Advances in point-contact spectroscopy: two-band superconductor MgB₂ (Review Article)

Advances in point-contact spectroscopy: two-band superconductor MgB₂ (Review Article)

Analysis of the point-contact spectroscopy (PCS) data on the new dramatic high-Tc superconductor magnesium diboride MgB₂ reveals quite different behavior of two disconnected σ and π electronic bands, deriving from their anisotropy, different dimensionality, and electron—phonon interaction. PCS...

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Hauptverfasser: Yanson, I.K., Naidyuk, Yu.G.
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Veröffentlicht: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2004
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spelling irk-123456789-1195082017-06-08T03:04:33Z Advances in point-contact spectroscopy: two-band superconductor MgB₂ (Review Article) Yanson, I.K. Naidyuk, Yu.G. Обзоp Analysis of the point-contact spectroscopy (PCS) data on the new dramatic high-Tc superconductor magnesium diboride MgB₂ reveals quite different behavior of two disconnected σ and π electronic bands, deriving from their anisotropy, different dimensionality, and electron—phonon interaction. PCS allows direct registration of both the superconducting gaps and electron—phonon interaction spectral function of the two-dimensional σ and three-dimensional π band, establishing correlation between the gap value and intensity of the high-Tc driving force — the E₂g boron vibration mode. PCS data on some nonsuperconducting transition-metal diborides are surveyed for comparison. 2004 Article Advances in point-contact spectroscopy: two-band superconductor MgB₂ (Review Article) / I.K. Yanson, Yu.G. Naidyuk // Физика низких температур. — 2004. — Т. 30, № 4. — С. 355-372. — Бібліогр.: 54 назв. — англ. 0132-6414 PACS: 74.25.Fy, 74.80.Fp, 73.40.Jn http://dspace.nbuv.gov.ua/handle/123456789/119508 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language English
topic Обзоp
Обзоp
spellingShingle Обзоp
Обзоp
Yanson, I.K.
Naidyuk, Yu.G.
Advances in point-contact spectroscopy: two-band superconductor MgB₂ (Review Article)
Физика низких температур
description Analysis of the point-contact spectroscopy (PCS) data on the new dramatic high-Tc superconductor magnesium diboride MgB₂ reveals quite different behavior of two disconnected σ and π electronic bands, deriving from their anisotropy, different dimensionality, and electron—phonon interaction. PCS allows direct registration of both the superconducting gaps and electron—phonon interaction spectral function of the two-dimensional σ and three-dimensional π band, establishing correlation between the gap value and intensity of the high-Tc driving force — the E₂g boron vibration mode. PCS data on some nonsuperconducting transition-metal diborides are surveyed for comparison.
format Article
author Yanson, I.K.
Naidyuk, Yu.G.
author_facet Yanson, I.K.
Naidyuk, Yu.G.
author_sort Yanson, I.K.
title Advances in point-contact spectroscopy: two-band superconductor MgB₂ (Review Article)
title_short Advances in point-contact spectroscopy: two-band superconductor MgB₂ (Review Article)
title_full Advances in point-contact spectroscopy: two-band superconductor MgB₂ (Review Article)
title_fullStr Advances in point-contact spectroscopy: two-band superconductor MgB₂ (Review Article)
title_full_unstemmed Advances in point-contact spectroscopy: two-band superconductor MgB₂ (Review Article)
title_sort advances in point-contact spectroscopy: two-band superconductor mgb₂ (review article)
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2004
topic_facet Обзоp
url http://dspace.nbuv.gov.ua/handle/123456789/119508
citation_txt Advances in point-contact spectroscopy: two-band superconductor MgB₂ (Review Article) / I.K. Yanson, Yu.G. Naidyuk // Физика низких температур. — 2004. — Т. 30, № 4. — С. 355-372. — Бібліогр.: 54 назв. — англ.
series Физика низких температур
work_keys_str_mv AT yansonik advancesinpointcontactspectroscopytwobandsuperconductormgb2reviewarticle
AT naidyukyug advancesinpointcontactspectroscopytwobandsuperconductormgb2reviewarticle
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last_indexed 2025-07-08T16:00:08Z
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fulltext Fizika Nizkikh Temperatur, 2004, v. 30, No. 4, p. 355–372 Advances in point-contact spectroscopy: two-band superconductor MgB2 (Review Article) I.K. Yanson and Yu.G. Naidyuk B.Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, 47 Lenin Ave., Kharkov 61103, Ukraine E-mail: yanson@ilt.kharkov.ua Received July 28, 2003 Analysis of the point-contact spectroscopy (PCS) data on the new dramatic high-Tc supercon- ductor magnesium diboride MgB2 reveals quite different behavior of two disconnected � and � elec- tronic bands, deriving from their anisotropy, different dimensionality, and electron—phonon in- teraction. PCS allows direct registration of both the superconducting gaps and electron—phonon interaction spectral function of the two-dimensional � and three-dimensional � band, establishing correlation between the gap value and intensity of the high-Tc driving force — the E g2 boron vibra- tion mode. PCS data on some nonsuperconducting transition-metal diborides are surveyed for com- parison. PACS: 74.25.Fy, 74.80.Fp, 73.40.Jn Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 355 1.1. Crystal structure . . . . . . . . . . . . . . . . . . . . 356 1.2. Electron band structure . . . . . . . . . . . . . . . . . 356 1.3. Critical magnetic field . . . . . . . . . . . . . . . . . . 357 1.4. Phonons and electron—phonon interaction . . . . . . . . . 357 1.5. Mechanism for high Tc in MgB2. . . . . . . . . . . . . . 358 2. Samples . . . . . . . . . . . . . . . . . . . . . . . . . . 359 3. Theoretical background of PCS . . . . . . . . . . . . . . . . 360 3.1. Nonlinearity of I–V characteristic . . . . . . . . . . . . . 360 3.2. Two-band anisotropy . . . . . . . . . . . . . . . . . . 361 4. Experimental results . . . . . . . . . . . . . . . . . . . . . 362 4.1. Superconducting energy gaps . . . . . . . . . . . . . . . 362 c-axis oriented thin films. . . . . . . . . . . . . . . . . 362 Single crystals . . . . . . . . . . . . . . . . . . . . . 363 4.2. Phonon structure in the I–V characteristics. . . . . . . . . 365 PC EPI spectra of nonsuperconducting diborides . . . . . . 365 PC EPI spectra of MgB2 in c-axis oriented films . . . . . . 366 PC EPI spectra of MgB2 in the ab direction . . . . . . . . 368 5. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . 370 Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . 370 Note added in proof . . . . . . . . . . . . . . . . . . . . . . 371 References . . . . . . . . . . . . . . . . . . . . . . . . . . 371 1. Introduction MgB2 was discovered to be superconducting only a couple of years ago [1], and despite that, many of its characteristics have now been investigated and a con- sensus exists about its outstanding properties. First of all, this refers to its high Tc (� 40 K) which is a re- cord-breaking value among the s—p metals and al- loys. It appears that this material is a rare example of © I.K. Yanson and Yu.G. Naidyuk, 2004 multiband (at least two) electronic structure, which are weakly connected with each other. These bands lead to very uncommon properties. For example, Tc is almost independent of elastic scattering, unlike for other two-band superconductors [2]. The maximal up- per critical magnetic field can be made much higher than that for a one-band dirty superconductor [3]. The properties of MgB2 have been comprehensively calcu- lated by the modern theoretical methods, which lead to a basic understanding of their behavior in various experiments. 