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PRL 96, 116402 (2006)

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Subgap Collective Tunneling and Its Staircase Structure in Charge Density Waves
Yu. I. Latyshev,1 P. Monceau,2 S. Brazovskii,3 A. P. Orlov,1 and T. Fournier
1 3

2

IRE RAS, Mokhovaya 11-7, 101999 Moscow, Russia 2 CRTBT-CNRS, B.P. 166, 38042 Grenoble, France ^ ґ LPTMS-CNRS, Batiment 100, Universite Paris-Sud, 91405 Orsay Cedex, France (Received 4 November 2005; published 23 March 2006)

Tunneling spectra of chain materials NbSe3 and TaS3 were studied in nanoscale mesa devices. Currentvoltage I -V characteristics related to all charge density waves (CDWs) reveal universal spectra within the normally forbidden region of low V , below the electronic CDW gap 2. The tunneling always demonstrates a threshold Vt 0:2, followed, for both CDWs in NbSe3 , by a staircase fine structure. T dependencies of Vt T and T scale together for each CDW, while the low T values Vt 0 correlate with the CDWs' transition temperatures Tp . Fine structures of CDWs perfectly coincide when scaled along V=. The results evidence the sequential entering of CDW vortices (dislocations) in the junction area with the tunneling current concentrated in their cores. The subgap tunneling proceeds via the phase channel: coherent phase slips at neighboring chains.
DOI: 10.1103/PhysRevLett.96.116402 PACS numbers: 71.45.Lr, 03.75.Lm, 71.10.Pm, 73.40.Gk

Electronic spectra of strongly correlated systems are formed self-consistently via interactions among electrons or between them and the host crystal. Commonly observed states with spontaneously broken symmetries are susceptible to external fields and are affected by even the measuring device. That can modify the ground state, being able to readjust to changes of the local concentration of electrons and even to added individual particles. We are concerned with so-called electronic crystals [1], especially charge density waves (CDWs) [2]. A usual temptation is to treat CDWs as a kind of semiconducting superlattices with a gap at the Fermi level. But contrary to usual crystals, here the number of unit cells (CDW periods) is not fixed and can be readjusted to absorb added or removed electrons to the extended ground state. Particularly intriguing is that this process goes on via topologically nontrivial defects: discommensurations, dislocations, solitons. Dynamically, they are created as instantons of field theories, i.e., phase slips in the language of incommensurate CDWs (ICDWs). In this Letter, together with the earlier publication [3], we show that all these effects appear in the experiments. We report measurements of the intrinsic interlayer tunneling in two quasi-one-dimensional (1D) materials, NbSe3 and o-TaS3 , below their Peierls transition temperatures Tp to CDW states. NbSe3 has two ICDWs which develop below their corresponding Tp : T1 145 K (CDW1) and T2 59 K (CDW2) opening the gaps 1 and 2 . In o-TaS3 , a single ICDW is formed below Tp 215 K. In general, the intrinsic tunneling is observed [4] in layered or chain crystalline materials with well decoupled elementary conducting planes. Experimentally, the interlayer transport is studied by the method [5,6] of nanoscale devices called stacked structures, or mesas, or overlap junctions (Fig. 1). We fabricated the devices by a focused ion beam (FIB) technique developed previously for a high-Tc superconductor [7] and extended to the layered 0031-9007= 06 =96(11)=116402(4)$23.00

CDW material NbSe3 [8]. The essence of the method is that all elements of the device are parts of the same single crystal. The typical size of the junctions is 1 m 1 m 0:05 m, so the depth includes only 20 ­ 30 atomic layers. We used two modifications of the FIB processing: (A) the double-sided one [7] (used here for NbSe3 ) and (B) the lateral one [5] (used here for TaS3 ). Technique A allowed us to fabricate stacked structures with connecting electrodes oriented across the chains -- Fig. 1, which prevents an interference between the interlayer tunneling and a possible CDW sliding within connecting channels. Figure 2 shows the tunneling spectra dI =dV of NbSe3 for both (a) upper and (b) lower CDW states and of (c) TaS3 . We performed experiments at higher T , Tp =2 & T & Tp , because at lower 4 K & T & 20 K, the low V spectrum of NbSe3 is dominated by a zero bias conducting peak (ZBCP), originated by specific carriers from remnant pockets [8]. Two features are common for both materials, including both CDW states in NbSe3 : the CDW gap peak at Vg T 2 and the much lower sharp threshold voltage Vt 0:2 for the tunneling onset. The peak at Vg is particularly pronounced at lower T , where Vg 0 reaches the
a*(z) c (y) b (x)
V+ I+ ~0.05 µm stack 1 µm
dislocation lines

I­ V­

FIG. 1 (color online). Scheme of the tunneling device of type A. Crystallographic axes of NbSe3 a ;b;c (b is the chain direction) correspond to coordinates z; x; y. Elliptic cylinders show the cores of dislocation lines; the curved arrow shows the current path.

