Spin-Fluctuation-Induced Non-Fermi-Liquid Behavior with suppressed superconductivity in LiFe 1−x Co x As
Y. M. Dai, H. Miao, L. Y. Xing, X. C. Wang, P. S. Wang, H. Xiao, T. Qian, P. Richard, X. G. Qiu, W. Yu, C. Q. Jin, Z. Wang, P. D. Johnson, C. C. Homes, H. Ding
SSpin-fluctuation induced non-Fermi liquid behavior with suppressed superconductivityin LiFe − x Co x As Y. M. Dai, H. Miao, L. Y. Xing, X. C. Wang, P. S. Wang, H. Xiao, T. Qian, P. Richard,
2, 4
X. G. Qiu, W. Yu, C. Q. Jin,
2, 4
Z. Wang, P. D. Johnson, C. C. Homes, ∗ and H. Ding
2, 4, † Condensed Matter Physics and Materials Science Department,Brookhaven National Laboratory, Upton, New York 11973, USA Beijing National Laboratory for Condensed Matter Physics,Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Department of Physics, Renmin University of China, Beijing 100872, China Collaborative Innovation Center of Quantum Matter, Beijing, China Department of Physics, Boston College, Chestnut Hill, Massachusetts 02467, USA (Dated: September 24, 2018)A series of LiFe − x Co x As compounds with different Co concentrations have been studied bytransport, optical spectroscopy, angle-resolved photoemission spectroscopy and nuclear magneticresonance. We observed a Fermi liquid to non-Fermi liquid to Fermi liquid (FL-NFL-FL) crossoveralongside a monotonic suppression of the superconductivity with increasing Co content. In paral-lel to the FL-NFL-FL crossover, we found that both the low-energy spin fluctuations and Fermisurface nesting are enhanced and then diminished, strongly suggesting that the NFL behavior inLiFe − x Co x As is induced by low-energy spin fluctuations which are very likely tuned by Fermi sur-face nesting. Our study reveals a unique phase diagram of LiFe − x Co x As where the region of NFLis moved to the boundary of the superconducting phase, implying that they are probably governedby different mechanisms.
PACS numbers: 74.70.Xa,74.25.Dw,71.10.HfKeywords: Condensed Matter Physics, Superconductivity
The normal state of high-temperature (high- T c ) super-conductors is very unusual, with the electrical resistivity(or quasiparticle scattering rate) varying with tempera-ture in a peculiar way that deviates significantly from thequadratic T dependence expected from Landau’s Fermi-liquid (FL) theory of metals [1–4]. Because this anoma-lous non-Fermi-liquid (NFL) behavior is often revealedexperimentally above a superconducting dome, there is abroad consensus that the origin of the NFL behavior mayhold the key to understanding the pairing mechanism ofhigh- T c superconductivity.Studies on high- T c cuprate superconductors [1–3],heavy-fermion metals [4, 5], organic Bechgaard salts [6] aswell as the newly-discovered iron-based superconductors(IBSCs) [7–10] have shown that the NFL behavior andhigh- T c superconducting dome favor proximity to mag-netic order. This fact has led to proposals ascribing boththe NFL behavior and high- T c superconductivity to spinfluctuations close to a magnetic quantum critical point(QCP) [11–13]. However, a growing number of experi-ments do not agree with these scenarios. For example, arecent magnetotransport study has shown that by dop-ing CeCoIn with Yb, the field-induced QCP is fully sup-pressed while both the NFL behavior and superconduc-tivity are barely affected [14]. At the current time, themicroscopic mechanism of the NFL behavior and its re-lationship to high- T c superconductivity are still a matterof considerable debate.IBSCs feature an intricate phase diagram with NFLbehavior, superconducting phase, magnetic order, struc- tural transition, possible QCP(s) and nested Fermi sur-faces interacting with each other. This complexity makesit quite challenging to distinguish the roles played by dif-ferent orders or interactions. The LiFe − x Co x As systempresents a simple phase diagram: LiFeAs exhibits su-perconductivity with a maximum transition temperature T c ≈