1.1. Crystal structure Magnesium diboride, like other diborides MeB2 (Me = Al, Zr, Ta, Nb, Ti, V etc.), crystalizes in a hex- agonal structure, where honeycomb layers of boron are intercalated with hexagonal layers of magnesium located above and below the centers of boron hexa- gons (Fig. 1). The bonding between boron atoms is much stronger than that between magnesium, and therefore the disordering in the magnesium layers ap- pears to be much easier than in the boron layers. This difference in bonding between boron and magnesium atoms hinders the fabrication of MgB2 single crystals of appreciable size. 1.2. Electron band structure The electron band structure of MgB2 has been cal- culated using different ab initio methods yielding ba- sically the same result [4–8]. The E k( ) curves are shown in Fig. 2. The dispersion relations are shown for boron p-character orbitals, which play a major role in transport and thermodynamic properties. The radii of the hollow circles are proportional to the �-band character, which is made from pz boron orbitals, while those of the filled circles are proportional to the �-band character, made from pxy orbitals. The most important is a quasi-two-dimensional dispersion rela- tion along the �A ( )� direction with a small Fermi en- ergy � 0.6 eV, and accordingly, with a moderate Fermi velocity. The corresponding sheets of the Fermi en- ergy form the cylindrical surfaces along the �A direc- tion seen in Fig. 5 below. The corresponding electron transport is very anisotropic (� �c ab/ � 3 5. [10]), with the plasma frequency for the � band along the c (or z) axis being much smaller than that in the ab ( )xy direction [11]. The hole branch along �A experiences a huge interaction with the phonon E g2 mode for car- riers moving along the ab plane (see below), although its manifestation is screened effectively by the much faster hole mobility in the � band [2]. In a dirty material, with prevailing disorder in the magnesium planes, the �-band conductivity is blocked by defects, and the � band takes over, implying greater electron—phonon interaction (EPI) than in the clean material. This constitutes a plausible expla- nation for the violation of the Matthiesen rule, which manifests itself in an increase of the residual resisti- vity together with an increase of the temperature coef- ficient at high temperatures [2]. At the same time, the critical temperature Tc does not decrease substantially in dirty materials [2], since 356 Fizika Nizkikh Temperatur, 2004, v. 30, No. 4 I.K. Yanson and Yu.G. Naidyuk Fig. 1. Crystal structure of MgB2. � M K � A L –15 –10 –5 0 5 10 � � � E n e rg y, e V Fig. 2. Band structure of MgB2 with the B p character. The radii of the hollow (filled) circles are proportional to the � (�) character and zero-line marks the Fermi energy. After Mazin et al. [9]. the superconductivity is induced by EPI in the � band, whose crystal order is much more robust. This consideration is very important in understand- ing the point-contact data, since the disorder at the surface of the native sample depends on the position of the contact spot, and because of the uncontrolled in- troduction of further disorder while fabricating the contact. 1.3. Critical magnetic field In a clean material the layered crystal structure dic- tates strong anisotropy of the upper critical magnetic fields B Bc ab c c 2 2 . Their ratio at low temperatures reaches about 6 while Bc c 2 is as low as 2–3 T [12]. If the magnetic field is not aligned precisely along the ab plane, the Bc2 value is strongly decreased. On the other hand, for a dirty material the aniso- tropy is decreased (to a ratio of about 1.6–2), but both the magnitudes of Bc ab 2 and Bc c 2 are strongly in- creased. For strongly disordered sample, it may be as high as 40 T [3]! It is interesting that this high value is achieved at low temperature, where the disordered � band is fully superconducting. Hence, we may expect that the value of the critical magnetic field at low temperatures is the smaller the cleaner is the part of the MgB2 volume near the con- tact, provided its T Tc c� bulk . This observation is im- portant in the classification of contacts with respect to their purity. 1.4. Phonons and electron—phonon interaction The phonon density of states (PDOS) is depicted in Fig. 3. The upper panel shows the measured PDOS at T 8 K, while the lower ones show the calculated DOS with the partial contribution from boron atoms moving in the ab plane and out of it. One can see the peak for boron atoms moving in the ab plane at � 75 meV, which plays a very important role in the electron—phonon interaction, as is shown in Fig. 4, measured by inelastic x-ray scattering [14]. This mode gives a weakly dispersion branch between 60 and 70 meV in the �A direction with E g2 symmetry at the � point. The linewidth of this mode is about 20–28 meV along the �A direction, while along the �M di- rection it is below the experimental resolution. The same phonon peak is active in Raman scattering [15–17]. It is located at the same energy with the same linewidth. This points to the very strong EPI for this particular lattice vibration mode. The same result fol- lows from theoretical considerations. Advances in point-contact spectroscopy: two-band superconductor MgB2 Fizika Nizkikh Temperatur, 2004, v. 30, No. 4 357 0 20 40 60 80 100 0.4 0.8 0 0.2 0.4 0.6 0.8 1.0 Total DOS B in-plane B out-of-plane Energy, meV Mg 11 B2 G D O S ,a rb .u n its e xp G D O S , a rb . u n its ca l Fig. 3. Upper panel: Phonon density of states in MgB2 de- termined experimentally by neutron scattering. Bottom panel: calculated curve (solid line) with decomposition on boron atoms vibrating out of ab plane (dotted curve) and parallel to it (dashed curve). After Osborn et al. [13]. 30 20 10 0 80 60 40 20 0 Fr e q u e n cy ,m e V W id th , m e V M � A L B1g E2g A2u E1u Fig. 4. Dispersion curves of phonons in MgB2 and the width of phonon lines determined by inelastic x-ray scat- tering (symbols) together with calculations (solid lines). After Shukla et al. [14]. Figure 5 shows the distribution of the supercon- ducting energy gap on the Fermi surface of MgB2 [18]. The maximum gap value is calculated along the �A direction due to the very strong EPI. Just in this direction is located 2D � band (cylinders along the �A direction). The 3D � band has a much smaller EPI, and, correspondingly, a smaller energy gap. The EPI parameter � can be decomposed between different pieces of the Fermi surface. It is shown [19] that the value of � on the � band amounts to 2–3. Moreover, �� can be decomposed between different phonon modes, and it appears that only the E g2 phonon mode along the �A direction plays a major role with a par- tial �� value of about � 25 [20], though concentrated in a very restricted phase space. 