NORTHERN ILLINOIS UNIVERSITY

Transcription

1 ABSTRACT Name: Jiong Hua Department: Physis Title: Commensurate Effet in Superonduting Niobium Films Containing Arrays of Nanosale Holes Major: Physis Degree: Dotor of Philosophy Approved by: Date: Dissertation Diretor NORTHERN ILLINOIS UNIVERSITY

2 ABSTRACT Commensurate effet is one of the intriguing properties observed in superonduting films ontaining periodi hole-arrays. It represents itself as minima in the magneti field dependene of the resistane or maxima in the field dependene of the ritial urrent when an integer number of flux quantum is ommensurate with the unit-ell of the artifiial hole-array in a superonduting film. Two mehanisms, vorties pinning enhanement by the hole-array or ritial temperature osillations of a wire network, an aount for this effet at temperatures lose to the zero-field ritial temperature. This thesis investigates the ommensurate effet of niobium superonduting films with periodi hole-arrays near the zero-field ritial temperature. Experimental approahes have been developed to distinguish the possible mehanisms. The field dependene of the resistane in various field diretions and the angular dependenes of the ritial temperature and resistane reveal that at temperatures lose to the zero-field ritial temperature superonduting niobium films with periodi hole-arrays behave like superonduting wire networks and the observed ommensurate effet originates from hole-indued ritial temperature suppression at noninteger flux quantum

3 fields rather than the widely believed pinning enhanement at integer flux quantum fields. These onlusions are supported by the results of patterned films with various thiknesses, hole-hole separations and the symmetries of the holearrays. For omparison, a ontinuous thin film and a thin strip are also investigated. This researh will signifiantly advane the understanding on superondutors with artifiial defets and the results an be diretly applied to different types of the defets suh as arrays of magneti dots.

4 NORTHERN ILLINOIS UNIVERSITY COMMENSURATE EFFECT IN SUPERCONDUCTING NIOBIUM FILMS CONTAINING ARRAYS OF NANOSCALE HOLES A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS BY JIONG HUA DEKALB, ILLINOIS MAY 8

5 Certifiation: In aordane with departmental and Graduate Shool poliies, this dissertation is aepted in partial fulfillment of degree requirements. Dissertation Diretor Date

6 ACKNOWLEDGEMENTS At the ompletion of this thesis I take great pleasure to thank all the people who have supported me over the past years. Above all, I would like to express my sinere and deep gratitude to my advisor Professor, Zhili Xiao, for giving me the opportunity to do my Ph.D. at Northern Illinois University. His guidane and areful proofreading of draft papers and the final thesis manusript were very muh appreiated. I would also like to thank all the other ommittee members, Dr. Ulrih Welp, Dr. Dennis Brown and Dr. Susan Mini, for their valuable input. The researh was onduted mainly at Argonne National Laboratory. I thank all members in the superondutivity and magnetism group for their help. Speial thanks go to Dr. Wai-Kwong Kwok for providing exellent working onditions and the opportunities to partiipate in several international onferenes, Dr. Alexandra Imre for her help on sample fabriation, and Dr. Alexei Koshelev for the theory part. In addition, I would like to aknowledge the finanial support of the NIU/Argonne Distinguished Graduate Fellowship. Last but not least, I would like to thank my family, in partiular my wife; without her support this researh would never have been ompleted.

7 TABLE OF CONTENTS Page LIST OF TABLES.. vi LIST OF FIGURES... vii Chapter 1. INTRODUCTION.. 1. THEORY 5 Angular Dependene of the Critial Field in Superonduting Films and Strips Fluxoid Quantization and the Little-Parks Effet.. 15 Commensurate Effet and Possible Origins METHODOLOGY... 8 Sample Fabriation PPMS Measurement System ANISOTROPY IN D AND 1D SUPERCONDUCTORS.. 39 Critial Field of a D Niobium Superonduting Thin Film. 39 Resistive Anisotropy of a 1D Niobium Strip ANISOTROPIC PROPERTIES IN SUPERCONDUCTING FILMS WITH PERIODIC HOLE-ARRAYS Relation of the Properties in Parallel and Perpendiular Fields 6 Angular Dependene of the Mathing Field.. 78

