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the interplay between disorder and superconductivity has attracted sustained interest over the past few decades . in bcs superconductors , according to anderson s theorem , @xcite the superconducting critical temperature @xmath1 and the energy gap @xmath2 remain unaffected by the presence of weak disorder , although in unconventional superconductors both @xmath1 and @xmath2 are strongly suppressed with weak disorder . @xcite in the strongly disordered regime , things become interesting even in bcs superconductors : in two - dimensional ( 2d ) systems , strong disorder leads to a universal superconductor - to - insulator transition at a critical sheet resistance of @xmath3 ; @xcite furthermore , it has been found that , as @xmath4 tends to zero , there emerges a broad range of magnetic field where the resistivity remains finite but is much smaller than the normal - state value , which arguably signifies the importance of quantum phase fluctuations . @xcite in three - dimensional ( 3d ) systems , strong disorder leads to a disruption of the global superconducting coherence through reductions in both the superfluid density @xmath5 and pairing interactions , which results in the superconducting order parameter to spatially fluctuate and gives rise to anomalous electronic states near the superconductor - to - metal transition . @xcite experimentally , while the 2d systems have been actively studied in the past , @xcite the role of strong disorder in 3d superconductors is just beginning to be addressed with a modern viewpoint , @xcite which naturally calls for explorations of suitable materials for such a study .    in this paper , we report that agsnse@xmath6 , which is a low-@xmath7 conventional superconductor having a cubic structure , offers an ideal playground for studying the effects of strong disorder in 3d superconductors . we show that one can synthesize high - quality samples of agsnse@xmath6 presenting a very sharp superconducting transition , which indicates that morphologically the system is homogeneous ; nonetheless , strong electron scatterings that are intrinsic to this system lead to an extremely type - ii superconductivity with the ginzburg - landau parameter @xmath8 of as large as 55 . this causes the emergence of a broad range of magnetic fields where the resistivity remains finite but is much smaller than the normal - state value as @xmath9 , which is very much reminiscent of the behavior observed in disordered 2d superconductors @xcite or in quasi-2d systems like high-@xmath7 cuprates . @xcite furthermore , such an anomalous metallic state is found in the magnetic field range _ above _ the upper critical field determined thermodynamically from the specific heat . since this anomalous state can be easily reached with the magnetic field of 3 t and the microscopic mechanism of superconductivity is well understood in conventional superconductors , future studies of agsnse@xmath6 would help elucidate the roles of quantum phase fluctuations which remain controversial to date . the superconductivity in agsnse@xmath0 was discovered by johnston and adrian @xcite in 1977 as a possible valence - skipping superconductor . in valence - skipping compounds , the nominal valence state of a constituent element happens to be the skipped valence for that element . this characteristic has been of particular interest for superconductivity , since it may provide a means to realize the so - called negative-@xmath11 mechanism to enhance the superconducting transition temperature @xmath7 . @xcite well - known examples of such valence - skipping superconductors are babi@xmath12pb@xmath13o@xmath14 , @xcite ba@xmath12k@xmath13bio@xmath14 , @xcite and pb@xmath12tl@xmath13te . @xcite in all these compounds , @xmath7 is relatively high for their low carrier density . at the same time , valence fluctuations lead to significant electron scattering , making those superconductors highly disordered even though the matrix is homogeneous . agsnse@xmath0 adopts the face - centered cubic structure , in which the cation sites are randomly occupied by equal amount of ag and sn atoms . since the nominal valence of sn is 3 + which coincides with its skipped valence state , the sn ions should be separated into 1:1 mixture of sn@xmath15 and sn@xmath16 and the chemical formula of this material can be more appropriately expressed as ( ag@xmath17)(sn@xmath15)@xmath18(sn@xmath16)@xmath18(se@xmath19)@xmath0 , as evidenced by the measurements of the magnetic susceptibility @xcite and the @xmath20sn mssbauer spectra . @xcite despite the valence skipping nature and the carrier density of the order of 10@xmath21 @xmath22 , its @xmath7 of 4.7 k is not particularly high , which suggests that the negative-@xmath11 mechanism is not at work here . therefore , in this compound one can explore manifestations of strong disorder associated with valence fluctuations in the context of conventional superconductivity without being bothered by strong electron interactions .    . note that ag and sn atoms are randomly distributed in the cation sites . ( b ) xrd pattern of the agsnse@xmath0 sample . all the diffraction peaks can be well indexed using a cubic cell with the f__m__3__m _ _ space group . ( c ) temperature dependence of @xmath23 of the agsnse@xmath0 sample . lower inset shows an enlarged view of the data near the superconducting transition for @xmath23 ( right axis ) and for the dc magnetic susceptibility measured upon zero - field - cooling ( left axis , shown in terms of the shielding fraction ) . upper inset shows the magnetic - field dependence of @xmath24 at 5 k ; its slope gives @xmath25 = 2.0 @xmath26 10@xmath21 @xmath22 . , width=328 ] we synthesized high - quality ingots of polycrystalline agsnse@xmath0 by using a two - step method as described in ref . . first , high - purity shots of ag ( 99.999% ) , sn ( 99.99% ) , and se ( 99.999% ) with the stoichiometric ratio of 1:1:2 were melted in sealed evacuated quartz tube at 800 @xmath27c for 48 h with intermittent shaking to ensure homogeneity , followed by quenching to room temperature . the quartz tube containing the melt - quenched ingot was subsequently annealed at 450 @xmath27c for two weeks , and then quenched into cold water . the structure of the resulting sample was characterized by powder x - ray diffraction ( xrd ) . as can be seen in fig . 1(b ) , the xrd pattern shows sharp diffraction peaks consistent with the face - centered cubic structure , giving the lattice parameter @xmath28 of 5.675(1 )  which agrees with the literature . @xcite    the ingot was cut into bar - shaped samples , and the temperature - dependent magnetization @xmath29 under various magnetic fields was measured with a vibrating sample magnetometer ( vsm ) , while the magnetization curves at fixed temperatures were measured with a commercial squid magnetometer ( quantum design mpms- 1 ) . the demagnetization effect was corrected for by considering the slab geometry . @xcite the resistivity @xmath23 and the hall resistivity @xmath24 were measured by using a standard six - probe method where the contacts were made by spot - welding gold wires . the specific heat @xmath30 was measured with a relaxation - time method using quantum design ppms-9 . for consistency reasons , all the data presented here were measured on exactly the same sample . figure 1(c ) shows the temperature dependence of @xmath23 , which exhibits only a weak metallic temperature dependence . this is qualitatively similar to that reported @xcite for ag@xmath31sn@xmath32se@xmath0 and signifies the presence of intrinsically strong disorder that does not change much with the ag / sn ratio . as shown in the upper inset of fig . 1(c ) , @xmath24 is linear in @xmath33 and its slope at 5 k corresponds to an electron density @xmath34 of 2.0 @xmath26 10@xmath21 @xmath22 . the superconducting transition in @xmath23 occurs sharply between 4.5 and 4.6 k , and the magnetic susceptibility shows the onset @xmath7 of 4.55 k , which corresponds to the midpoint of the resistive transition . the superconducting shielding faction achieves nearly 100% after the demagnetization correction . there is no resistivity upturn at low temperatures , giving no clear sign of anderson localization . measured in 0 and 9 t. inset shows @xmath35 plotted as a function of @xmath36 for the 9-t data , where the superconductivity is completely suppressed ; the solid line presents the best fit to the data using the debye model ( see ref . ) . ( b ) temperature dependence of @xmath37 in 0 t. the dash - dotted line is the calculated @xmath37 curve given by the weak - coupling bcs theory ; the solid line is the best fit using a modified bcs theory with @xmath38 = 1.925 . the horizontal solid line denotes @xmath39 . , width=328 ]      figure 2(a ) shows the temperature dependences of @xmath35 measured under 0 and 9 t. application of a 9 t field completely suppresses the superconductivity , enabling us to determine the phononic contribution and the normal - state parameters such as the electronic specific - heat coefficient @xmath39 and the effective mass @xmath40 . @xcite figure 2(b ) shows the temperature dependence of the electronic specific heat @xmath37 in 0 t obtained by subtracting the phononic contribution determined from the 9-t data . one can see that @xmath37 quickly approaches zero at low @xmath4 , indicating a fully - gapped nature . as for the precise temperature dependence , although the simple weak - coupling bcs model ( dash - dotted line ) fails to describe the @xmath37 data , a modified bcs model @xcite that allows the coupling constant @xmath41 to vary [ @xmath42 is the gap at 0 k ] can fit our data well with @xmath38 = 1.925 ( solid line ) , which is only slightly larger than the weak - coupling bcs value of 1.764 .    