1.5. Mechanism for high Tc in MgB2 The commonly accepted mechanism for high Tc in MgB2 is connected with the strong interaction be- tween charge carriers and phonons in the E g2 mode.This mode is due to antiparallel vibration of at- oms in the boron planes. The key issue is that along the �A direction the electron band structure is such that the Fermi energy of the hole carriers is only 0.5–0.6 eV, which shrinks even more when the borons deviate from the equillibrium positions. Together with the 2D structure of the corresponding sheet of the Fermi surface, this leads to a constant density of states at the Fermi energy and, correspondingly, to very large EPI with partial �� (the EPI parameter in the � band) of about ~ 25 [20]. Cappelluti et al. [21] point out that the small Fermi velocity for charge carriers along the �A direction leads to a large nonadiabatic correction to Tc (about twice as much compared with the adiabatic Migdal—Eliashberg treatment). Al- though this interaction is a driving force to high Tc in this compound, it does not lead to crystal structure in- stability, since it occupies only a small volume in the phase space. The role of the � band is not completely clear. On the one hand, the � and � bands are very weakly con- nected, and for some crude models they can be thought as being completely disconnected. On the other hand, the energy gap of the � band goes to zero at the same Tc as in the bulk, and correspondingly 2 0 1 4� �( ) ./kTc , which is much less than the value predicted by the weak-coupling BCS theory. One can think of the � band as having intrinsically much lower Tc � 10 K than the bulk [22], and at higher tempera- tures its superconductivity is induced by a proximity effect in k space from the � band [23]. This proximity effect is very peculiar. On the one hand, this proxim- ity is induced by the interband scattering between the � and � sheets of the Fermi surface. On the other, the charge carriers connected with the � band are mainly located along the magnesium planes, which can be considered as a proximity effect in coordinate space for alternating layers of S—N—S structure, although on a microscopic scale. Moreover, many of the unusual properties of MgB2 may be modeled by an alternating S—N—S layer structure, the limiting case to the crys- tal structure of MgB2. In other words, MgB2 presents a crossover between two-band superconductivity and a simple proximity-effect structure. 358 Fizika Nizkikh Temperatur, 2004, v. 30, No. 4 I.K. Yanson and Yu.G. Naidyuk � � � � � � � M M M M K K H L L A � , meV �� � 0 2 4 6 8 Fig. 5. Superconducting energy gap distribution over the Fermi surface (FS) of MgB2. The gap value around 7 meV corresponds to cylinderlike sheets of the FS centered at � points, while the small gap value around 2 meV corre- sponds to the tubular FS network. After Choi et al. [18]. 2. Samples We have two kind of samples supplied for us by our colleagues from the Far East*. The first is a thin film with a thickness of about several hundred nanometers (Fig. 6) [24]. Similar films have been investigated by several other groups with different methods. These films are oriented with their c axis perpendicular to the substrate. The resid- ual resistance is about several tens of �� � cm with a residual resistance ratio (RRR) � 2 2. . This means that on average the films have a disorder between crystallites. It does not exclude the possibility that on some spots the films contain clean enough small single crys- tals on which we occasionally may fabricate a point contact; see Fig. 6. Normally, we make a contact by touching the film surface by noble metal counter elec- trode (Cu, Au, Ag) in the direction perpendicular to the substrate. Thus, nominally the preferential cur- rent direction in the point contact is along the c axis. Nevertheless, since the surface of the films contains terraces with small crystallites, point contact to the ab plane of these crystallites is also possible. Some- times, in order to increase the probability of making the contact along the ab plane, we broke the substrate with the film and made contact to the side face of the sample. The second type of sample is single-crystal [26], which also was measured by other groups [10,27]. Crystals are platelike (flakes) and have submillimeter size (see Fig. 7). They were glued by silver epoxy to the sample holder by one of their side faces. The noble metal counter electrode was gently touched in liquid helium by another (the opposite) side face of the crys- tal. In this way we try to preferentially make a con- tact along the ab plane. On average, in the bulk, the single crystals are cleaner than the films, but one should be cautious, since the properties of the crystal surface differ from the properties of the bulk, and fab- rication of a point contact may introduce further un- controlled defects into the contact area. Thus, a priori one cannot define the structure and composition of the contacts obtained. Nevertheless, much of that information can be ascertained by mea- suring various characteristics of a contact. Among those the most important is the Andreev-reflection nonlinearities of the I–V curves in the superconduct- ing energy-gap range. The magnetic-field and temper- ature dependences of the superconducting nonlinearities supply us with additional information. And finally, much can be extracted from the I–V nonlinearities in the normal states (the socalled point-contact spectra). The more information we can collect about the electrical conductivity for different Advances in point-contact spectroscopy: two-band superconductor MgB2 Fizika Nizkikh Temperatur, 2004, v. 30, No. 4 359 15 kV 150� 100 m 122200� Fig. 7. Scanning electron microscopy image of MgB2 sin- gle crystals. After Lee et al. [26]. Fig. 6. Scanning electron microscopy image of MgB2 films. After Kang et al. [25]. * The films were provided by S.-I. Lee from National Creative Research Initiative Center for Superconductivity, Department of Physics, Pohang University of Science and Technology, Pohang, South Korea. The single crystals were provided by S. Lee from Superconductivity Research Laboratory, ISTEC, Tokyo, Japan. conditions of the particular contact, the more detailed and defined picture of it emerges. It is not an easy task, since a contact has limited lifetime, due to elec- trical and mechanical shocks. Let us make a rough estimate of the distance scales involved in the problem. The crystallite size of the films is of the order of 100 nm (see [25]). The contact size d in the ballistic regime equals d l/R� � (the Sharvin for- mula). Taking �l � � � �� �(7 10 7 107 6� cm)( cm) = = 4,9 10 cm–12 2� �� [10], we obtain d � 7 nm along both the ab and c directions for a typical resistance of 10 �. If we suppose that a grain is dirty (with a very short mean free path), then we apply the Maxwell for- mula d /R� � with the results for d values of about 0.7 nm and 2.6 nm for the ab and c directions, respec- tively, taking � for the corresponding directions from the same reference [10]. Thus, the contact size can be of the order of or smaller than the electronic mean free path (lab 70 nm and lc 18 nm, according to [10]), which means that we are working admittedly in the spectroscopic regime, probing only a single grain. Rowell [28], analyzing a large amount of experi- mental data for the resistivity and its temperature de- pendence, came to the conclusion that for highly resis- tive samples only a small part of the effective cross section should be taken into account. The reason is that the grains in MgB2 are to great extent discon- nected by oxides of magnesium and boron. For point- contact spectroscopy previous analysis leads us to the conclusion that the contact resistance is frequently measured only for a single grain or for several grains, with their intergrain boundaries facing the contact in- terface. This is due to the current spreading on a scale of the order of the contact size dnear the constriction. 