18 K in its stoichiometric form [15]. The substitu-tion of Fe by Co results in a monotonic lowering of T c ;neither magnetic nor structural transitions have been de-tected in the temperature–doping ( T – x ) phase diagramof LiFe − x Co x As [16]. The normal state of LiFeAs is aFL, as evidenced by the quadratic T dependence of thelow-temperature resistivity [17, 18]. Such a simple phasediagram makes LiFe − x Co x As an excellent system to elu-cidate the origin of the NFL behavior and its relationshipto superconductivity.In this article, through a combined study of trans-port, optical spectroscopy, angle-resolved photoemissionspectroscopy (ARPES) and nuclear magnetic resonance(NMR) on LiFe − x Co x As, we found that while super-conductivity is monotonically suppressed with increasingCo concentration, the transport and optical propertiesreveal a prominent FL-NFL-FL crossover which closelyfollows the doping evolution of low-energy spin fluctu-ations (LESFs) and Fermi surface nesting. Our obser-vations provide clear evidence that LESFs, which arelikely tuned by FS nesting, dominate the normal-statescattering, and are thus responsible for the FL-NFL-FLcrossover in LiFe − x Co x As. A unique phase diagram ofLiFe − x Co x As derived from our studies shows that theTypeset by REVTEX a r X i v : . [ c ond - m a t . s up r- c on ] S e p T (K) ( c m ) ( c m ) ( c m ) T (K) T (K) T (K) T (K) ( c m ) ( c m ) (a) LiFe Co x Asx = 0 ~ T ~ T (b) x = 0.06 ~ T (c) x = 0.12 ~ T (d) x = 0.26 ~ T (e) x = 0.4 ( c m ) (f) n Figure 1. (color online) (a)–(e) Resistivity as a functionof temperature ρ ( T ) (open circles) for LiFe − x Co x As at fiveselected Co concentrations. For each material, ρ ( T ) is fit to asingle power law ρ ( T ) = ρ + AT n (solid lines). (f) Evolutionof the exponent n (solid circles) and residual resistivity ρ (solid diamonds) with Co substitution x . NFL behavior is decoupled from superconductivity, sug-gesting that they do not share the same origin.High quality single crystals of LiFe − x Co x As withdifferent Co concentrations were grown by a self-fluxmethod [15]. Details of the sample synthesis and exper-imental methods for all the techniques we used in thiswork are included in Appendixes A to E.Figures 1(a)–1(e) show the T -dependent resistivity ρ ( T ) for five representative dopings. For each doping, ρ ( T ) is fit to a single power law ρ ( T ) = ρ + AT n , re-turning the exponent n and the residual resistivity ρ .The evolution of n and ρ with doping are summarizedin Fig. 1(f) as solid circles and solid diamonds, respec-tively. In LiFeAs ( x = 0), n = 2, i.e. the resistivity variesquadratically with temperature, indicating a FL normalstate, in agreement with previous transport studies onLiFeAs [17, 18]. With increasing Co doping, n decreases,reaching a minimum of 1.35 at x ≈ .
12. With furtherdoping ( x > . n begins to increase and recovers toa value of 2 again at about x = 0 . ρ ( T ) and the singlepower-law fitting results for more dopings are displayed inAppendix B (Fig. 5). To present the crossover behaviormore clearly, ρ ( T ) is plotted as a function of T n , as shownin Appendix B (Fig. 6), where a linear behavior can beseen for all the dopings. This doping dependence of n is an explicit indication of a doping-induced FL-NFL-FLcrossover in LiFe − x Co x As. In addition, we note that, asshown in Fig. 1(f), ρ increases with doping all the wayto x = 0 .
4, while n exhibits a FL-NFL-FL crossover inthe same doping range. This indicates that the FL-NFL-FL crossover in LiFe − x Co x As is not tied to the impuritylevel.Further evidence for the FL-NFL-FL crossover can berevealed by the T dependence of the quasiparticle scatter-ing rate obtained via optical spectroscopy [19, 20]. Fig-ure 2(a) displays the real part of the optical conductivity σ ( ω ) for LiFeAs ( x = 0) at 100 K. The low-frequency -1 ) / c o ( c m - ) / c o ( c m - ) / c o ( c m - ) / c o ( c m - ) / c o ( c m - ) () ( - c m - ) LiFe Co x As T = 100 K(a) x = 0 0 50 100 1500100200 (b) co ~ T x = 0 co ~ T x = 0.06 T (K) T (K) T (K) T (K)(d) co ~ T x = 0.12 T (K)0 50 100 150150200250 (e) co ~ T x = 0.26 co ~ T x = 0.4 Figure 2. (color online) (a) The thick blue curve is the realpart of the optical conductivity σ ( ω ) of LiFeAs ( x = 0) mea-sured at 100 K. The thin red curve through the data is theDrude-Lorentz fit which consists of the contributions from acoherent narrow Drude (red shaded region), a nearly incoher-ent broad Drude (green shaded region) and series of Lorentzcomponents (blue shaded region). (b)–(f) T dependence ofthe quasiparticle scattering rate 1 /τ co derived from the co-herent narrow Drude component for five Co concentrations. σ ( ω ) is dominated by the well-known Drude-like metallicresponse, where the width of the Drude peak at half max-imum gives the value of the quasiparticle scattering rate.In order to accurately extract the quasiparticle scatter-ing rate, we fit the measured σ ( ω ) to the Drude-Lorentzmodel: σ ( ω ) = 2 πZ (cid:88) k ω p,k τ k ( ω + τ − k ) + (cid:88) j γ j ω Ω j ( ω j − ω ) + γ j ω , (1)where Z (cid:39)
