## 1 Introduction

The layered organic conductors, α-(BEDT-TTF)_{2}MHg(SCN)_{4} where M = K, Tl or Rb, have been of high interest since the early 1990s due to their anomalous electronic properties at low temperatures (see e.g. [1–3] and references therein). While the exact nature of some anomalies is still a matter of debate, most of the features are associated with a density-wave instability of the quasi-one-dimensional (Q1D) part of the electronic system and its interaction with the quasi-two-dimensional (Q2D) conducting band. The 2k_{F}-superstructure, observed originally in the angle-dependent magnetoresistance experiments [4], was recently confirmed by X-ray studies [5], thus suggesting a charge-density-wave (CDW) formation in these compounds. Current–voltage characteristics taken on the M = K salt below the transition temperature [6,7] reveal the existence of a threshold electric field whose behaviour can be interpreted in terms of a CDW with a d-wave order parameter [8,9].

The critical temperature of the CDW transition in these compounds, T_{CDW} ≈ 8–10 K, and, therefore, the relevant energy gap are much smaller than it is usually met in CDW systems. One of the important consequences of this fact is a strong influence of a magnetic field on electronic properties. On the other hand, the ground state of these materials turns out to be very sensitive to the hydrostatic [10,11] or uniaxial [12,13] pressure. This gives rise to a rich phase diagram including a number of new field-induced CDW (FICDW) subphases and a superconducting phase coexisting with the CDW. This paper is aimed to overview recent results obtained on the best-studied M = K salt. In the next section we will briefly introduce two basic mechanisms of the magnetic field effect on a CDW system and illustrate how they are manifested in α-(BEDT-TTF)_{2}KHg(SCN)_{4}. In Section 3 two different types of FICDW transitions will be presented. Finally, in Section 4 the superconducting state and its coexistence with the CDW state under quasi-hydrostatic pressure will be considered.

## 2 Pauli and orbital effects of a magnetic field on the CDW state

Generally, the magnetic field affects the CDW system via coupling to the spins and to the orbits of charge carriers.

Since the CDW interaction involves electron–hole pairs from the same spin subband [14,15], an external field **B**, lifting the spin degeneracy, leads to different values of the optimal nesting vector for the subbands with spins parallel and antiparallel to the field:

$${Q}_{\text{opt},x}\left(\mathit{B}\right)={Q}_{x0}\pm 2{\mu}_{\text{B}}\mathit{B}/\u0127{v}_{\text{F}}$$ | (1) |

_{x}

_{0}≈ 2k

_{F}is the nesting vector in the absence of the field, μ

_{B}is the Bohr magneton, v

_{F}is the Fermi velocity, and the sign +(−) stands for the spins parallel (antiparallel) to the applied field. Thus, the CDW nesting conditions become worse in the magnetic field.

This so-called Pauli effect causes a gradual suppression of the CDW state with increasing field [16–19]. The relevant ‘magnetic field–temperature’ (B–T) phase diagram [17–19] is largely similar to that of a spin-Peierls system or a clean superconductor in the absence of the orbital pair-breaking. In particular, it includes the high-field, low-temperature phase CDW_{x} corresponding, respectively, to the soliton lattice [20,21] or spatially modulated Larkin–Ovchinnikov–Fulde–Ferrel [22,23] phases in the latter two cases. It is mainly the Pauli effect which determines the shape of the B–T phase diagram of α-(BEDT-TTF)_{2}MHg(SCN)_{4} at ambient pressure, in magnetic field perpendicular or almost perpendicular to the highly conducting ac-plane [24–26]. An example of the phase diagram of the M = K salt obtained from magnetization measurements [26] is presented in Fig. 1. The experimental data show a good agreement with the theoretically predicted phase lines [18] (lines in Fig. 1). The steeper, in comparison with the theory, slope of the experimentally determined border between the low-field CDW_{0} state and metallic state is caused by the second, orbital effect of the magnetic field.