3. Theoretical background of PCS 3.1. Nonlinearity of I–V characteristic The nonlinearities of the I–V characteristic of a metallic contact, when one of the electrodes is in the superconducting state, can be written as [29,30] I V V R I V I VN( ) ( ) ( )� 0 � �� ph exc . (1) Here R0 is the contact resistance at zero bias in the normal state. �I VN ph( ) is the backscattering inelastic current, which depends on the electron mean free path (mfp) l. For ballistic contact this term is equal in order of magnitude to �I V I VN d lph in ( ) ( )� , (2) where lin is the inelastic electron mfp, and d is the characteristic contact diameter. If the electron flow through the contact is diffusive (l del �� , lel being an elastic mfp) but still spectroscopic, since l l din el , then the expression (2) should be multiplied by l /del . This decreases the characteristic size for which the in- elastic scattering is important from d to lel (d l� el), and for short lel makes the inelastic current very small. We notice that the inelastic backscattering current �I VN ph( ) in the superconducting state is ap- proximately equal to the same term in the normal state. Its second derivative turns out to be directly proportional to the EPI function � � �2( ) ( )F [31,32] � � d I dV ed v F F 2 2 28 3� � � �( ) ( ) (3) where � describes the strength of the electron interac- tion with one or another phonon branch, and F( )� stands for the phonon density of states. In point-con- tact (PC) spectra the EPI spectral function � � �2( ) ( )F is modified by the transport factor, which strongly increases the backscattering processes contribution. In the superconducting state the excess current Iexc (1), which is due to the Andreev reflection of electron quasiparticles from the N—S boundary in an N—c—S point contact (c stands for «constriction»), can be written as I V I I Vexc exc exc( ) ( ) �0 � (4) where I /Rexc const0 0� �� for eV � (� being the superconducting energy gap). The nonlinear term in the excess current (4) can in turn be decomposed in two parts, which depend in dif- ferent ways on the elastic scattering of electron quasi- particles: � � �I V I V I Vexc exc el exc in( ) ( ) ( ) � (5) where � �I Vexc el is of the order of ( )�/eV Iexc 0 , and �I V d/l Iexc in in exc� ( ) 0 . Notice that the latter behaves very similarly to the inelastic backscattering current �I VN ph( ), namely, it disappears if lel � 0, while the first term in the right-hand side of expression (5) does not depend on lel in the first approximation. This enables one to distinguish the elastic term from the inelastic. Finally, all excess current terms disap- pear when the superconductivity is destroyed, while �I VN ph( ) remains very similar in both the supercon- ducting and normal states. The expression for the elastic term in the excess current was calculated for ballistic N—c—S contacts by Omelyanchuk, Beloborod’ko, and Kulik [33]. Its first derivative equals ( )T 0 : 360 Fizika Nizkikh Temperatur, 2004, v. 30, No. 4 I.K. Yanson and Yu.G. Naidyuk dI dV R eV eV eVNcS exc el ballistic � � � � � ! ! � � 1 0 2 2 � � ( ) ( ) (eV) " " "" " " "" 2 . (6) For the diffusive limit ( ),l di �� Beloborod’ko et al. derived the current—voltage characteristic (see Eq. (21) in Ref. 34), which for the first derivative at T 0 gives [35]: R dI dV eV eV eV e NcS 0 1 2 exc el diffusive � � � � � ! ! � � ln ( ) ( � � V eV eV eV eV eV ) Re ( ) ( ) / Re ( ) ( ) " " " " " "� � � # $ % % & ' ( ( �2 2 2 2� � � ( ) . eV # $ % % & ' ( ( (7) For the sake of comparison, the similar expression of the nonlinear term in NIS tunnel junctions (I stands for «insulator»), due to the self-energy super- conducting energy gap effect, has the form [36]: dI dV R eV eV eVNIS � � � � ! � # $ % % & ' ( ( 1 0 2 2 Re ( ) ( )� . (8) Equations (6), (7), and (8) are identical in their structure and take into account the same effect, viz., the renormalization of the energy spectrum of a super- conductor in the vicinity of characteristic phonon en- ergies. From the expressions (1), (2), (4), and (5) it becomes clear that only on the relatively clean spots can one observe the inelastic backscattering current �I VN ph( ), provided that the excess current term �I Vexc in ( ) is negligible. The latter can be canceled by suppression of superconductivity either with magnetic field or temperature. On the contrary, in the super- conducting state, for dirty contacts, all the inelastic terms are very small, and the main nonlinearity is provided by the �( )eV dependence of the excess current (7). 3.2. Two-band anisotropy Brinkman et al. have shown [11] that in the clean case for an NIS MgB2 junction, the normalized con- ductance is given by � � � �� � � ( ) / ( ) ( ) ( V dI dV dI dV V NIS NIN p p � � � � ! � � � � ! �2 ) ( ) ( ) ( ) 2 2 2 � � � � � � V p p� where �� � p ( ) is the plasma frequency for the � �( ) band and �� �( )( )V is the normalized conductivity of the � �( ) band separately. The calculated tunneling con- ductance in the ab plane and along the c axis are [11] � � �� �ab V V V( ) . ( ) . ( ) , �0 67 0 33 (10) � � �� �c V V V( ) . ( ) . ( ) . �0 99 0 01 (11) Hence, even along the ab plane the contribution of the � band is less than that of the � band, to say nothing about the direction along the c axis, where it is negligible small. The calculation predicts that if the «tunneling cone» is about several degrees from precise the ab plane, then the two superconducting gaps should be visible in the tunneling characteristics. In other directions only a single gap, corresponding to the � band, is visible. We will see below that this pre- diction is fulfilled in a point-contact experiment, as well. Things are even worse when one tries to measure the anisotropic Eliashberg function by means of super- conducting tunneling. The single-band numerical in- version program [36,37] gives an uncertain result, as was shown in Ref. 38. Point-contact spectroscopy in the normal state can help in this deadlock situation. It is known that the inelastic backscattering current is based on the same mechanism as an ordinary homogeneous resistance, provided that the maximum energy of the charge carri- ers is controlled by an applied voltage. The electrical conductivity of MgB2 can be considered as a parallel connection of two channels, corresponding to the � and � bands [2]. The conductivity of the � band can be blocked by disorder of the Mg atoms . This situation is already obtained in experiment, when the temperature coefficient of resistivity increases simultaneously with an increase of the residual resistivity, which leads to Advances in point-contact spectroscopy: two-band superconductor MgB2 Fizika Nizkikh Temperatur, 2004, v. 30, No. 4 361 –15 –10 –5 0 5 10 15 1,0 1.5 2�L 2�S R –1 d V/ d I Voltage, mV 1 2 3 4 . Fig. 8. Typical shapes of dV/dI (experimental dots) for 4 contacts between MgB2 thin film and Ag with the corres- ponding BTK fitting (lines). �L S( ) stand for large (small) superconducting energy gap. After Naidyuk et al. [40]. violation of Matthiessen’s rule (see Fig. 3 in [2]). In this case we obtain direct access to the �-band conduc- tivity, and the measurements of the PC spectra of the EPI for the � band is explicitly possible in the normal state. Below we will see that this unique situation happens in single crystals along ab plane. 