377 Ω is the impedance of free space. Thefirst term describes a sum of delocalized (Drude) car-rier responses with ω p,k and 1 /τ k being the plasma fre-quency and scattering rate in the k th Drude band, re-spectively. In the second term, ω j , γ j and Ω j are theresonance frequency, width and strength of the j th vibra-tion or bound excitation. As shown in Fig. 2(a), the thinred line through the data represents the fitting result forLiFeAs at 100 K which is decomposed into a coherent nar-row Drude, a nearly incoherent broad Drude, and seriesof Lorentz components, consistent with previous opticalstudies on IBSCs [19–22]. Fitting results for other dop-ings at several representative temperatures can be foundin Appendix C [Fig. 7(f)–7(j)]. Tu et al. suggest thatit is more appropriate to describe the broad Drude com-ponent as bound excitations [21] because the mean freepath l = v F τ ( v F is the Fermi velocity) associated withthe broad Drude component is close to the Mott-Ioffe-Regel limit. In any event, since the broad Drude com-ponent only gives rise to a T -independent backgroundcontribution to the total σ ( ω ), the T dependence ofthe optical response is governed by the coherent nar-row Drude component. As a result, the nature of thebroad Drude term does not affect our analysis of the co- -0.50.00.5 k y ( π / a ) x = 0.26-0.50.00.5 k y ( π / a ) x = 0-0.50.00.5 k y ( π / a ) k x ( π /a) x = 0.4-0.50.00.5 k y ( π / a ) x = 0.12 -0.5 0.0 0.5 k x ( π /a) -0.50.00.5 k y ( π / a ) x = 0.06 0.50.40.30.20.1 / T T ( s - K - ) x : T T | T = 20 K : F ( Q ) (Normalized)0.90.60.3 / T T ( s - K - ) T (K) H//ab LiFe Co x As x = 0 x = 0.06 x = 0.12 x = 0.26 x = 0.4 F ( Q ) ( N o r m a li z ed ) LiFe Co x As -0.50.00.5 k y ( π / a ) k x ( π /a) θ θ k Fhole ( θ ) k Fele ( θ )Q = ( π , 0) 1.0 0.8 0.6 0.4 0.2 (a) (f)(b) (g)(c) (h)(d) (i)(e) (j) (k)(l)(m) Figure 3. (color online) (a)–(e) FS contour of LiFe − x Co x Asfor five representative Co concentrations, determined by inte-grating the ARPES spectral intensity within ±
10 meV withrespect to E F . (f)–(j) Extracted FSs from correspondingARPES measurements for each doping. (k) Definition of k hole F ( θ ), k ele F ( θ ) and Q in the k -space. (l) Spin-lattice relax-ation rate As 1 /T T as a function of temperature for fiveCo concentrations measured by NMR. (m) Evolution of As1 /T T at 20 K (solid squares) and FS nesting factor (solidcircles) with increasing Co concentration. herent narrow Drude component and the T dependenceof σ ( ω ). The application of the Drude-Lorentz analysisat all the measured temperatures for five representativedopings yields the T dependence of the scattering rateof the coherent narrow Drude component 1 /τ co , shownin Fig. 2(b)–2(f). For each doping, 1 /τ co follows the ex-pression 1 /τ co = 1 /τ + BT α with the exponent α ≈ n ,where n is the exponent determined from the fit to ρ ( T )for the corresponding dopings. Again, we plot 1 /τ co asa function of T α in Appendix C [Fig. 7(k)–7(o)], whichreveals distinct linear behavior for all the dopings. Suchcrossover behavior of α provides further evidence for thedoping-induced FL-NFL-FL changes in LiFe − x Co x As.In order to gain insight into the origin of the anomalousFL-NFL-FL crossover in LiFe − x Co x As, we examine theevolution of LESFs with Co concentration by looking intothe nuclear spin-lattice relaxation rate 1 /T T , reflect-ing the summation of all different q modes of sub-meVLESFs weighted by a nearly uniform form factor, whichcan be determined from As nuclear magnetic resonance(NMR) measurements [8, 23]. Figure 3(l) displays the T dependence of 1 /T T for five different dopings. In LiFeAs( x = 0), 1 /T T is almost T independent, and a negligibleupturn at low temperature indicates weak LESFs. How-ever, in sharp contrast to LiFeAs, 1 /T T for x = 0 .
12 in- creases rapidly upon cooling and a prominent upturn de-velops at low temperature, implying that LESFs are sig-nificantly enhanced. For x = 0 .