8 v Chapter Page Angular Dependene of the Critial Temperature. 83 Angular Dependene of the Magnetoresistane CONCLUSION 11 REFERENCES 14

9 LIST OF TABLES Table Page 1. Niobium thin films ontaining different types of strutures. 35

10 LIST OF FIGURES Figure Page -1 Coordinate system of a superonduting thin film in a magneti field 6 - Coordinate system of a superonduting strip in a magneti field 13-3 A superondutor ontaining holes and different paths for fluxoids 16-4 Experimental setup for the Little-Parks effet Shemati v s and - Δ T (suppression of T ) with respet to the applied field in the Little-Parks experiment Field dependene of the resistane and ritial urrent for a Nb film with a square array of holes Resistane as a funtion of the temperature under various fixed magneti fields for a Nb film with a square array of holes Magneti field dependene of the redued ritial temperature in a wire network Proedures of sample preparation SEM images of a triangular hole-array in a niobium film Optial mirosope image of the referene niobium film SEM image of the niobium thin strip SEM images of niobium films ontaining hole-arrays with different types of latties Sample rod with the horizontal rotator Resistive transition of the niobium referene thin film at zero-field 4 4- Phase diagrams of the niobium referene film in parallel and perpendiular fields. 4

11 viii Figure Page 4-3 Analysis of the phase diagram in the parallel field for the niobium referene film Angular dependene of the ritial field for the niobium referene film at a fixed temperature Phase diagrams of the niobium referene strip in parallel and perpendiular fields Angular dependene of the ritial field for the niobium referene strip at a fixed temperature Saling behavior of R (H ) urves at various field diretions for the niobium referene strip at a fixed temperature Shemati outline of R (θ ) to R (H ) onversion Reonstrution of R (H ) at from R (θ ) urves R (θ ) fitting for the referene strip at different temperatures in a fixed field (H = 4 G) Shemati outline to identify the origin of the ommensurate effet H (T ) phase diagrams of niobium film with different strutures in the perpendiular field Baseline reonstrution for the R (H ) urve ( T =.99T or 5.9 K) and H (T ) phase diagram (inset) Baseline reonstrution for the R (H ) urve at different temperatures Baseline reonstrution of the R (H ) urve and H (T ) phase diagram (inset) for different samples Magneti field region of the ommensurate effet Comparison of T osillation region of the phase diagram in perpendiular fields for niobium films with different hole-arrays

12 ix Figure Page 5-8 Shemati outline for the angular dependene of the mathing field Angular dependene of the first mathing field for Sample C Angular dependene of the normalized first mathing field for niobium films ontaining different hole-arrays Phase diagram at different angles for Sample C Calulated strip bakground of the T (θ ) urve for Sample C at a fixed field (H=5 G) Δ T osillation in various perpendiular fields for Sample C Experimental method to fit the T (θ ) for Sample C at a fixed field (H=5 G) Experimental fitting for T (θ ) urves at various fixed fields for niobium films with different hole-arrays R (θ ) urves in various fixed fields at T= 5.9 K for Sample C Calulated strip bakground of the R (θ ) urve in a fixed field (H=5 G) at T= 5.9 K for Sample C Shemati outline of alulating the resistane hange due to T suppression Experimental fitting for R (θ ) urves in various fixed fields at T=5.9 K for Sample C 1

13 CHAPTER 1 INTRODUCTION When the applied magneti field is higher than the lower ritial field but below the upper ritial field, a Type II superondutor allows magneti flux to penetrate it in the form of vorties a tiny normal area irulated by supperurrent. Driven by the Lorentz fore of a passing external urrent or by thermal ativation, vorties an move. Their motion indues energy dissipation and eventually an destroy the superondutivity. During the past deades, a lot of efforts have been devoted to introduing artifiial pinning enters into superondutors to stabilize and pin the vortex latties against the external driving fore, thus giving rise to higher ritial urrents [1-1]. It is pratially important sine superondutors are required to maintain high ritial urrents for potential tehnologial appliations. Generally there are two different kinds of artifiial pinning enters. The first one is the random imperfetions, for example heavy-ion radiation damage [1], old work indued disloations [], disordered hole-arrays [3], et. The other one is periodi defet arrays suh as antidots (holes) [4-6], blind holes [7], and magneti dots [8-1].