4. Experimental results 4.1. Superconducting energy gaps c-axis oriented thin films. Our measurements of the superconducting energy gap by means of Andreev reflection from about a hundred NS junctions yield two kinds of dV/dI curves, shown in Fig. 8. The first one clearly shows two sets of energy gap minima located, as shown in distribution graph of Fig. 9 (upper panel), at 2.4 ) 0.1 and (7.1 ) 0.4) meV. These curves are nicely fitted by BTK [39] theory (with small � parameter) for two conducting channels with an adjusted gap weighting factor [40]. The sec- ond kind is better fitted with a single gap provided an increased depairing parameter � (Fig. 9 (middle panel)). Certainly, the division of the gap structure into the two kinds mentioned is conventional, and de- pends upon the circumstance that the larger energy gap is explicitly seen. These two kinds of gap struc- ture comprise about equal parts of the total number of junctions. Usually the contribution of the large gap in the double-gap spectra is an order of magnitude lower than that of the small one, which is in line with the small contribution of the � band to the conductivity along the c axis (see Eq. (11)). It is important to note that the critical temperature of the material around the contact is not more than a few K below Tc in the bulk material. This is deter- mined by the extrapolating the temperature depend- ence of PC spectra up to the normal state. Such an in- sensitivity of Tc on the elastic scattering rate is explained in Ref. 2. Nevertheless, we stress that the gap structure (either double- or single-gap feature, and the position of the single-gap minimum on dV/dI) depends very much on random variation of the scattering in the contact region. Moreover, since the main part of the junction conductivity is due to the charge carriers of the � band, even the background conductance quite often follows the «semiconductive» behavior, namely, the slope of the dV/dI curve at large biases is negative (Fig. 10). That means that the carriers in the � band are close to localization [41]. In the lower panel of Fig. 9 the theoretical predic- tion of the energy gap distribution [18] is shown. One can see that the theoretical positions of the distribu- tion maxima coincide approximately with the experi- mental values. Only the low-lying maximum is not seen in the experiment. It should be noted that accord- 362 Fizika Nizkikh Temperatur, 2004, v. 30, No. 4 I.K. Yanson and Yu.G. Naidyuk –60 –40 –20 0 20 40 60 Voltage , mV B, T 8 6 4 2 1 0.5 0.2 0 d V /d I, ar b .u n its Fig. 10. Negative slope of dV/dI at large biases for a 36 � contact between MgB2 single crystal and Ag showing the magnetic-field gap-structure evolution at 4.2 K. 0 2 4 6 8 10 2 4 6 8 100 10 20 PCS data double gap C o u n ts 2 4 6 8 100 10 PCS data single gap Gap value, meV Theory Choi et al. Fig. 9. Superconducting energy gap distribution of � 100 different junctions prepared on a MgB2 film. On the lower panel the theoretical distribution is shown. After Naidyuk et al. [40]. ing to Mazin et al. [42] variation of the superconduct- ing gaps inside the � and � bands can hardly be ob- served in real samples. The distribution of the different gaps over the Fermi surface is shown in Fig. 5. One can immediately see that for a c-oriented film the main structure should have a smaller gap, which is approximately isotropic. Only if the contact touches the side face of a single crystallite (Fig. 6), is the larger gap visible, since it corresponds to the cylindrical parts of the Fermi sur- face with Fermi velocity parallel to the ab plane. Single crystals. The same variety of energy gap structure is observed for single crystals as well, but with some peculiarity due to preferential orientation along the ab plane. The most amazing of them is the observation of dV/dI-gap structure in Fig. 11 with vi- sually only the larger gap present. This gap persists in a magnetic field of a few tesla, unlike the smaller gap, which according to [43,44], vanishes above 1 T. Spectra of that kind were not observed in thin films. This means that the conductivity is governed only by the � band. This may be caused by the circumstance that the � band is blocked completely by Mg disorder or by oxi- dation of Mg atoms on ab side surface of the crystal. At the same time, in a single crystal there is much less scattering in the boron planes, due to the robustness of the B—B bonds. We will see below that just this case Advances in point-contact spectroscopy: two-band superconductor MgB2 Fizika Nizkikh Temperatur, 2004, v. 30, No. 4 363 –15 –10 –5 0 5 10 15 B, T 6 5 4 3 2 1 0.8 0.6 0.4 0.2 0 Voltage, mV d V /d I, ar b .u n its Fig. 12. Magnetic field dependences of dV/dI curves (solid lines) for a single-crystal MgB2—Cu 2.2 � junction along the ab plane with their BTK fittings (thin lines). Two separate sets of gap minima are clearly seen at low fields. –15 –10 –5 0 5 10 15 T, K 24 21 19 17 15 13 11 9 7 4.3 Voltage, mV d V /d I, ar b .u n its Fig. 13. Temperature dependences of dV/dI curves (solid lines) for the same junction as in Fig. 12 with their BTK fittings (thin lines). 5 10 15 20 25 300 2 4 6 8 � m e V T, K , Fig. 14. Temperature dependences of large and small superconducting energy gaps obtained by BTK fitting from Fig. 13. The solid lines represent BCS-like behavior. –60 –40 –20 0 20 40 60 dV /d I ar b. Voltage, mV B, T 8 5 4 3 2 1 0 u n its , Fig. 11. Large gap structure evolution for single crystal MgB2—Au 87 � junction in magnetic field at 4.2 K. The curves are shifted vertically for clarity. enables us to observe directly the most important E g2 phonon mode in the electron—phonon interaction within the � band. In single crystals the negative slope in dV/dI curve at large biases is observed quite often, which confirms that the disorder in the � band leads to quasi-localization of charge carriers. An example of this is already shown in Fig. 10. Figures 12 and 13 display a series of magnetic-field and temperature dependence of dV/dI curves with their BTK fittings. Here the two gaps are clearly visi- ble, corresponding to the theoretical prediction, in the ab direction Eq. (11). The temperature dependence of both gaps follows the BCS prediction (see Fig. 14). For temperatures above 25 K their behavior is un- known because this particular contact did not survive the measurements, likely due to thermal expansion of the sample holder. Figure 15 displays the magnetic-field dependences of large and small gaps. Surprisingly, the small gap value is not much depressed by a field of about 1 T, and the estimated critical field about 6 T is much higher, as stated in [44,45], although the intensity of the small-gap minima is suppressed rapidly by a field of about 1 T. Correspondingly, the small-gap contri- bution w* to the dV/dI spectra is decreased by mag- netic field significantly, from 0.92 to 0.16 (see Fig. 15), while w versus temperature even increases slightly from 0.92 at 4.3 K to 0.96 at 24 K (not shown). The area under the energy-gap minima in dV/dI V( ) is approximately proportional to the excess current Iexc (see Eq. (4)) at eV � (or roughly to the superfluid density). The excess current depends on the magnetic field with a positive overall curvature (Fig. 16). I Bexc( ) decreases abruptly at first and then more slowly above 1 T. This corresponds to a drastic depressing of the dV/dI V( ) small-gap-minima inten- sity by a magnetic field of about 1 T and to robustness 364 Fizika Nizkikh Temperatur, 2004, v. 30, No. 4 I.K. Yanson and Yu.G. Naidyuk 1 2 3 4 5 6 70 1 B, T I ,a rb .u n its e xc Fig. 16. I Bexc( ) (squares) for a MgB2—Cu junction from Fig. 12. The dashed lines show the different behavior of I Bexc( ) at low and high fields. 