38 K towards an insulating ground state [3, 4]. The phase transition at 54 K is manifested as a drop in the conductivity by a factor of about two and at T_L = 38 K a first-order transition towards an insulating ground state occurs. Between 54 and 38 K, charge density waves (CDWs) successively develop first in the TCNQ and than in the TTF stacks. These transitions have been ascribed to the instability of the one-dimensional electronic gas due to the Peierls mechanism. All CDW phase transitions of TTF–TCNQ investigated in detail by x-ray diffuse scattering experiments [13] and elastic neutron scattering [14] have been recently identified using a low temperature scanning tunnelling microscope (STM) [15] that provides direct experimental proof for the existence of phase-modulated and amplitude-modulated CDWs.

The present Hall effect investigations of TTF–TCNQ single crystals were performed in a broad temperature range 30–300 K. Although the primary goal was to achieve the best possible experimental conditions in order to provide information regarding the variance of the old Hall effect results (concerning the sign and value of the Hall coefficient at high temperatures above phase transitions), even more important was whether the change in the Hall coefficient could be observed in the temperature range of the phase transitions, which would then give experimental proof about the successive development of CDWs first in the TCNQ and then in TTF stacks. In what follows we shall therefore discuss separately the obtained data for R_H(T) (regarding the sign and value) for the high (150 K < T < 300 K) and low (30 K < T < 150 K) temperature ranges.

Taking into consideration the quasi-one-dimensionality of TTF–TCNQ, the standard expression for the low field Hall coefficient for an anisotropic tight binding band should be used, namely R_H = (1/ne)(k_Fb/tank_Fb), where b is lattice constant along the b axis. As a partial charge is transferred from TTF to TCNQ stacks and the charge density ρ potentially available for transport is determined by the value of k_F at which the bonding TCNQ band crosses the antibonding TTF band, leading to 2k_F = ρπ/b in the 1D band picture, we obtain k_Fb = 0.29π (taking a band filling of 0.58 el/TCNQ) [13]. The calculated Hall coefficient value is then R_H =± 3.2 × 10⁻³ cm³ C⁻¹, represented by the dashed line in figure 4 (' − ' for electrons only, and ' + ' for holes only). The old data from [10] are not far from this calculated value for electrons only. However, our results for all configurations used and for T > 150 K are around R_H = 0 (the tiny inclination towards negative values above 250 K is probably irrelevant), indicating that the Hall coefficient results do not depend on the geometry conditions during measurements. If our data are compared with those from [10] (where a negative Hall coefficient was obtained) and [11] (where a positive Hall coefficient was obtained for another geometry), apparently the difference could not be solely attributed to the geometry configuration used for measurements. It would be correct to note that the possible cause could also be differing sample quality, which is now difficult to confirm but cannot be excluded. Namely, the thermoelectric power (TEP) results (from the same period as [10, 11]) have shown that at high temperatures the TEP is negative and approximately linear in temperature, indicating the metallic state, while at T < 54 K positive and negative signs were found in samples from different sources [16, 17]. On the other hand, our results are quite consistent with the measurements of ¹³ C NMR in TTF–TCNQ (obtained more than 20 years ago as well [18]) that somehow contradicted the expected 'narrower' TTF band due to the previous suggestions that the dominant carriers are electrons located on TCNQ stacks [10]. To conclude this part, we state that our Hall coefficient results above 150 K show R_H ≈ 0 for all geometries studied, and do not confirm dominance of either electrons or holes in transport properties [19]. In other words, they show the possibility that electron-like (TCNQ stacks) and hole-like (TTF stacks) contributions effectively cancel each other, giving R_H ≈ 0, which indicates that both chains equally contribute to the conductivity.