14 Reent advanes in miro- and nanofabriation tehnologies have made it possible to produe superondutors with arefully ontrolled arrays of artifiial defets [4-15] with sizes and separations omparable to the relevant length sales of superondutors suh as the oherene length ξ (T ) and the penetration depth λ (T ), whih are generally under submiron sales. As one of the most effiient and easiest methods, introduing periodi hole-arrays into superonduting films has attrated muh interest [4-7]. Furthermore, superonduting films ontaining periodi hole-arrays also provide a unique platform to understand vortex motion and pinning in the presene of regular pinning enters. The interplay between the periodi pinning fores and the elasti repulsive vortex-vortex interation generates a variety of novel vortex phenomena suh as the ommensurate effet [11, 1], the retifiation and phase loking of the vorties [13], omposite fluxline latties [14], et., whih are not observed in the ontinuous superonduting films or superondutors with random pinning enters. The ommensurate effet is one of the intriguing properties in superonduting films with periodi hole-arrays [4-6, 14, 16]. It appears as minima in the magneti field dependene of the resistane, R (H ), or as maxima in the field dependene of the ritial urrent, I (H ), when an integer number of flux quantum is ommensurate with the unit-ell of the artifiial hole-array in a superonduting film. This effet is normally interpreted as a result of the pinning enhanement [17-4], i.e., vorties are most diffiult to move at ommensurate

15 3 fields. Thus, a pinning enhanement ours and high ritial urrent (or resistane minima) an be ahieved. However, a similar effet has also been observed in superonduting wire networks [5-8], whih are -dimensional (D) multionneted superonduting thin strips whose width w is omparable to the superonduting oherene length ξ (more preisely w < 1.84ξ ). Here, the resistane minima at integer flux quantum fields are interpreted as the result of the additional suppression of the superonduting ritial temperature T by magneti fields at non-integer flux quantum values arising from the fluxoid quantization effet (similar to the Little- 1/ Parks effet) [9-34]. Sine ( ) ~ (1 / ξ T T T is temperature dependent, a superonduting film ontaining a periodi hole-array should resemble a ) superonduting wire network at temperatures lose to T. Both of these mehanisms have been used to interpret the ommensurate effet observed in transport measurements on a superonduting film ontaining an array of holes at temperatures lose to T. For example, in aluminum films ontaining periodi hole-arrays, Fiory et al. assoiated the osillation in the magneti field dependene of resistane to ommensurate vortex pinning [34], whereas Pannetier et al. attributed it to wire network properties [35]. In order to understand the related physis and for possible appliations, it is neessary to distinguish these two mehanisms and identify the real origin of the ommensurate effet. But there is no effetive method so far.

16 This thesis investigates the effet of a periodi hole-array on the transport 4 properties of a niobium superonduting film, with emphasis on the ommensurate effet observed near T. Experiments on the field dependene of the resistane in various field diretions, the angular dependenes of the ritial temperature and resistane, reveal that the observed ommensurate effet originates from the fluxoid quantization effet. This researh will signifiantly advane the understanding on superondutors with artifiial defets and the outome an be diretly applied to different types of defets suh as arrays of magneti dots. For omparison, a ontinuous thin film and a thin strip are also investigated. Chapter presents the relevant theories, inluding the angular dependene of the ritial field for a superonduting thin film and a thin strip; fluxoid quantization and the Little-Parks effet; and the onept of the ommensurate effet and possible origins. Chapter 3 provides details on sample preparations and haraterizations. In Chapter 4, the angular dependene of the ritial field and resistane of a referene film and a thin strip are investigated. Chapter 5 presents results on the ommensurate effet in patterned films, inluding a new approah to identify the flux quantization effet as the origin of the ommensurate effet, the angular dependenes of the ritial temperature and the resistane whih provide additional evidenes on the wire network nature of the patterned films. The last hapter summarizes the results of this thesis and gives suggestions for future researh.