using @xmath40 and @xmath38 obtained from @xmath43 , @xcite the pippard coherence length @xmath44 can be estimated as @xmath44 = @xmath45 = 181 nm . this is to be contrasted with the mean free path @xmath46 = @xmath47 = 0.9 nm , where @xmath48 is the residual resistivity . this @xmath46 is less than @xmath49 and is very short , pointing to the existence of strong scattering which is consistent with the valence fluctuations of sn . the ratio @xmath50/@xmath44 = 0.005 means that agsnse@xmath0 is in the very dirty limit . yet , @xmath51 is estimated to be @xmath528 and hence the system is still a good metal . note that the estimations of @xmath40 , @xmath53 , @xmath44 , and @xmath46 in this paper are based on the free electron theory ; once the band structure of agsnse@xmath0 is known , those estimations can be made more accurate . curves measured after zero - field cooling to various temperatures . the dashed line marks the initial linear behavior . ( b ) plot of @xmath54 vs @xmath4 ; the solid line is a fit to the data using the local dirty limit formula . the @xmath55(0 ) obtained after the demagnetization correction is marked by an arrow . inset shows the @xmath33 dependences of @xmath56 @xmath57 @xmath58 , where @xmath59 denotes the initial meissner contribution ; the data are shifted vertically for clarity . the determined @xmath60 value for each temperature is shown by arrows . ( c ) temperature dependences of @xmath61 under various magnetic fields . the @xmath62 values are indicated by arrows . ( d ) @xmath63 vs @xmath4 phase diagram ; the solid line is a whh fit to the data . , width=328 ]      figure 3 summarizes the results of the magnetization measurements to extract the values of the upper and lower critical fields . figure 3(a ) shows @xmath64 curves measured after zero - field cooling to various temperatures . we define @xmath54 at each temperature as the value at which the @xmath64 data deviates from its initial linear behavior , as can be more clearly seen in the inset of fig . . extrapolation @xcite of the resulting @xmath65 data to @xmath4 = 0 k yields @xmath66 = 1.18 mt which , after correcting for the demagnetization effect , @xcite gives the lower critical field at @xmath4 = 0 k , @xmath67 , of 1.58 mt . figure 3(c ) shows the temperature dependences of the magnetization near the superconducting transition under various magnetic fields . a paramagnetic background with a temperature - independent susceptibility @xmath68 = 4.2 @xmath26 10@xmath69 emu / g has been subtracted from the data , @xcite and what is plotted is @xmath70 . a gradual suppression of superconductivity with increasing magnetic field is evident . for each magnetic field , we determined the onset temperature of the transition , @xmath62 , from the intersection of the linear extrapolation of the @xmath71 data to the @xmath72 = 0 line , as indicated by the arrows in fig . the obtained @xmath62 data shown in fig . 3(d ) give the temperature dependence of the magnetically - determined upper critical field , @xmath73 , which is extrapolated to @xmath4 = 0 k using the werthamer - helfand - hohenberg ( whh ) theory @xcite to yield @xmath74 = 2.26 t.     vs @xmath4 phase diagram ; the solid line is a whh fit to the data . for comparison , @xmath73 and its whh fitting are shown with filled squares and a dashed line , respectively . , width=226 ]    to evaluate the upper critical field at low temperatures in a more direct way , we measured the superconducting transition in specific heat under various magnetic fields . figure 4(a ) shows representative data of @xmath37 vs @xmath4 measured in 0 , 0.25 , 1.0 , and 1.5 t , where the superconducting transition is well resolved . in higher magnetic fields , the specific - heat anomaly becomes weaker , as shown in fig . 4(b ) for 1.75 , 1.9 , and 2.0 t ; here , the transition can still be seen in 1.9 t , but no clear anomaly is discernible at 2.0 t. these data demonstrate that the upper critical field is thermodynamically well - defined in agsnse@xmath0 at low temperature and that it lies around 2 t near @xmath4 = 0 k.    to quantitatively analyze the superconducting transition in the specific - heat data , for each magnetic field we determined three characteristic temperatures : onset of the transition ( below which the specific heat grows ) , midpoint of the transition , and the peak in @xmath75 . it is useful to note that the transition width @xmath76 , which we define by the difference between the onset and peak temperatures , remains reasonably narrow with @xmath77 k at all temperatures . we choose to use the midpoint of the specific - heat transition for defining the transition temperature @xmath78 for each magnetic field , which in turn gives the temperature dependence of the upper critical field @xmath79 . the solid line in fig . 