While the Pauli effect is operative in any CDW system, provided the field is strong enough, the orbital effect becomes important when the Fermi surface nesting is sufficiently imperfect. In the layered conductors, the imperfectness of nesting can be described by a finite second-order transfer integral t_{y}′, corresponding to the effective next-nearest-chain transfer in the plane of conducting layers, in the dispersion law for the charge carriers near the Fermi level [27]:

$$\varepsilon \left(\mathbf{k}\right)=\u0127{v}_{\text{F}}\left(\left|{k}_{x}\right|-{k}_{\text{F}}\right)-2{t}_{y}\phantom{\rule{0.25em}{0ex}}\mathrm{cos}\left({k}_{y}{a}_{y}\right)-2{t}_{y}^{\prime}\phantom{\rule{0.25em}{0ex}}\mathrm{cos}\left(2{k}_{y}{a}_{y}\right)\text{,}$$ | (2) |

_{F}= ħv

_{F}k

_{F}is the Fermi energy of the Q1D electrons, and a

_{y}is the interchain period. In Eq. (2) the dispersion in the least conducting z direction is assumed to be negligibly small which is usually well justified for layered organic conductors [27].

If t_{y}′ is comparable to the characteristic CDW energy t′^{∗} = Δ_{0}/2 (Δ_{0} is the CDW gap at t_{y}′ = 0), the zero-field transition temperature T_{CDW}(0) becomes considerably lower than for a perfectly nested Fermi surface. In this case, a magnetic field applied perpendicular to the layers restricts electron orbits to the chain direction, i.e. enhances their one-dimensionality, and restores the density-wave state [18,28].

In the experiment, the usual parameter for controlling the antinesting term t_{⊥}′ is the external pressure. For α-(BEDT-TTF)_{2}KHg(SCN)_{4} the transition temperature is strongly reduced by applying the quasi-hydrostatic pressure of ∼2 kbar [29,30]. Under this pressure the magnetic field of ∼5–10 T perpendicular to the layers considerably increases T_{CDW} [30], as can be seen from Fig. 2 in which the lines separating the metallic and CDW states are shown for different pressures. This is a direct manifestation of the orbital effect. At ambient pressure T_{CDW} monotonically decreases in the field. However, as mentioned above the corresponding phase line is steeper than predicted for a perfectly nested CDW system. This steeper slope is obviously due to the orbital effect, indicating that the nesting is considerably imperfect already at P = 0.

Obviously, the increase of T_{CDW} due to the orbital effect is limited: in any case the critical temperature value cannot exceed that of the perfectly nested system. Moreover, the orbital effect competes with the suppressing Pauli effect that further reduces T_{CDW}. Indeed, as seen from Fig. 2, the field-induced increase of T_{CDW} at 1.8 and 2.3 kbar saturates by ∼15 T and is followed by a decrease at higher fields.

## 3 Field-induced charge-density-wave (FICDW) transitions

### 3.1 Re-entrant CDW state under high pressure

If the antinesting parameter t_{y}′ exceeds the critical value t′^{∗}, the CDW state is completely suppressed in the absence of a magnetic field. In this case, a series of first-order phase transitions to FICDW subphases are expected at low temperatures [18,31,32]. These FICDW subphases are characterized by quantized (at a fixed field strength) values of the Q_{x} component of the nesting vector [32]:

$${Q}_{x}^{N}={Q}_{x0}\pm \frac{2{\mu}_{\text{B}}\mathit{B}}{\u0127{v}_{\text{F}}}+N\frac{2\pi}{{\lambda}_{x}},\phantom{\rule{0.5em}{0ex}}N=0,\pm 1,\pm 2,\text{\u2026}$$ | (3) |

_{x}is the magnetic length along the x axis:

$${\lambda}_{x}=\frac{\pi \u0127}{e{a}_{y}{\mathit{B}}_{z}}\text{,}$$ | (4) |

_{z}is the magnetic field component perpendicular to the layers. The last term in Eq. (3) originates from the quantization of the orbital motion of the charge carriers under a strong magnetic field. Different subphases are characterized by different quantum numbers N. This is similar to the well known field-induced spin-density wave (FISDW) phenomenon observed on the Bechgaard salts (TMTSF)

_{2}X (see Refs. [27,33] for a review). However, unlike the latter case, the FICDW is influenced by the paramagnetic Pauli effect that is expressed by the second term in the right side of Eq. (3). In particular, this effect leads to a considerably lower coupling constant for the FICDW formation as compared to that of FISDW [32]. Nevertheless, recent experimental data [30,34] provide substantial arguments for the existence of FICDW in α-(BEDT-TTF)

_{2}KHg(SCN)

_{4}under pressure.