17 CHAPTER THEORY Angular Dependene of the Critial Field in Superonduting Films and Strips When a type-ii superondutor is plaed in a magneti field, the field penetrates the sample and the mixed state is formed. The superondutor is driven ompletely into the normal state at the upper ritial field H. It is the highest field at whih superondutivity an nuleate in the interior of a large sample in a dereasing external field. Unlike homogeneous and isotropi superondutors, the behavior of superonduting thin films in a magneti field depends markedly on the diretion of that field. A shemati outline of the oordinate system for a thin film in the magneti field is shown in Figure -1. The film loates between d z. The magneti field H v lies in the yz plane at an angle θ from the normal diretion of the film ( z axis). First, two extreme situations are onsidered; i.e., the magneti field is parallel (along the y axis) or perpendiular (along the z axis) to the film.

18 z 6 H θ H y x H // Figure -1. Coordinate system of a superonduting thin film in a magneti field. When a superonduting thin film is plaed in a magneti field parallel to its surfae, it is antiipated to have a seond-order phase transition sine the order parameter ψ and therefore, the magnetization goes smoothly to zero with the inrease of the magneti field [36]. The Gibbs free energy per unit area of the film is G = d / d hh f dz / 4π (-1) where * β 4 1 h e h f = f n + αψ + ψ + A ψ + * m i is the Ginzburg- 8π Landau free energy density; e * = e and m * = m stands for the harge and mass for Cooper pairs; h is the field inside the film and H is the applied field. The parameters α and β are defined in Ginzburg-Landau theory [37] as

19 m H e m H e λ π β λ α = = (-) where λ refers to the effetive penetration depth at zero field and H is the thermodynami ritial field. In the thin film limitation, ξ < λ, d, the sreening effet may be negleted and the vetor potential an be hosen as z H A x = (-3) With this hoie, h in the film an be approximated by H and the phase ϕ as well as the amplitude ψ of the ordered parameter are onstants. Thus, the Gibbs free energy per unit area of the film is ( ) π ψ ψ β αψ π ψ ψ β αψ π π ψ ψ β αψ / / 4 / / * 4 d H m d H e f d dz H z H e m f dz hh h A e i m f G n d d n d d n = = = h (-4) Minimizing this expression with respet to ψ, the density of the Cooper pairs omes out to be + = m d H e α β ψ (-5) Using the definition in Eq.-, the above expression an be further simplified to

20 ψ 3 α e H d α = d 1 1 β 1m α β 4 H λ = H 8 (-6) The film beomes normal, i.e. ψ, at the parallel ritial field: H λ H // = 6 (-7) d Eq.-7 indiates that a thin film in parallel fields an have a ritial field muh higher than the bulk thermodynami ritial field. This ours beause the penetration of the magneti field leads to inomplete flux exlusion, reduing the diamagneti energy for a given magneti field. In order to ompare with the result of the perpendiular ritial field derived below, this result an be written in terms of the Ginzburg-Landau oherene length [37], whih is defined as ξ ( T ) = Φ π H. Thus, λ H // 3 Φ = (-8) π d ξ ( T ) For a magneti field normal to the superonduting thin film, it penetrates the thin film by way of individual vorties. Hene, the order parameter, ψ, varies inside the film and the above proedure an not be used to obtain the ritial field. Instead, a more generalized method has to be applied, i.e. solving the Ginzburg- Landau (GL) equation: * 1 h e αψ + β ψ ψ + = * A ψ m i (-9)