26, the upturn in 1 /T T at low temperature becomes less prominent, suggestingthat the LESFs diminish again. The low-temperatureupturn in 1 /T T can be empirically described by theCurie-Weiss expression 1 /T T = A + BT + C/ ( T + θ )[solid lines in Fig. 3(l)], in good agreement with previ-ous NMR studies [8, 23]. To quantify the doping depen-dence of the LESFs, we take the value of 1 /T T at 20 K(just above T c of LiFeAs), 1 /T T | T =20 K , for each dop-ing and plot them as a function of x [solid squares inFig. 3(m)]. Upon doping, 1 /T T | T =20 K first grows butthen drops, resulting in a peak at x = 0 .
12. The link be-tween the LESFs and the NFL behavior can be revealedby comparing the doping dependence of n [solid circles inFig. 1(f)] and 1 /T T | T =20 K [solid squares in Fig. 3(m)].Below x = 0 .
12, the enhancement of LESFs (increasein 1 /T T | T =20 K ) leads to a more conspicuous deviationfrom a FL (decrease in n ) while above x = 0 .
12, the re-duction of LESFs (decrease in 1 /T T | T =20 K ) results in agradual recovery of the FL behavior (increase in n ). Themost robust NFL behavior ( n ≈ .
35) occurs at x = 0 . /T T | T =20 K peaks).These observations strongly suggest that the FL-NFL-FL crossover in LiFe − x Co x As is governed by LESFs.Since an investigation into the FS may provide infor-mation on the nature of the LESFs, we then studied theevolution of the FSs in LiFe − x Co x As. The FS contourof LiFe − x Co x As is traced out from the ARPES intensityplot near the Fermi energy ( E F ) for five representativedopings [Fig. 3(a)–3(e)]. The extracted FSs for each dop-ing are shown in Fig. 3(f)–3(j). For LiFeAs ( x = 0), twohole and two electron FS pockets are observed at the Γand M points, respectively [Fig. 3(a)], in accord withprevious ARPES studies [24, 25]. The inner hole pocketis quite small, while the outer hole pocket is much largerthan the electron pockets, resulting in a poor nestingcondition in LiFeAs [Fig. 3(f)]. With Co doping, theelectron pockets expand while the hole pockets shrink.Consequently, the FS nesting is improved. As shown inFig. 3(h), the shape of the hole FS matches the outercontour of the two electron FSs at x ≈ .
12. FurtherCo doping ( x > .
12) makes the electron and hole pock-ets mismatched again, e.g. x = 0 . Q F ( Q ) = (cid:88) i,j (cid:90) π (cid:107) k ele i F ( θ ) − k hole j F ( θ ) − Q (cid:107) + δ dθ (2)where the definitions of k F ele ( θ ), k F hole ( θ ) and Q areillustrated in Fig. 3(k); δ is a small positive number toavoid singular behavior; for an n -dimensional vector x , || x || = √ x · x = (cid:112)(cid:80) ni =1 ( x i ) . F ( Q ) increases as theFS nesting is improved, and is maximized when the holeFS matches the outer contour of the two electron FSs.Assuming v F is uniform on all FSs, the nesting factor F ( Q ) is proportional to the non-interacting single-orbitalmagnetic susceptibility at vector Q : χ (0) ( Q ) = (cid:88) p f p − f p + Q ε p + Q − ε p (3)where f p is the Fermi distribution and ε p is the quasi-particle kinetic energy. F ( Q ) is calculated by Eq. (2)for each doping and normalized by its value at x = 0 . F ( Q ). Remarkably, F ( Q ) follows exactly the samedoping dependence as 1 /T T | T =20 K , indicating that theLESFs probed by NMR are closely related to the FSnesting. The FS structure naturally suggests that thespin fluctuations are of the antiferromagnetic type withlarge wave vectors close to the nesting vectors ( ± π, , ± π ). This is indeed consistent with our NMRdata. The Knight shift that measures the uniform sus-ceptibility becomes T independent below 30 K (AppendixE, Fig. 11), indicating that the low-temperature upturnin 1 /T T comes from large-momentum spin fluctuations.Note that since the NMR form factor for As is known tobe broadly distributed in momentum space in IBSCs [26],both commensurate (close to FS nesting) and incommen-surate (away from nesting) LESFs are captured by thespin-lattice relaxation rate 1 /T T .In Fig. 4 we summarize our experimental results inthe T – x phase diagram of LiFe − x Co x As. With in-creasing Co concentration x that monotonically sup-presses the superconducting transition temperature T c by electron-doping: (i) The T dependence of the re-sistivity ρ ( T ) ∝ AT n and the optical scattering rate1 /τ ( T ) ∝ BT α deviates from a FL ( n, α = 2) observednear x = 0, reaching the most pronounced NFL power-law behavior ( n (cid:39) α (cid:39) .