1 2 3 4 5 6 70 1 2 3 4 5 6 7 8 0 0.5 1.0 � L� L� S� S � , � , m eV B, T P ar tia lc o n tr ib u tio n o f s m al l g ap Fig. 15. Magnetic field dependences of the large and small superconducting energy gaps (solid triangles) ob- tained by BTK fitting from Fig. 12. Open triangles show the � value for large and small gap, respectively. The cir- cles demonstrate the depression of the small-gap contribu- tion to the dV/dI spectra by magnetic field. The lines connect the symbols for clarity. * w inversely depends on � value, therefore nearly constant w value between 0 and 1 T is due to the fact that � rises by factor 4 at 1 T. 0.2 0.4 0.6 0.8 1.00 0.5 1.0 � L2 ( 0 )/ � L2 (T ) T/ Tc Fig. 17. Temperature dependence of the penetration depth in the model of two independent BCS superconducting bands (dashed and dotted line) with different supercon- ducting gaps. The resulting penetration depth (solid line) clearly shows a non-BCS temperature behavior. The low-temperature behavior will be dominated by the band with the smaller superconducting gap. After Golubov et al. [46]. of the residual superconducting structure against fur- ther increase of magnetic field. This is a quite differ- ent dependence from what is expected for Iexc, which is in general proportional to the gap value (4). In contrast, I Texc( ) has mostly negative curvature and shape, similar to the BCS dependence. Often a positive curvature appears above 25 K (see e.g. Fig. 18). This kind of anomaly can be due to the two-band nature of superconductivity in MgB2, since the mag- netic field (temperature) suppresses the superconduc- tivity more quickly in the � band and then, at higher field (temperature), in the � band. The same consider- ation is valid for 1/ L� , which is roughly proportional to the «charge density of superfluid condensate». In the case of zero interband scattering, the simple model [46] predicts the temperature dependence shown in Fig. 17 for � and � parallel channels, which will yield a smooth curve with general positive curvature, taking into account the small interband scattering occurring in reality. If the �-band conductivity is blocked by a short mean free path, then the curvature of I Texc( ), being proportional to � �( )T , should be negative, which supplies us with additional confirmation of single- band conductivity along the � band. Thus, measuring the magnetic field and temperature dependences of Iexc can elucidate the contact structure. Figure 18 displays the temperature dependence of the gap for the dV/dI curves with a single-gap struc- ture, which vanishes around 25 K. A magnetic field of 1 T suppresses the gap minima intensity by factor of two, but the minima are clearly seen even at 4 T (not shown), the maximal field in this experimental trial. This excludes an origin of these gap minima due to a small gap. According to the calculation in [11] a large amount of impurity scattering will cause the gaps to converge to � � 4.1 meV and Tc to 25.4 K. Therefore these single-gap spectra reflect a strong interband scattering due to impurities, which likely causes a «semiconducting-like» behavior of dV/dI above Tc (see Fig. 18, inset). I Texc( ) behaves nearly as �( )T except in the region T 25 K, where Iexc is still non- zero because of a residual shallow zero-bias minimum in dV/dI above 25 K. 4.2. Phonon structure in the I—V characteristics PC EPI spectra of nonsuperconducting diborides. We have studied the PC EPI spectra d V/dI d I/dV2 2 2 2� � (see also Eq. (3)) of non- Advances in point-contact spectroscopy: two-band superconductor MgB2 Fizika Nizkikh Temperatur, 2004, v. 30, No. 4 365 R = 5.5 , T = 4.2 K� –120 –80 –40 0 40 80 120 Voltage, mV 1.0 0.5 0 –0.5 –1.0 d V /d I ,a rb .u n its 2 2 Fig. 19. Raw PC EPI spectrum for a ZrB2 5.5 � point contact at 4.2 K. After Naidyuk et al. [47]. –30 –20 –10 0 10 20 30 7 8 9 5 10 15 20 25 300 1 2 3 T, K 35 30 25 22 20 19 16 15 14 12 10 8 4.5 d V/ d I � Voltage, mV �BCS I exc � � m e V , I e xc T, K , ar b .u n its , , Fig. 18. Temperature dependence of a single supercon- ducting energy gap (squares) obtained by BTK fitting of dV/dI curves from inset. The solid lines represent BCS-like behavior. The triangles show the dependence of the excess current. Inset: dV/dI curves for a MgB2—Cu 8 � contact at different temperatures. 0.5 1.0 1.5 0 50 100 1.5 1.0 0.5 0 Energy, meV ZrB 2 2 0 50 100 W a ve ve ct o r, A – 1 � � p cF ( ), a rb . u n its K M � NbB2 Fig. 20. Comparison of high-resolution electron-en- ergy-loss spectroscopy measurements of surface phonon dispersion (bottom panels, symbols) [49] with the PC spectra for ZrB2 and NbB2 after subtraction of the rising background (upper panels). superconducting diborides MeB2 (Me = Zr, Nb, Ta) [47]. The cleanest sample we have is a ZrB2 single crystal, and its PC EPI spectrum is shown in Fig. 19. One recognizes a classical PC EPI spectrum from which one can estimate the position of 3 main phonon peaks and obtain the lower limit of EPI parameter �PC [47]. Essentially similar spectra were observed for other diborides, taking into account their purity and in- creased EPI, which leads to a transition from the spec- troscopic to a non-spectroscopic (thermal) regime of current flow [47]. The positions of the low-energy peaks are proportional to the inverse square root of the masses of the d metals [47], as expected. For these compounds the phonon density of states is measured by means of neutron scattering [48] and the surface phonon dispersion is derived by high-resolution elec- tron-energy-loss spectroscopy [49]. The positions of the phonon peaks or d /dq� = 0 for the dispersion curves correspond to maxima of the PC spectra (Fig. 20, Fig. 21). PC EPI spectra of MgB2 in c-axis oriented films. From the above considerations we had anticipated that one could easily measure the EPI spectral func- tion of MgB2 in the normal state, provided that the superconductivity is destroyed by magnetic field. Un- fortunately, that was not the case. The stronger we suppress the superconductivity in MgB2, the less traces of phonon structure remain in the I—V charac- teristic and its derivatives (Fig. 22) [23]. This is in odd in relation to the classical PCS, since the inelastic phonon spectrum should not depend on the state of electrodes in the first approximation (see Sec. Theoreti- cal background). Instead, most of the MgB2 spectra in the supercon- ducting state show reproducible structure in the phonon energy range (Fig. 23) which was not similar to the expected phonon maxima superimposed on the rising background. This structure disappears upon transition to the normal state. Quite interestingly the 366 Fizika Nizkikh Temperatur, 2004, v. 30, No. 4 I.K. Yanson and Yu.G. Naidyuk –80 –40 0 40 80 6 5 4 3 2 1MgB2–Ag B II ab Voltage, mV T, K B, T 4.2 0 4.2 4 10 4 20 4 30 4 40 0 V , ar b .u n its 2 Fig. 22. Phonon singularities in the PC spectra of a MgB2 thin-film—Ag junction as a function of magnetic field and temperature. T and B are shown beside each curve. After Yanson et al. [23]. –100 –50 0 50 100 –6 –4 –2 0 2 –20 –10 0 10 20 0.8 1.0 1.2 14 2 3 1 Voltage, mV 3 2 1 b V 2 � V R 0 � � mV 3 – 111; 4.86 2 – 43; 2.6 1 – 45; 2.1 V1 mV 1 – 3.31 2 – 2.78 3 – 2.5 3 2 1 a d V/ d I/ R N , , ,, Fig. 23. Superconducting gap features (upper panel) and phonon structure (bottom panel) in the spectra of thin- film MgB2—Ag junctions with different resistances at T 42. K, B 0. After Yanson et al. [23]. 