Fig. 3 shows the interlayer magnetoresistance of a high-quality sample of α-(BEDT-TTF)_{2}KHg(SCN)_{4} in the field perpendicular to the layers recorded at T = 0.1 K at different pressures [34]. The strong rapid oscillations of the magnetoresistance originate from the Shubnikov–de Haas effect due to the Q2D carriers, which remain metallic in both the normal and CDW states [1–3]. We, however, focus here on slow oscillations observed in the field interval between ∼4 and 10 T under pressure. It is important that these new features emerge right at the critical value P_{c} ≈ 2.5 kbar necessary for the complete suppression of the CDW state in the absence of the field. This pressure range corresponds to the antinesting parameter t_{y}′ slightly exceeding the critical value t′^{∗} that exactly matches the theoretical condition for the FICDW transitions [18]. As discussed in Refs. [34,35], the periodicity of the oscillations in the inverse field scale, F ≅ 20 T, as well as their evolution with changing the pressure and temperature are consistent with the FICDW scenario.

As mentioned above, transitions between subsequent FICDW subphases are characterized by discontinuous changes of the wave vector Q_{x}; thus they are the first-order phase transitions. This is corroborated by the hysteretic structure of the field-dependent magnetoresistance: Fig. 4a shows the magnetoresistance measured under the pressure of 3 kbar, at increasing and decreasing magnetic fields; the difference between the up and down sweeps plotted in Fig. 4b exhibits a clear structure correlated with the slow oscillations. The peaks in the hysteresis are observed at the fields corresponding to the expected border between subsequent FICDW states.

### 3.2 FICDW transitions in the CDW_{x} state

The FICDW transitions presented above are the consequence of the quantizing orbital effect of the magnetic field. In this respect they are very much alike FISDW phenomenon. The necessary condition for these effects is the complete suppression of the zero-field density-wave ground state. As we have shown, the latter can be achieved in α-(BEDT-TTF)_{2}KHg(SCN)_{4} by applying sufficiently high pressure and thus increasing the antinesting parameter t_{y}′ above the critical value. Now we will consider another mechanism for the FICDW transitions which can be realized even at ambient pressure. This mechanism is associated with the superposition of the orbital quantization and the Pauli effect. The new transitions can be observed only in CDW systems since the SDW state is insensitive to the Pauli effect.

Fig. 5 shows the interlayer magnetoresistance (upper panel) and magnetic torque (lower panel) of α-(BEDT-TTF)_{2}KHg(SCN)_{4} measured at zero pressure in a magnetic field strongly tilted with respect to the conducting layers [36]. The orientation is determined by the tilt angle θ as shown in the inset. The anomaly at B_{k} ≈ 12.5 T corresponds to the transition between the low-field CDW_{0} state and the high-field CDW_{x} state. This transition is caused by the suppressing Pauli paramagnetic effect as discussed in Section 2. Additionally, both the magnetoresistance and torque display several anomalies above B_{k}. The hysteretic character of the anomalies suggests that they are associated with multiple first-order phase transitions. The transitions emerge at tilt angles above ≅65°, their positions in field being strongly dependent on θ. As shown in Fig. 6, they rapidly shift down, approaching B_{k} as θ increases towards 90°.

The increasingly high sensitivity of the new transitions to changes in the tilt angle θ near 90° suggests that the orbital effect determined by the field component B_{z} = B cos θ plays a crucial role. On the other hand, it is important that they occur at B > B_{k}, i.e. in the fields producing a very strong Pauli effect. Therefore it is very likely that the new transitions originate from an interplay between the Pauli and orbital effects on the high-field CDW_{x} state.