35) at x (cid:39) .
12 and then grad-ually returns for x > .
12 to the FL values at x = 0 . /T T showsthat LESFs, small at x = 0, gradually enhance and be-come strongest at x = 0 .
12, but diminish for x > . x = 0, the nesting improves with doping, is optimizednear x = 0 .
12, and then degrades with further electrondoping for x > .
12; (iv) No long range magnetic orderis observed in the T – x phase diagram of LiFe − x Co x Asup to x = 0 .
4, which is consistent with previous stud-ies [15, 16, 27, 28].A comparison between the above observations (i) and(ii) strongly suggests that the FL-NFL-FL crossover inLiFe − x Co x As is induced by LESFs. Point (iii) in combi-nation with (i) and (ii) implies that the integrated LESFsprobed by NMR are dominated by, or at least scale with c T T| T=20 K T ( K ) Doping x / T T ( s - K - ) LiFe Co x As Figure 4. (color online) Temperature–doping ( T – x ) phase di-agram of LiFe − x Co x As. Superconductivity (yellow regime)is monotonically suppressed with increasing Co concentration x and terminates at a critical value x ≈ .
17. The normalstate of LiFeAs is a Fermi liquid (blue regime at x = 0),where the T -dependent resistivity follows ρ ( T ) = ρ + AT n with n = 2. A crossover from Fermi liquid to non-Fermiliquid is induced by Co doping. At x = 0 . n reachesits minimum value of 1 .
35, indicating the most robust non-Fermi liquid behavior (red regime). Further doping resultsin a reversal of this trend until by x = 0 .
4, the Fermi liq-uid behavior is fully recovered (blue regime at x = 0 . /T T | T =20 K ) measured by NMR for several representativedopings. 1 /T T | T =20 K reaches the maximum at x ≈ .
12, sig-nifying that low-energy spin fluctuations are optimized at thisdoping. The three inset panels depict the extracted Fermi sur-faces for three representative Co concentrations: x = 0 (left), x = 0 .
12 (middle) and x = 0 . x = 0 and 0 .
4, the nesting conditionis significantly improved at x = 0 . those near q ∼ Q which are most likely tuned by FSnesting. The fact (iv) does not directly support a mag-netic QCP in the T – x phase diagram of LiFe − x Co x As.However, the pronounced NFL behavior observed near x = 0 .
12, which is usually considered as a signature ofquantum criticality [2, 5, 13], in conjunction with thestrong tendency to diverge in 1 /T T upon cooling at thesame doping, points to an incipient QCP near x = 0 . x = 0 .
12, resulting in anactual magnetic QCP associated with other tuning pa-rameters. Finally, the NFL behavior is observed at theboundary of the superconducting phase, implying thatthey are likely to be governed by different mechanisms.Y. M. Dai, H. Miao and L. Y. Xing contributed equallyto this work. We thank A. Akrap, S. Biermann, P. C. Dai,J. P. Hu, W. Ku, R. P. S. M. Lobo, A. J. Millis, A. vanRoekeghem, W. G. Yin, I. Zaliznyak and G. Q. Zhengfor valuable discussion. Work at BNL was supported bythe U.S. Department of Energy, Office of Basic EnergySciences, Division of Materials Sciences and Engineeringunder Contract No. de-sc0012704. Work at IOP wassupported by grants from CAS (XDB07000000), MOST(2010CB923000, 2011CBA001000 and 2013CB921700),NSFC (11234014, 11274362, 11220101003 and 11474344).Work at RUC was supported by the National Ba-sic Research Program of China under Grants No.2010CB923004 and 2011CBA00112 and by the NSF ofChina under Grants No. 11222433 and 11374364. Workat BC was supported by U.S. Department of Energy, Of-fice of Science, Basic Energy Sciences, under Award DE-FG02-99ER45747.
APPENDIX A: SAMPLE SYNTHESIS ANDCHARACTERIZATION
Sample synthesis
High-quality single crystals ofLiFe − x Co x As were grown with the self-flux method.The precursor of Li As was prepared by sintering Lifoil and an As lump at about 700 ◦ C for 10 h in a Titube filled with argon (Ar) atmosphere. Fe − x Co x Aswas prepared by mixing the Fe, Co and As powdersthoroughly, and then sealed in an evacuated quartztube, and sintered at 700 ◦ C for 30 h. To ensure thehomogeneity of the product, these pellets were regroundand heated for a second time. The Li As, Fe − x Co x As,and As powders were mixed according to the elementalratio Li(Fe − x Co x ) . As. The mixture was put into analumina oxide tube and subsequently sealed in a Nbtube and placed in a quartz tube under vacuum. Thesample was heated at 650 ◦ C for 10 h and then heatedup to 1000 ◦ C for another 10 h. Finally, it was cooleddown to 750 ◦ C at a rate of 2 ◦ C per hour. Crystals witha size up to 5 mm were obtained. The entire process ofpreparing the starting materials and the evaluation ofthe final products were carried out in a glove box purgedwith high-purity Ar gas.