0 0.01 0.02 0.03 0.04 PCS neutron NbB 2 Energy, meV 1/ m e V 0 20 40 60 80 100 120 140 0.01 0.02 0.03 0.04 PCS neutron TaB 2 G D O S , Fig. 21. Comparison of phonon DOS neutron measure- ments after Heid et al. [48] (symbols) with PC spectra for TaB2 and NbB2 after subtraction of the rising background (solid curves). intensity of this structure increases with increase of the value of the small gap, which means that the gap in the � band and observed phonon structure is con- nected [23]. Based on the theoretical consideration mentioned in the Introduction, we conclude that the disorder in the � band is so strong that it precludes ob- servation of the inelastic current, and the phonon non- linearities of the excess current (see Eq. (6)) play the main role, which does not depend on the scattering. Very rarely we did see signs that the observed char- acteristics indeed satisfy the conditions imposed on the inelastic PC spectra. One such example is shown in Figs. 24, 25. For this particular junction the super- conducting peculiarities are almost completely sup- pressed above 20 mV in moderate field (4 T). What remains is a very weak structure (� 1 %) from the rather high value of the gap (Fig. 24). The back- ground in dV/dI V( ) rises nearly quadratically up to a few % of R0 at large biases (� 100 mV). This leads to a linear background in V d V/dI V2 2 2� ( ) with phonon peaks superposed above the background both in nega- tive and positive bias polarity (compare with Fig. 19). The structure observed corresponds reasonably in shape above 30 meV to the phonon density of states (Fig. 25). At low voltages (below 30 meV), most probable, the gap peculiarities still prevail over d V/dI V2 2( ) structure. Thus, for this contact we as- sume to observe the inelastic PC spectrum for the � band, which should be compared to the Eliashberg EPI function for the same band calculated in Ref. 51 Advances in point-contact spectroscopy: two-band superconductor MgB2 Fizika Nizkikh Temperatur, 2004, v. 30, No. 4 367 20 40 60 80 1000 2 4 6 8 20 40 60 80 1000 1 2 3 �2 F (� ) Energy, meV �� �� �� �� Fig. 26. Calculated Eliashberg functions for the � and � bands (inset). After Golubov et al. [51]. 0 20 40 60 80 100 120 0.4 0.8 0 0.1 0.2 d (l n R )/ d V V – 1 ) PCS data Yildirim et al. Osborn et. al. Energy, meV , P h o to n d e n si ty o f s ta te s Fig. 25. Comparison of the PC EPI spectrum (upper panel) from Fig. 24 (after subtraction of the linear back- ground and zero-bias maxima below 25 meV) with the phonon DOS measured by neutron scattering [8,13] (bot- tom panel). 9.6 10.0 T = 4.2 K H = 4 T ad V /d I, � –120 –80 –40 0 40 80 120 –4 –2 0 2 4 b V 2 , � V Voltage, mV Fig. 24. dV/dI and V d V/dI2 2 2� curves for a thin-film MgB2—Ag junction revealing the inelastic PC spectrum for the � band. After Bobrov et al. [50]. (Fig. 26). Both the experimental spectrum and the �-band Eliashberg function do not show anomalously high intensity of the E g2 phonon mode, since only the Eliashberg function for � band is the principal driving force for high Tc in MgB2. The same conclusion should be ascribed to the excess-current phonon struc- ture, since it also corresponds to the � band. This band has much larger Fermi velocity and plasma frequency along the c axis compared to the � band [11]. Thus, in order to register the principal EPI with the E g2 phonon mode, we are faced with the necessity of measuring the PC spectra for only the � band. This can be done in a single crystal along the ab plane with blocked �-band conductivity. PC EPI spectra of MgB2 in the ab direction. The desired situation was described in Ref. 52 for a single crystal oriented in the ab plane. As was mentioned above, the nominal orientation of the contact axis to be parallel to ab plane is not enough to be sure that this situation occurs in reality. Moreover, even if one establishes the necessary orientation (i.e., contact axis parallel to ab plane) the spectra should reflect both bands with a prevalence of the undesired � band, be- cause due to spherical spreading of the current the orientational selectivity of a metallic point contact is much worse than that for the plane tunnel junction, where it goes exponentially. The large mixture of �-band contribution is clearly seen from the gap struc- ture in Fig. 27. Beyond the wings at the biases corres- ponding to the large gap (supposed to belong to the �-band gap) the deep minima located at the smaller gap (correspondingly to the �-band gap) are clearly seen (see bottom panel of Fig. 27). The EPI spectrum of the same junction is shown in the upper panel. One can see that the nonlinearities of the I–V characteristic at phonon biases are very small, and a reproducible struc- ture roughly corresponding to the Eliashberg EPI func- tion of the � band [38,51] appears in the bias range � 20–60 mV. Above 60 mV the PC spectrum broadens sufficiently hidden higher-lying phonon maxima. 368 Fizika Nizkikh Temperatur, 2004, v. 30, No. 4 I.K. Yanson and Yu.G. Naidyuk 0 50 100 150 0.3 0.6 40 K, 0T 4.5K, 9T(+) 4.5K, 8T 4.5K, 9T(–) –150 –100 –50 0 50 100 150 1.0 1.1 1.2 Voltage, mV 7.5 mV 4.5K, 9T 40 K, 0T 4.5K, 1.2T 4.5K, 0T V 2 � V , R 0–1 d V/ d I Fig. 28. V d V/dI2 2 2� (for two bias voltage polarities at 9 T) and dV/dI curves for a single-crystal MgB2—Cu junction (R0 72 . �) along the ab plane. Here the conduc- tivity along the � band prevails, as is shown by the pro- nounced large-gap structure for the zero-field dV/dI-curve at 4.5 K. After Naidyuk et al. [52]. –60 –40 –20 0 20 40 60 0.9 1.0 1.1 50 100 1500 0.2 0.4 0.6 2.7 mV Voltage, mV 40 K, 0T 4.5K, 8T 4.5K, 1T 4.5K, 0TR 0–1 d V/ d I T = 4.5 K, B = 8 T T = 40 K, B = 0 T �2 F(�) V 2 � V , Fig. 27. V d V/dI2 2 2� (for two bias voltage polarities) and dV dI/ curves for a single-crystal MgB2—Cu junc- tion ( . )R0 15 � along the ab plane. Here the conductivity along the � band prevails, as is shown by the pronounced small-gap structure for the zero-field dV/dI curve at 4.5 K. The � �2F( ) curve is the theoretical prediction for the �-band Eliashberg function from Fig. 26 (inset) smeared similarly to the experimental data. After Naidyuk et al. [52]. Even in the normal state (T Tc* ), where the excess current disappears, one can see the kink at � 20–30 meV, where the first peak of the phonon DOS and the first maximum of the Eliashberg EPI function of the � band occurs. At eV � 90–100 meV the PC EPI spectrum of Fig. 27 saturates just where the phonon DOS ends. At T Tc* intermediate phonon peaks are hardly seen, since the thermal resolution, which equals 5 44. k TB , amounts about 20 meV, and the re- gime of current flow is far from ballistic, due to the high background observed. No prevalence of the E g2 phonon mode is observed, like a big maximum of EPI at � 60–70 meV or a kink at T Tc* for these biases. A quite different spectrum is shown in Fig. 28, which is our key result. Consider first the dV/dI V( ) characteristics (see bottom panel). The energy gap structure shows the gap minima corresponding to the large gap (�-band gap). The increase of dV/dI V( ) at larger biases is noticeably larger than in the previous case (Fig. 27). One can notice that the relatively small magnetic field (�1 T) does not decrease the in- tensity of gap structure substantially, unlike those for Fig. 12, and even less than for Fig. 27. According to [43,44] a field of about 1 T should depress the small- gap intensity completely. All these facts evidence that we obtain a contact in which only the �-band channel in conductivity is oper- ated. Let us turn to the PC EPI spectra d V/dI V2 2( ), which are connected via the following expression to the second harmonic signalV2 recorded in experiment: 1 2 2 2 2 2 2 1 2R d V dI V V . Here R dV/dI and V1 is the rms value of the modu- lation voltage