4(c ) is the fitting of the whh theory to the @xmath79 data , which gives the @xmath4 = 0 k value @xmath80 of 2.11 t. for comparison , we reproduced in fig . 4(c ) the @xmath81 data obtained from the magnetization measurements . the difference in the whh fittings for the specific - heat and magnetization data is only 7% , which indicates that the two thermodynamical measurements are essentially consistent . in the following discussions , we take @xmath82 as the thermodynamically - determined @xmath83 .      now we estimate various superconducting parameters of agsnse@xmath0 by using @xmath84 as the zero - temperature upper critical field @xmath85 . note that since agsnse@xmath0 has cubic symmetry , no anisotropy is expected . the ginzburg - landau ( gl ) coherence length @xmath86 is estimated as @xmath87(0 ) = @xmath88 = 12.5 nm , where @xmath89 = 2.07 @xmath26 10@xmath90 wb is the flux quantum . alternatively , one may also estimate @xmath87 from @xmath86 @xmath91 @xmath92 for dirty superconductors , which yields @xmath87 @xmath91 12.7 nm . the consistency is gratifying . the equation @xcite @xmath55(0)/@xmath93(0 ) = @xmath94 gives the gl parameter @xmath95 = 55 , which in turn allows us to calculate the effective penetration depth @xmath96 = 685 nm through @xmath97 = @xmath98 . this gives the london penetration depth @xmath99 @xmath91 @xmath100 = 48.3 nm .    since agsnse@xmath0 is a very dirty superconductor , the superfluid density @xmath101 is expected to be reduced by a factor of @xmath52 @xmath102 . @xcite indeed , the calculation of @xmath101 based on @xmath101 = @xmath103 yields @xmath101 @xmath104 9 @xmath26 10@xmath105 @xmath22 , which is close to the estimation of @xmath101 @xmath104 @xmath106 = 1 @xmath26 10@xmath107 @xmath22 , reassuring the applicability of the bcs theory to agsnse@xmath0 and the soundness of our @xmath108 measurements which can be adversely affected by flux pinning .     that is used for determining @xmath109 , @xmath110 , and @xmath111 are indicated by dashed lines . , despite the conventional nature of the superconductivity in agsnse@xmath0 , nontrivial physics becomes evident when one looks at the magnetic - field - induced resistive transition at fixed temperatures ( fig . one can easily see that the resistive transition is very sharp at temperatures close to @xmath7 , but it becomes increasingly broader at lower temperatures . we determined the values of the magnetic field at which @xmath23 recovers a certain percentage of the normal - state resistivity @xmath112 for each temperature , and denote them @xmath113 , @xmath114 , _ etc . _ we also determined the irreversibility field @xmath115 to mark the onset of a finite resistivity within our sensitivity limit , which corresponds to 0.15% of @xmath112 . the results are shown in fig . 6 , where we also plotted the thermodynamically - determined @xmath116 with a solid line . usually , either @xmath114 or @xmath117 are considered to give a measure of the upper critical field . @xcite in the present case , for temperatures down to @xmath118 , @xmath114 agrees reasonably well with the thermodynamically determined @xmath83 , but at lower temperatures the resistive transition occurs largely above @xmath83 . this means that at low temperature , there is a wide range of magnetic field above @xmath83 where the resistivity is finite but is noticeably smaller than @xmath112 . the existence of such an anomalous metallic state above @xmath83 has never been documented for a conventional 3d superconductor , and this is the main finding of the present work . note that the pauli paramagnetic limit is estimated to be 8.4 t , which is well above the anomalous metallic regime observed here . determined thermodynamically from specific - heat measurements ( solid line ) together with @xmath119 , @xmath109 , @xmath110 , and @xmath111 obtained from the resistive transition data . , width=245 ]    similar broadening of the magnetic - field - induced resistive transition at low temperatures has been observed in high - temperature superconductors such as bi@xmath6sr@xmath6cuo@xmath120 , @xcite tl@xmath6ba@xmath6cuo@xmath120 , @xcite mgb@xmath6 , @xcite and ba@xmath12k@xmath121bio@xmath122 . @xcite this phenomenon is generally believed to be caused by enhanced vortex fluctuations and the resulting vortex liquid phase due to a short coherence length that is inherent to high - temperature superconductors . in light of this common belief , the present observation in agsnse@xmath0 is striking . here the coherence length is not so short , 12.1 nm , and the system is 3d , both of which work against strong vortex fluctuations . therefore , our result calls for a reinterpretation of the low - temperature broadening of the magnetic - field - induced resistive transition .      