Determination of the doping level
The molar ratioof Co and Fe of the LiFe − x Co x As single crystals waschecked by energy dispersive x-ray spectroscopy (EDS)at several points on one or two selected samples for eachCo concentration. For each doping, the Co concentrationmeasured by EDS is consistent with the nominal value.
APPENDIX B: TRANSPORT
Resistivity measurements
The electrical transportmeasurements of LiFe − x Co x As were carried out in acommercial physical properties measurement system(PPMS) using the four-probe method. To preventsample degradation, the electrical contacts were pre- pared in a glove box and then the sample was protectedby n -grease before transferring to the PPMS. Eachsample was cut into a rectangular piece, so that itsdimensions could be measured more accurately with amicroscope. With these precisely-measured geometryfactors, the resistivity can be easily calculated from themeasured resistance. The resistivity determined fromthe transport measurements is then compared withthe values determined from the optical conductivity toensure the consistency between different techniques. Single power law fitting
Figure 5 displays the resis-tivity as a function of temperature ρ ( T ) (open circles) upto 70 K for all 8 samples. For each substitution, ρ ( T ) isfit to a single power law expression, ρ ( T ) = ρ + AT n ,from ∼ T c , whichever is greater) up to 70 K. Thesolid lines through the data in each panel denote the fit-ting results. The power n , determined from the fitting,is shown for the stoichiometric material and all the Cosubstitutions in the corresponding panels; the crossoverbehavior of n can be seen clearly.In order to present the power-law behavior of the ρ ( T )curve more clearly, we plot ρ ( T ) as a function of T n foreach substitution in Fig. 6, where n is the power deter-mined from the single power law fitting. In this case,all the ρ ( T ) curves can be perfectly described by linearbehavior, shown by the solid line in each panel. APPENDIX C: OPTICAL SPECTROSCOPY
Reflectivity
The temperature dependence of theabsolute reflectivity R ( ω ) of LiFe − x Co x As has beenmeasured at a near-normal angle of incidence for thestoichiometric material and 4 representative substi-tutions using an in situ overcoating technique [29].For each sample, data were collected at 17 differenttemperatures from 5 K to room temperature over a widefrequency range ( ∼ − x Co x As samples, the sample mounting and cleav-ing were done in a glove bag purged with high-purityAr gas. Immediately after the cleaving, the sample wastransferred to the vacuum shroud (also purged withAr) with the protection of a small Ar-purged plasticbag. The reproducibility of the experimental resultswas checked by repeating the R ( ω ) measurements 2 or3 times for each doping. Figure 7(a)–7(e) show R ( ω ) inthe far-infrared region at 4 selected temperatures for 5different Co concentrations. For all the materials, R ( ω )approaches unity at zero frequency and increases uponcooling, indicating a metallic response. Kramers-Kronig analysis
The real part of thecomplex optical conductivity σ ( ω ) is determined froma Kramers-Kronig analysis of the reflectivity. Given T (K) T (K) T (K) T (K) T (K) T (K) T (K) ( c m ) ( c m ) T (K) ~ T LiFe Co x Asx = 0 (a) (b) x = 0.03 (c) ~ T x = 0.06 (d) ~ T x = 0.09 (e) ~ T ~ T x = 0.12 (f) ~ T x = 0.16 (g) ~ T x = 0.26 (h) ~ T x = 0.4 Figure 5. (color online) Resistivity of LiFe − x Co x As as a function of temperature for all the Co values. For each sample, theresistivity curve is fit to the single power law expression ρ ( T ) = ρ + AT n . The open circles denote the measured resistivityand the solid lines in each panel are the fitting results. The power n derived from the fitting is shown in each panel. Co x As ( c m ) ( c m ) ~ T ~ T ~ T ~ T ~ T ~ T ~ T ~ T T (K ) T (K ) T (K ) T (K ) T (K ) T (K ) T (K ) T (K ) (a) (b) x = 0.03 (c) x = 0.06 (d) x = 0.09 (e) x = 0.12 (f) x = 0.16 (g) x = 0.26 (h) x = 0.4 Figure 6. (color online) Resistivity of LiFe − x Co x As as a function of T n for all the Co values. n is the power determined fromthe single power law fit to the resistivity as a function of T . The straight solid line in each panel is a guide to the eye. the metallic nature of the LiFe − x Co x As materials,the Hagen-Rubens form [ R ( ω ) = 1 − A √ ω ] is used forthe low-frequency extrapolation, where A is chosento match the data at the lowest-measured frequency.Above the highest-measured frequency, R ( ω ) is assumedto be constant up to 1.0 × cm − , above which afree-electron response (cid:2) R ( ω ) ∝ ω − (cid:3) was used. Optical conductivity