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-Vt V

t

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140 135 130 125 120 115 110 105

K K K K K K K K

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FIG. 2 (color online). Tunneling spectra dI =dV as a function of the voltage V normalized to the CDW gap 2 at different T : (a) CDW1 in NbSe3 , (b) CDW2 in NbSe3 , (c) o-TaS3 .

expected values 20 of the CDW gaps (Vg 50 mV, 150 mV for CDWs in NbSe3 and Vg 180 mV for TaS3 ). These values are in reasonable agreement with those measured for NbSe3 in optics [9], angle-resolved photoemission spectroscopy [10], tunneling, and STM [11], point contact spectroscopy [12], and for TaS3 in optics [13]. The T dependence Vg T in NbSe3 is well fitted by the BCS law [8,14]; hence, at any T this peak can be attributed to the tunneling across the CDW gap 2T . The gap peak in o-TaS3 is much broader. To define its position at T > 120 K, we subtracted from the original spectra the parabolic background fitted at high biases. That type of background has been confirmed by measurements above Tp . Figure 2 demonstrates that the ratio Vt T =T 0:2 is temperature independent for all cases; hence, the origin of Vt is linked to the CDW gap. The next enlightening step is to compare Vt with the 3D ordering scale kTp among the different CDWs. We find a surprisingly good linear relation

Vt 0 1:3kTp , in a wide range of Tp from 60 to 215 K. It proves that the origin of the threshold voltage is related also with the phase decoupling of CDWs in adjacent layers. Figure 3(a) shows the fine structure in NbSe3 at T 130 K. We see clear steps in dI =dV and corresponding sharp peaks in d2 I =dV 2 . Figure 3(b) proves that this is not an artifact: For each CDW, the structure is well reproduced for both bias polarities. Moreover, Fig. 3(b) shows that for the normalized V=21;2 the peak positions coincide with a remarkable accuracy for both CDWs at so different T 130 K; 120 K; 50 K. It proves the universality of the fine structure and its common origin with Vt , viewed as the first step position. Similarly to the threshold feature, the fine structure was well resolved only at high enough temperatures where the ZBCP was suppressed. The model presented below aims to explain these unexpected observations. Let (see Fig. 1) x be the chain direction along the junction; z is the transverse direction across the junction; y is the direction along the junction but perpendicular to chains. dx ;dy ;dz are the corresponding unit cell sizes. The Fermi scales of the parent metal are the energy EF , the momentum kF , and the velocity vF ; then the screening radius r0
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80 60 40 20 0 -20 0.0 0.1 0.2 0.3 0.4 0.5 0.6

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FIG. 3 (color online). Fine structure of the tunneling spectra in NbSe3 within the magnified threshold region. (a) dI =dV and its derivative d2 I =dV 2 as a function of the voltage V normalized to the CDW gap, at T 130 K. (b) Comparison of d2 I =dV 2 for the two voltage polarities for both CDWs, at T 130 and 50 K and the positive polarity at T 120 K (lines are guides for the eye). The peaks are interpreted as a sequential entering of dislocation lines into the junction area.

only one spacing. The minimal model treats the interlayer decoupling as an internal incommensurability transition. It takes into account only two layers 1 and 2 kept at potentials V=2 as given by the energy functional (cf. [15]) Z @vF V dx '02 '02 '0 Ъ '0 Ъ Jz cos'1 Ъ '2 : 2 2 2 1 4 1 Here the three terms are the CDWs' elastic energy, the interchain charge transfer gain, and the interchain cou-