The temperature dependence of the Hall coefficient in the 30 K < T < 150 K temperature range for our best sample and for the j || a, B || c geometry is presented in figure 5. As discussed previously, the other two geometries (j || b, B || a and j || b, B || c) were not able to give reliable data due to the geometric limitations for anisotropic samples. For 100 K < T < 150 K, R_H is small and positive indicating that the TTF stack (holes) contributes more to the conductivity than the TCNQ stack (electrons). Here, we would like to recall the ¹³C NMR measurements [18], which have shown that the density of states of TCNQ stacks starts to decrease appreciably below about 150 K: this may explain the fact that R_H was not detectable above 150 K indicating the equal contribution of TTF and TCNQ stacks in the transport, while below 150 K a small positive R_H may be correlated to the decrease of the density of states on TCNQ stacks. We do not give special significance to the obtained R_H values regarding the accurate determination of the number of electrons and/or holes that participate in transport properties, as there is substantial experimental evidence that Coulomb interaction plays an essential role in the electronic structure of TTF–TCNQ [4, 7–9] and a purely band theoretical description may be inadequate. The smooth increase of positive R_H below 100 K down to T_H ≈ 54 K—which is clearly seen in the inset of figure 5—can also be ascribed to a pre-transitional behaviour. However, as the temperature is lowered in the phase transition range, our Hall data show features which have not been observed up to now: all three phase transitions of TTF–TCNQ are identified from the Hall effect measurements, as indicated in figure 5. Around T_H = 54 K there is a maximum in positive R_H(T) that decreases with further cooling and changes its sign around T_I = 49 K. The strong temperature dependence of R_H(T) (a pronounced upturn) close to the phase transition at T_L = 38 K marks the first-order phase transition (where, as known, the period in the a direction jumps discontinuously to 4a [4]) towards an insulating ground state. Here it is worth pointing out that such a result for R_H(T) was not foreseen. Namely, the phase transition at 54 K is driven by the CDW Peierls instability in the TCNQ chains, that is manifested in a drop of the conductivity by a factor of about two [4]. We expected that in the temperature region T_L < T < T_H the dominance of the TTF stacks (i.e. the contribution of the hole-like carriers to the transport) in Hall coefficient measurements may be found. The gradual decrease of positive R_H below T_H (where a reduction of the number of electron-like carriers is expected) and the observed change of sign around T_I suggest that both kinds of carriers contribute to the conductivity in this transitional region. For the Hall coefficient in a two-band model the resultant value and the sign are determined by the carrier concentrations as well as their mobilities, which both have strong and probably different temperature dependences in this temperature region. R_H(T) then comes from a balance of the hole and electron terms [20]. These facts are even more pronounced with the well-marked upturn close to the phase transition at T_L = 38 K. This kind of behaviour is a known feature in semiconductors [19]. As the Hall fields created by electrons and holes are opposing each other, the galvanomagnetic effects can have unusually strong temperature variations in regions where the resultant Hall field is nearly zero and where the relative electron–hole population is temperature dependent. Below phase transitions, for T < 38 K, R_H(T) is negative with the value that rapidly increases with further temperature decrease following the activated behaviour that corresponds to that of resistivity.

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In summary, we have reported measurements at ambient pressure of the Hall coefficient R_H(T) for the quasi-one-dimensional organic conductor TTF–TCNQ at 30 K < T < 300 K and in magnetic fields up to 9 T. We have applied the equivalent isotropic sample approach, suggested a method for choosing the best geometry for Hall effect measurements of highly anisotropic samples, and proposed that for the TTF–TCNQ system the j || a, B || c configuration is the best choice.

Above 150 K, we could not confirm the dominance of either electrons or holes in the transport properties and we suggest the possibility that both chains, electron-like (TCNQ stacks) and hole-like (TTF stacks), make an equal contribution to the conductivity giving R_H(T) ≈ 0. Also, based on our data for other measurement geometries, we indicate that the differences in geometry are not important regarding the sign and/or value of the Hall coefficient in this temperature range. For T < 150 K our results show a small and positive R_H(T) that may indicate the dominance of holes in this region, and an increase below 100 K down to phase transition at T_H = 54 K that could be ascribed to a pre-transitional behaviour. For the temperature region 30 K < T < 54 K we have shown that all three phase transitions of TTF–TCNQ can be identified from the Hall effect measurements: (i) around T_H = 54 K there is a maximum positive R_H(T); (ii) around T_I = 49 K, R_H(T) changes sign, becoming negative for lower temperatures; (iii) around T_L = 38 K, a pronounced upturn in R_H(T) marks the first-order phase transition towards the insulating ground state for T < T_L that is characterized by activated R_H(T) with the activation energy corresponding to that of resistivity.

The work was supported by the Croatian Ministry of Science, Education and Sports grants 119-1191458-1023 and 035-0000000-2836.