To explain the origin of the observed transitions, we consider qualitatively the field dependence of the Q_{x} component of the nesting vector of a CDW system with a moderately imperfect nesting [36]. At zero-field, Q_{x}_{0} = 2k_{F} corresponds to the optimal nesting; the CDW energy gap covers the whole Fermi surface. At a finite magnetic field, the optimal nesting conditions modify according to Eq. (1). This is illustrated by dashed lines in Fig. 7. Nevertheless, both spin subbands remain fully gapped and the system as a whole maintains the constant nesting vector Q_{0x} up to the critical field B_{k} ∼ Δ(B = 0)/2μ_{B}. Above B_{k}, Q_{x}_{0} is no longer a good nesting vector as it leads to ungapped states in either subband. In this case the CDW energy can be minimized by introducing a field-dependent nesting vector ${Q}_{x}^{\text{Pauli}}\left(B\right)$ [18] which is schematically represented in Fig. 7 by the thin solid line asymptotically approaching the value Q_{x}_{0} + 2μ_{B}B/ħv_{F}. This obviously improves the nesting conditions for one of the spin subbands (with spins parallel to the field) at the cost of an additional ‘unnesting’ of the other (spin-antiparallel). Now the situation for the second, unnested subband resembles that of the FICDW case considered above; the only difference is that now the nesting is destroyed by the Pauli effect rather than by pressure. Therefore, the orbital quantization condition in the form of Eq. (3) sets on the nesting vector. The corresponding quantized levels ${Q}_{x}^{N}\left(\mathit{B}\right)$ of the nesting vector for the subband with spins antiparallel to the external field are shown by dotted lines in Fig. 7. As a result, the most favorable values of the CDW wave vector above B_{k} are determined by the intersections of the continuous curve ${Q}_{x}^{\text{Pauli}}\left(\mathit{B}\right)$ with the straight lines ${Q}_{x}^{N}\left(\mathit{B}\right)$. Thus, with changing the field we obtain a series of first-order transitions between CDW subphases characterized by different quantized values of the wave vector as schematically shown by thick lines in Fig. 7.

The proposed model qualitatively explains the major features of the behaviour of α-(BEDT-TTF)_{2}KHg(SCN)_{4} at high fields [36]. The multiple FICDW transitions can be observed when the distance 2π/λ_{x} between the quantized levels [see Eqs. (3) and (4)] is smaller than μ_{B}B/ħv_{F}. This condition is not fulfilled for the present compound at the field perpendicular to the layers. With tilting the field, the perpendicular field component B_{z} = B cos θ reduces, thus decreasing the value 2π/λ_{x} ∝ B_{z}, whereas the Pauli effect remains unchanged. Therefore, the transitions emerge starting from some threshold angle θ_{c}. From the experimentally determined angle θ_{c} ≈ 65° one can estimate the upper limit for the Fermi velocity of the Q1D carriers [36], v_{F} ≤ 2μ_{B}/3ea_{y} cos θ_{c} ≈ 9 × 10^{4} m/s; in good agreement with the independently obtained [37] value v_{F} = 6.5 × 10^{4} m/s.

## 4 Superconductivity and its coexistence with the CDW state

Under the quasi-hydrostatic pressure above the critical value, P ≥ P_{c} ≈ 2.5 kbar, the zero-field CDW state in α-(BEDT-TTF)_{2}KHg(SCN)_{4} is completely suppressed and the normal metallic state undergoes the superconducting (SC) transition at low temperatures [38]. The maximum critical temperature T_{c} = (110 ± 5) mK at P = P_{c} and decreases at the rate dT_{c}/dP ≈ 30 mK/kbar with further increase in the pressure. It is interesting to note that both T_{c} and dT_{c}/dP are an order of magnitude lower than those found earlier in uniaxial strain experiments [39]. This dramatic difference is not understood at present and needs further investigation.

When the pressure is decreased below P_{c}, the CDW state emerges; however, the superconductivity does not disappear completely. As shown in Fig. 8, the resistive SC transition shifts down in temperature and broadens at lowering the pressure but even at P = 0 the resistance slightly decreases, indicating a partial transition to the SC state. The latter observation is in agreement with earlier reports by Ito et al. [40,41].

The shape and magnitude of the SC transition below P_{c} are strongly sample dependent. To demonstrate this, examples of the zero-pressure transition on three different samples are presented in the inset in Fig. 8. The resistance of sample 3 is nearly zero at the lowest temperature. This, however, does not mean that the whole sample is in the SC state: d.c. magnetization measurements performed on the same sample have not shown any Meissner signal down to below 10 mK [42]. One can, therefore, conclude that below P_{c} the bulk superconductivity is suppressed and the SC phase exists in the form of minor inclusions into the CDW matrix.