to understand the above results , an interesting possibility is the scenario proposed by spivak , oreto and kivelson @xcite that the strong disorder leads to formations of superconducting  puddles " that are josephson - coupled to each other , and quantum fluctuations of the relative phase between those puddles give rise to a broad range of magnetic fields where the resistivity remains finite and smaller than @xmath112 as @xmath9 . if this is indeed the case , our data give evidence for the important role of quantum phase fluctuations in disordered 3d conventional superconductors . since the superconductivity mechanism is well understood here ( as opposed to the case of high - temperature superconductors ) and the required magnetic field to access the quantum fluctuation regime is low ( only @xmath523 t ) , agsnse@xmath0 provides a convenient platform for studying the role of quantum fluctuations in the magnetic - field - induced superconductor - to - metal transition . another anomalous behavior observed here is that at low temperatures , not only @xmath111 but also @xmath119 show little tendency toward saturation . a similar behavior was previously observed in strongly - disordered amorphous superconductors @xcite and was discussed to be due to the weak - localization effect . @xcite however , the discrepancy between the resistive transition and the thermodynamically - determined @xmath83 makes the conclusion of those old studies @xcite questionable . hence , the physical meaning of the resistive transition in disordered superconductors had better be reconsidered by taking into account the important role of quantum phase fluctuations . we have fully characterized the agsnse@xmath0 superconductor where its valence - skipping nature leads to intrinsically strong electron scattering without any obvious enhancement of @xmath7 . at low temperature , we observed an anomalous broadening of the magnetic - field - induced resistive transition which occurs largely above the thermodynamically - determined @xmath83 . this means the existence of an anomalous metallic state above @xmath83 where quantum phase fluctuations associated with a magnetic - field - induced superconductor - to - metal transition are likely to be playing an important role . this makes agsnse@xmath0 a useful platform for studying the role of quantum fluctuations in 3d disordered superconductors .                                                                        as can be seen in the inset of fig . 2(a ) , the 9 t data below 5 k can be well described by the debye model @xmath35 = @xmath39 + @xmath123@xmath36 + @xmath124 , where @xmath39 is the normal - state electronic specific - heat coefficient and @xmath125 ( @xmath126 = 3 , 5 ) denote the coefficients of the phononic contribution . the obtained parameters are @xmath39 = 3.5 mj / molk@xmath127 , @xmath123 = 0.7 mj / molk@xmath128 , and @xmath129 = 0.009 mj / molk@xmath130 . the @xmath123 value gives the debye temperature @xmath131 = 220 k via @xmath131 = ( 12@xmath132@xmath128@xmath133/5@xmath123)@xmath134 with @xmath135 = 4 and @xmath136 = 8.314 j / molk . in addition , if one assumes a single spherical fermi surface , the effective mass @xmath40 can be estimated as @xmath40 = @xmath137 @xmath91 1.5@xmath138 , where @xmath139 = 55 @xmath140/mol and @xmath138 is the free electron mass .
we report superconducting properties of agsnse@xmath0 which is a conventional type - ii superconductor in the very dirty limit due to intrinsically strong electron scatterings . 
 while this material is an isotropic three - dimensional ( 3d ) superconductor with a not - so - short coherence length where strong vortex fluctuations are _ not _ expected , we found that the magnetic - field - induced resistive transition at fixed temperatures becomes increasingly broader toward zero temperature and , surprisingly , that this broadened transition is taking place largely _ above _ the upper critical field determined thermodynamically from the specific heat . 
 this result points to the existence of an anomalous metallic state possibly caused by quantum phase fluctuations in a strongly - disordered 3d superconductor .