pling. Its minimization gives an array of interplane discommensurations which develops starting from the isolated line (in the y direction) of 2 solitons in the phase difference '1 Ъ '2 . Then Vt is identified as the energy necessary to create the first soliton: Es 4@vF Jz =1=2 [17]. At higher V > Es , the solitons overlap [their width is ls 4@vF =Jz 1=2 [17]], and at high V Es the phase difference changes almost linearly, '1 Ъ '2 2Vx=@vF , which means complete decoupling. The above model is only a transparent illustration: Actually, the decoupled planes are the boundaries for two semivolumes z _ 0 with the voltage V=2 applied at the distant outer boundaries of the junction, while the decoupling happens in between. The former lattice of discommensurations must be generalized to a sequence of dislocation lines (DLs -- the CDW vortices [18]). Vt is to be identified as the DL entry energy, in analogy to the Hc1 field for entering the Josephson vortex in layered superconductors. The DLs are lying along the y direction, presumably in the median junction plane x; y; 0 [19]. Close to the plane z 0, the sequence of DLs will look almost like a solitonic lattice with e charged phase increments of ( for passing above or below the DL). The distribution becomes more diffuse at distant planes and eventually overlaps, but invariantly the total phase difference 2 is accumulated between the lower and the upper halves of the junction, hence the charge transfer 2e per chain per DL. The two-plane interaction is generalized as a distributed shear stress energy f@x '2 @y '2 g characterized by the dimensionless parameter of anisotropy  dz Jz =@vF 1=2 1. There is also the Coulomb energy, which is particularly costly for charged phase variations '0 along the chain x; hence, they must be slow relative to other directions. Thus [20], the DL core has the atomic width dz in z and a longer length l 100 A in the x direction: l d2 =r0 z dz !p =Tp dz . The electric field Ez is concentrated close to the DL plane x; y; 0: Ez 0 =dz l=jxj1=2 expЪz=dz 2 l=jxj [20]. The potential drop across the junction width is 0 !p per each entering DL, which determines the threshold voltage Vt 0 and gives the same quantization for further steps, as is seen in I V [Fig. 3(b)]. Coulomb interactions increase the energy cost to create DLs but also enlarge their efficiency in building the potential: Only a few DLs are sufficient to cover the whole gap interval of V < 2T . Recall that T , hence 0 T , and finally Vt T must have the same T dependence governed by the factor T , which is in accordance with the experiment [21]. The tunneling takes p place over the distance 2zx x=l between matching points zx of surfaces x; z . The probability p expЪ2zx=dz expЪ2 x=l is exponentially enhanced towards smaller x where zx is small; i.e., the tunneling is confined within the DL core. Only here are the potential changes fast enough to give a short path for tunneling.

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d I/dV , arb. units

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The interchain coupling energy is proportional to the square of the CDW amplitude Jz T 2 ; then T , hence 0 T , and finally Vt T must have the same T dependence governed by the factor T . It explains the clear scaling of all spectral features with T . At low T , 2 Jz 0 Tp =@vF , then the saturated value of Vt scales with Tp , which is another observation [21]. The final step is to understand the nature of the microscopic dynamical processes of tunneling at V > Vt [22]. Several plausible mechanisms can be excluded. (a) Our special construction of the device eliminates the interference with the threshold for CDW sliding. (b) The usual tunneling through creation of e Ъ h pairs cannot take place below Vg 2. (c) Dressed single electron states, the ``amplitude solitons,'' reduce the energy by 2=3, as confirmed by experiments [3], but the scale is still too high for Vt = 0:2. (d) Contribution of normal carriers gives an opposite dependence I V : (i) In NbSe3 , Vt appears only when the ZBCP is suppressed; (ii) the concentration of the potential drop upon one layer can only reduce the normal current. We are left with a fascinating, while firmly based, picture that the excess tunneling conductivity above Vt can be provided only by the low energy phase channel. Empirically, one can already recognize the necessary scale from very low activation energies Ea for the on-chain conductivity measured in gapful CDWs in contrast to high 0 for the transverse one (e.g., in TaS3 [23] Ea 200 K, while 0 800 K). This conductivity is associated with 2 phase solitons, which correspond to stretching/squeezing of a chain by one period, ' 2, with respect to the surrounding ones. Contrary to their aggregated form of static DLs, the solitons exist as single chain items: elementary particles with the charge 2e and the energy Es Tp . Their dynamic creation might be very sensitive to the threshold proximity V V Ъ 2Es and to the chain number M 2z=dz to tunnel through: The tunneling rate drops as [24] V =Es M , where the index vF =u 1 is big because of the low phase velocity u vF . Hence, the pair of 2 solitons can be created by tunneling almost exclusively within the DL core, which process can be interpreted as a quantum excitation of the DL string. In conclusion, we have measured intrinsic interlayer tunneling in nanostructures of the quasi-1D materials NbSe3 and TaS3 . Enhanced tunneling occurs above a threshold voltage Vt which scales with the CDW gap as Vt 0:2. For NbSe3 , the tunneling spectrum exhibits a staircase structure. We have presented a theoretical model in which the tunneling occurs through a single layer in the core of dislocation lines in the junction plane. Sequential formation of a DL grid at V > Vt gives rise to the observed steps. The resolved, for the first time, tunneling in the normally forbidden subgap (2 >V > Vt ) region recovers collective quantum processes such as coherent phase slips at adjacent chains.

Studies were supported by the Russian Fund of Basic Research (Grants No. RFBR-CNRS 03-02-22001 and No. RFBR 05-02-17578-a) and by the RAS program ``New materials and structures.'' The work was partially performed in the frame of the CNRS-RAS Associated European Laboratory between CRTBT and IRE ``Physical properties of coherent electronic states in condensed matter.''

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