Figure 7(f)–7(j) display σ ( ω )at 4 selected temperatures for the stoichiometric ma-terial and 4 different Co concentrations. The metallicbehavior of these materials can be recognized by thepronounced Drude-like peak centered at zero frequency.The zero-frequency value of σ ( ω ) represents the dc conductivity σ dc which is in good agreement withthe values determined from transport measurements;the width of the Drude peak at half maximum yieldsthe quasiparticle scattering rate. As the temperaturedecreases, σ dc increases and the Drude peak narrows.This indicates that the quasiparticle scattering ratedecreases upon cooling, dominating the temperaturedependence of the electrical transport properties. Quasiparticle scattering rate
Figure 7(k)–7(o)show the quasiparticle scattering rate of the coherentnarrow Drude component 1 /τ co as a function of T α ,where α is the power determined from the single powerlaw fit to 1 /τ co as a function of T . Linear behavior R e f l e c t i v i t y LiFe Co x As x = 0 (a) 0 400 800(b)x = 0.06 20 K 100 K 150 K 200 K
20 K 100 K 150 K 200 K 0 400 800(d)x = 0.26
20 K 100 K 150 K 200 K 0 400 800(e)x = 0.4
20 K 100 K 150 K 200 K0 500 1000051015 (f) 20 K 100 K 150 K 200 K () ( - c m - ) / c o ( c m - ) co ~ T co ~ T (l) co ~ T (m) co ~ T (n) T (K )T (K ) co ~ T (o)T (K )T (K ) Wave number (cm -1 )Wave number (cm -1 )Wave number (cm -1 )Wave number (cm -1 )Wave number (cm -1 ) Wave number (cm -1 )Wave number (cm -1 )Wave number (cm -1 )Wave number (cm -1 ) Wave number (cm -1 )T (K ) Figure 7. (color online) Reflectivity, optical conductivity and quasiparticle scattering rate of LiFe − x Co x As. (a)–(e) Temper-ature dependence of the reflectivity of LiFe − x Co x As in the far-infrared region at several temperatures for the stoichiometricmaterial and 4 representative Co concentrations. (f)–(j) Real part of the optical conductivity derived from the reflectivity.The thick solid curves are the experimental data and the thin solid curves through the data denote the Drude-Lorentz fittingresults. (k)–(o) Scattering rate of the coherent narrow Drude component (1 /τ co ) as a function of T α . The straight solid line ineach panel is a guide to the eye. can be clearly observed in each panel as guided by thestraight solid lines. Comparison between transport and optics
Fig-ure 8(a) compares the doping dependence of ρ (solidcircles) and 1 /τ (solid triangles), where ρ is determinedby fitting ρ ( T ) to ρ ( T ) = ρ + AT n , and 1 /τ is derivedby fitting 1 /τ co ( T ) to 1 /τ co ( T ) = 1 /τ + BT α for eachdoping. Both ρ and 1 /τ grow as the Co concentra-tion increases, indicating that impurities are introducedinto the compounds by the Co substitution. The singlepower law fit also returns the coefficients A for transportand B for optics, respectively. As shown in Fig. 8(b), A (solid circles) and B (solid triangles) follow identicaldoping dependence, suggesting that the T dependencein ρ ( T ) and 1 /τ co ( T ) are governed by the same physics.Note that while the coefficients A and B follow exactlythe same trace across doping, the detailed behaviors of ρ and 1 /τ are slightly different. This is because 1 /τ isthe residual scattering rate of the coherent narrow Drudecomponent, which contributes to ρ in parallel with theincoherent broad Drude component. However, as a com-mon feature in all Fe-base superconductors [19, 20], the T dependence of the transport properties is dominatedby the coherent narrow Drude component. Since A and B are the prefactors that describe the properties of the T dependence in ρ ( T ) and 1 /τ co ( T ), respectively, it isnatural to expect similar behaviors for A and B if theyare governed by the same physics. APPENDIX D: ANGLE-RESOLVEDPHOTOEMISSION SPECTROSCOPY
Measurements