The inhomogeneous nature of the superconductivity below P_{c} is also reflected in the critical field behaviour [35,42]. Fig. 9 shows the temperature dependence of the critical field, for three pressures: P = 2.0, 2.5, and 3.0 kbar. At zero-field, the critical temperature does not vary by more than 10% within this pressure interval. By contrast, the critical field shows a considerable enhancement at lowering the pressure, i.e. at entering the CDW region of the phase diagram. Moreover, the slope of the temperature dependence, linear above P_{c}, exhibits a clear positive curvature at the lowest pressure. This behaviour does not depend on the criterion chosen for the definition of the critical field [42]. Moreover, it seems to be a common feature of the compounds exhibiting a coexistence of the SC and density-wave states. Indeed, very similar effects have been reported for the CDW material NbSe_{3} [43] and for the SDW organic conductor (TMTSF)_{2}PF_{6} [44]. In all these compounds the SC and the density-wave phases are most likely separated in space, so that the superconductivity occurs only in a minor volume fraction.

The presented above results are fully consistent with the generally accepted concept of a competition between the CDW and SC instabilities in low-dimensional conductors [27,45]: the fraction of the SC phase rapidly decreases upon entering the CDW region and T_{c} shifts to lower temperatures. However, it was recently found [42] that the CDW surprisingly has a certain stimulating effect on the superconductivity in α-(BEDT-TTF)_{2}KHg(SCN)_{4}.

Fig. 10 shows, in an enlarged scale, the temperature dependence of the interlayer resistance at P = 2 kbar, at temperatures right above the main transition [42]. One can clearly see that the resistance starts deviating downwards from the normal metallic behaviour considerably above the temperature of the main resistive transition. This contrasts to the weak linear temperature dependence of the resistance above the critical pressure as demonstrated by the 3 kbar curve in Fig. 10. As can be seen from the figure, the accelerated decrease of the resistance at 2 kbar strongly depends on the level of the applied current; it also has been shown to be suppressed by a tiny magnetic field, ∼10 mT [42] that reveals its superconducting nature.

As a whole, the data on the SC transition can be summarized in the low-temperature P–T phase diagram as shown in Fig. 11. Above the critical pressure, the sharp transition indicates the bulk superconductivity in α-(BEDT-TTF)_{2}KHg(SCN)_{4}. At lower pressures the SC phase exists in the form of minor inclusions in the CDW matrix. The most surprising result is the dramatic increase of the SC onset temperature which is hardly consistent with the existing conventional theories of the competition between the SC and CDW states. One can speculate that in the neighborhood of the CDW phase the superconductivity is partially stimulated by charge fluctuations as proposed in several theoretical works [46–48]. Certainly, further studies are necessary to verify this hypothesis.

## 5 Conclusion

The extremely low CDW transition temperature and, hence, the low energy gap in the organic conductor α-(BEDT-TTF)_{2}KHg(SCN)_{4} provide a unique opportunity for comprehensive studies of the influence of a magnetic field on the CDW state. The experiment reveals strong, competing Pauli (paramagnetic) and orbital effects of the field:

- • the Pauli effect gradually suppresses the CDW state and, at low temperatures, causes the transition into the new, field-dependent CDW
_{x}state; - • the orbital effect enhances the CDW instability in an imperfectly nested Q1D system; in the case of a strongly imperfect nesting, t
_{y}′ > t′^{∗}, it leads to a cascade of quantized FICDW subphases; - • the superposition of the Pauli effect and orbital quantization gives rise to a novel kind of FICDW transitions in tilted magnetic fields.

At very low temperatures, the SC state is found to exist in α-(BEDT-TTF)_{2}KHg(SCN)_{4} in the broad pressure range, down to ambient pressure:

- • at 0 ≤ P < P
_{c}the SC and CDW phases coexist but are most probably spatially separated; the SC phase occupies only a minor volume fraction of the sample; - • the SC onset temperature drastically increases in the coexistence region; further studies are needed to clarify the mechanism of this unusual stimulating effect of the CDW on superconductivity.

## Acknowledgements

The presented experimental data have been obtained on high-quality samples provided by H. Müller and N.D. Kushch. The authors are grateful to E. Balthes, A.G.M. Jansen, I. Sheikin, and P. van der Linden for their kind assistance during experiments in Grenoble High Magnetic Field Laboratory. The work was partially supported by the DFG-RFBR grant 436 RUS 113/592.