for the standard techniques of tunnel- ing and point-contact spectroscopy. The PC EPI spectra for this contact are shown in Fig. 28 (upper panel) for the highest field attainable in our experiments [52]. One can see that 8–9 T is still not enough to destroy completely the superconducti- vity in the energy-gap low bias range (0–30 meV), which can be taken as characteristic for a strongly superconducting � band. On the other hand, at larger biases no influence of field is noted, which evidences that this part of I–V characteristic does not contains superconducting peculiarities, likely due to the high current density in the contact. Except for a small asymmetry, the spectrum is reproduced for both polar- ities. Before saturation at biases * 100 meV, where the phonon DOS ends, a well-resolved wide bump occurs, which is located at about 60 meV. Further on, we will concentrate on this. First, we rescale it to the spectrum in R dR/dV0 1� units, in order to compare with the theoretical estima- tion. We will show that the bump is of spectroscopic Advances in point-contact spectroscopy: two-band superconductor MgB2 Fizika Nizkikh Temperatur, 2004, v. 30, No. 4 369 0 50 100 150 S pe ct ra �2F after Choi et al. harm. anharm. Voltage, mV PC data Calcul. thermal regime Fig. 29. Comparison of the experimental spectrum of Fig. 28 with the thermal spectrum for a model spectral function in the form of a Lorentzian at 60 meV with a width of 2 meV (dashed line) and with the theoretical EPI spectra (bottom curves). After Naidyuk et al. [52]. 0 50 100 150 0 0.2 0.4 –100 0 100 1.0 1.1 1.2 8T 9T 10 mV Voltage, mV 9T 1T 0T V 2 � V , R 0–1 d V/ d I Fig. 30. Thermal limit for the � band (as is shown by the pronounced large-gap structure for the zero-field dV/dI curve at 4.5 K) in the PC spectrum of a MgB2 single crys- tal along the ab plane. After Naidyuk et al. [52]. origin, i.e., the regime of current flow through the contact is not thermal, although the background at large biases (V * 100 meV) is high. To do so, we com- pare this bump with a PC spectrum in the thermal re- gime for a model EPI function, which consists of a Lo- rentzian at 60 meV with small (2 meV) width. Calculated according to Kulik [53], the thermal PC EPI spectrum is much broader, shown in Fig. 29 as a dashed line. Any further increase of the width of the model spectra will broaden the curve obtained. Com- paring the experimental and model spectra enable us to conclude that, in spite of the large width, the maxi- mum of the experimental spectra still corresponds to the spectroscopic regime. The high-temperature (T Tc* ) spectrum in Fig. 28 shows the smeared kink at about 60 meV, unlike that of Fig. 27. Introducing greater disorder in the boron plane by a fabrication procedure or by trying other spots on the side-face sur- face, the smeared thermal spectra were observed, coin- ciding in shape with the dashed curve in Fig. 30. In this figure another junction is shown, where the en- ergy gap structure also points to the �-band channel. Other junctions display the kink at about 30–40 meV, like the high-temperature spectrum in Fig. 27, which together with their energy-gap structure can be as- cribed to the thermal limit mainly in the � band, de- spite the rather low bath temperature. A PC spectrum with broad maxima including one at about 60 mV was observed in [45] on polycrystal- line MgB2 samples driven to the normal state by ap- plying moderate magnetic field and increasing the temperature. The large width of the EPI peak connected with the E g2 phonon mode (Fig. 28) is not surprising. Shukla et al. [14] measured the phonon dispersion curves along the �A and �M directions by means of inelastic x-ray scattering (see Fig. 4). The full width at half maximum for the E g2 mode along the �A direction amounts about 20–28 meV, which corresponds well to what we observe in the point-contact spectrum. If the phonon lifetime corresponds to this (inverse) energy, then the phonon mean free path is about equal to the lattice constant [52], and due to phonon reabsorption by accelerating electrons, we should anticipate a large background in the PC spectra as observed. If we com- pare the position of the bump (� 60 meV) with what is predicted for isotropic Eliashberg EPI function [19] (see Fig. 29), then we, together with Shukla et al., should admit that the phonon—phonon anharmoni- city is inessential for this mode, and its high width is due completely to the EPI. Now turn to the nonlinearity of the I–V curves due to the electron—phonon interaction, which can be es- timated from the dV/dI curves as about 10% for con- tact with the E g2 phonon modes in Fig. 28. This is comparable with the nonlinearity observed for nonsuperconducting diborides [47] with a small electron—phonon coupling constant. The reason for the relatively low nonlinearity of the I–V curves and low intensity of the principal E g2 phonon modes in the spectra for the MgB2 contacts can be the fact that anomalous strong interaction is characteristic for re- stricted group of phonons with sufficiently small wave vector [9], whereas in point-contact spectroscopy the large-angle scattering is underlined. 5. Conclusions We made an overview of the PCS investigations of c-axis oriented thin films and single crystals of MgB2. Our conclusions are as follows: 1. There are two different superconducting gaps in MgB2, which are grouped at 2.4 and 7.0 meV. Roughly, in half of all point contacts studied for c-axis oriented films the two gap structure merges together due to strong elastic scattering remaining a single gap at about 3.5 meV. 2. Anomalous temperature and especially magnetic field dependences of excess current in point-contact junctions reflect the two-band structure of the super- conducting order parameter in MgB2. 3. There are two mechanisms of revealing phonon structure in the point-contact spectra of MgB2: i) through the inelastic backscattering current, like for ordinary point-contact spectroscopy, and ii) through the energy dependence of the excess current, like in the similar tunneling spectroscopy of the elec- tron—phonon interaction. They can be discriminated by destroying the superconductivity with a magnetic field and/or temperature and by varying the electron mean free path. 4. The prevailing appearance of the E g2 boron mode, which mediates the creation of Cooper pairs, is seen in the PC spectra only along the a–b direction in accordance with the theory. The relatively small in- tensity of this mode in the PC spectra is likely due to their small wave vector and restricted phase volume. 5. Related diborides (ZrB2, NbB2, and TaB2) have d V/dI2 2 spectra proportional to the electron—pho- non interaction spectral function like that in common metals and a small EPI constant corresponding to their nonsuperconducting state. Acknowledgements The authors are grateful to N.L. Bobrov, P.N. Chubov, V.V. Fisun, O.E. Kvitnitskaya, L.V. Tyut- rina for collaboration during the MgB2 investigation. IKY thanks the Institute of Solid State Physics in For- 370 Fizika Nizkikh Temperatur, 2004, v. 30, No. 4 I.K. Yanson and Yu.G. Naidyuk schungzentrum Karlsruhe for hospitality, and Prof. H. von Löhneysen for constant support. The work in Ukraine was supported by the State Foundation of Fundamental Research under Grant F7/528-2001. Note added in proof After the paper was completed we have learned of the paper by Koshelev and Golubov [54], where the magnetic field dependence of � � and � � was pre- sented. 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