ARPES measurements were per-formed with a high-flux He discharge lamp. The energyresolution was set at 12 meV and 3 meV for the Fermisurface mapping and high resolution measurements,respectively. The angular resolution was set at 0.2 ◦ .Fresh surfaces for the ARPES measurements wereobtained by an in situ cleavage of the crystals in aworking vacuum better than 4 × − Torr. The Fermienergy (E F ) of the samples was referenced to that of agold film evaporated onto the sample holder. ARPES intensity plots and energy/momentumdistribution curves along the Γ– M direction ARPES intensity plots across the Γ point for fiverepresentative dopings and their corresponding energydistribution curves (EDCs) are shown in Fig. 9(a)–9(e) (b) 050100150 B ( c m - K - ) A ( c m K - n ) ( c m ) ( c m - ) Figure 8. (color online) (a) Doping dependence of ρ (solidcircles) and 1 /τ (solid triangles), respectively. (b) Coeffi-cients A for transport (solid circles) and B for optics (solidtriangles) determined from the single power law fit as a func-tion of Co concentration. and Fig. 9(f)–9(j), respectively. The same plots butcrossing the M point for each doping are shown inFig. 9(k)–9(o) and Fig. 9(p)–9(t), respectively. Thered curves in Fig. 9(a)–9(e) and Fig. 9(k)–9(o) are themomentum distribution curves (MDCs) at the Fermilevel. The peak positions on MDCs correspond to the k F positions along the high symmetry line. Since theEDC and MDC peaks, as well as the band dispersions,are well defined at all dopings, the k F positions can beaccurately determined for each Co concentration. Theuncertainty of the k F position is mainly from the energyand momentum resolutions of our system settings. Forthese measurements the angular and energy resolutionsof the system were set at 0.2 ◦ and 3 meV, respectively,which lead to a typical uncertainty of ± . π/a for k F at 21.2 eV photon energy. Evolution of the Fermi surface volume with dop-ing
The evolution of the Fermi surface volume withdoping is controlled by the Luttinger theorem: the totalalgebraic Fermi surface volume is directly proportionalto the carrier concentration. We checked that this is thecase in our study and we plot the results in Fig. 10, whichconfirm that the total volume of the Fermi surface (opencircles) satisfies the Luttinger theorem at each dopingif we assume a rigid chemical potential shift caused bythe introduction of one additional electron carrier per Featom substituted by Co (dashed line), as also observedtheoretically [30] and experimentally [31, 32] for the 122family of iron pnictides. -100 -50 0 50 E - E F (meV) Γ -100 -50 0 50 E - E F (meV) Γ -100 -50 0 E - E F (meV) Γ -100 -50 0 E - E F (meV) Γ I n t en s i t y ( a r b . un i t s ) -100 -50 0 50 E - E F (meV) Γ -0.5 0.0 k x ( π /a) k x ( π /a) -0.5 0.0 k x ( π /a) -100-50050 E - E F ( m e V ) k x ( π /a) k x ( π /a) k x ( π /a) k x ( π /a) k x ( π /a) k x ( π /a) -100-50050 E - E F ( m e V ) -0.5 0.0 k x ( π /a) -100 -50 0 50 E - E F (meV)M -100 -50 0 50 E - E F (meV)M -100 -50 0 E - E F (meV)M -100 -50 0 E - E F (meV)M I n t en s i t y ( a r b . un i t s ) -100 -50 0 50 E - E F (meV)Ma b c d ef g h i jk l m n op q r s tx= 0 x= 0.06 x= 0.12 x= 0.26 x= 0.4 HighLowHighLow
Figure 9. (color online) (a)–(j) ARPES intensity plots andcorresponding EDCs at the Brillouin zone center. (k)–(t)ARPES intensity plots and corresponding EDCs at the M point. All data are taken at 20 K along the Γ– M direction.The red curves shown in the intensity plots are MDCs at theFermi level. APPENDIX E: NUCLEAR MAGNETICRESONANCE
Measurements
We perform the As nuclear magneticresonance (NMR) measurements with the external fieldparallel to the a-b plane. The spin-lattice relaxation rateis measured by inversion-recovery method on the centraltransition and the recovery curve is fit with a standarddouble exponential form for an S=3/2 spin1 − m ( t ) m (0) = 0 . (cid:18) − tT (cid:19) + 0 . (cid:18) − tT (cid:19) . (4) Knight shift
Figure 11 shows the Knight shifts of x = 0 and x = 0 .
12 samples, respectively. The Knightshift, which measures the uniform susceptibility, be-comes temperature independent below 30 K, indicatingthe low-temperature upturn in 1 /T T comes from large-momentum spin fluctuations. Note that since the NMRform factor for As is known to be broadly distributedin momentum space in iron-based superconductors [26], n e l e - n ho l e LiFe Co x As Figure 10. (color online) Total electronic carrier concentra-tion (open circles) determined from the total algebraic Fermisurface volume at each Co concentration x . The dashed linecorresponds to the carrier concentration assumed by addingone extra electron for each Fe substituted by Co. K s ( % ) T (K) : x = 0: x = 0.12 Figure 11. (color online) The Knight shifts of LiFe − x Co x Asfor the x = 0 and x = 0 .
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