21 In the thin film limit, the linearized GL equation is appropriate for 9 determining the ritial field sine the term in ψ ψ in the full GL equation (Eq.-9) is negligible ompared to the linear term. Thus, the GL equation beomes the linearized form: r * πa m α ψ ) ψ = ψ = i Φ h ξ ( ) ( T (-1) where A r is the vetor potential, h Φ = =.7 1 e 7 G m is the flux quantum and ξ (T ) is the GL oherene length, whih represents the sale of variation of the wave funtion ψ in the material. Sine the magneti field is long the z axis, a onvenient gauge hoie is A y = H x (-11) Then the linearized GL funtion an be speified to 4π i + Φ πh H x + y Φ x ψ ψ = ξ (-1) Considering the effetive potential depends only on x, it is reasonable to look for a solution of the form: ik y y ikz z ψ = e e f (x) (-13) Substituting it into Eq.-1 and rearranging terms, the equation reahes a further simplified form: πh 1 f ( x) + z Φ ξ ( x x ) f ( x) = k f ( x) (-14)

22 1 where x k yφ =. πh By noting that Eq.-14 is very similar to the Shrodinger s equation for a harmoni osillator with a fore onstant 1 m * π h H, the solution an be Φ obtained by using the result of the harmoni osillator eigenvalue: ε n 1 1 eh h 1 n + h ω = n + h = k z (-15) m m ξ = Thus, H = π Φ k ( + 1) z n ξ 1 (-16) Evidently, it has the highest value when k = and n =. This value is defined as the perpendiular ritial field: H z Φ = (-17) πξ ( T ) Comparing with the parallel ritial field (Eq.-8), whih depends on the thikness of the film, a signifiant anisotropy is expeted. To desribe the angular dependene behavior of an ultra thin superonduting film in a magneti field, Tinkham proposed his famous model, whih is well aepted nowadays. It an be onsidered as a straightforward onsequene of GL theory [38-4]. I shall show it briefly below. Again I start with the linearized GL equation and hoose a oordinate system whih is shown in Fig. -1. The magneti field H v lies in the yz plane at

23 an angle θ from the normal diretion of the film. So both parallel and 11 perpendiular omponents exist in this ase. The vetor potential is hosen to have only an x ) omponent, whih is given by ( z sinθ osθ ) A x = H y (-18) Inserting this expression into Eq.-1 and putting all terms to the left-hand side, the linearized GL equation an be speified to 4π i + H Φ ( z sinθ y osθ ) πh + x ( z sinθ y osθ ) Φ 1 ψ = ξ ( T ) (-19) Sine both the magneti field and the vetor potential are not dependent on x, by hoosing a trial funtion independent to x, this equation beomes ψ ψ πh + z y Φ ψ ξ ( T ) ( z sinθ y osθ ) ψ = (-) ψ In the limit of a very thin film, =, whih automatially satisfies the z ψ boundary ondition =, and the first term an be negleted. Thus, ψ is z z= ± d a funtion only of y. Integrating both sides of Eq.- over the whole region of the thin film: d / + + ψ πh + y Φ ψ ξ ( T ) ( z sinθ + y osθ ) ψ dxdydz = d / (-1) and noting that d / 3 d d / z dz = and d / 1 z dz =, Eq.- an be onverted to d /

24 1 ) ( 1 1 sin os = Φ + Φ + + dy T d H y H dy d Xd ψ ξ θ π ψ θ π ψ (-) where X is the width in the x ) diretion and d is the thikness of the film. To make Eq.- valid, the part inside the integration has to be equal to zero. Thus, the ordinary differential equation is obtained with the form of ψ θ π ξ ψ θ π ψ Φ = Φ + 3 sin ) ( 1 os Hd T y H dy d (-3) This is again an eigenvalue problem whih has a form similar to the Shrodinger equation for the harmoni osillator from quantum mehanis. Applying the lowest eigenvalue Φ os θ πh to the right side yields the angular dependene formula of the ritial field: 3 sin ) ( ) ( 1 os ) ( Φ = Φ θ θ π ξ θ θ π d H T H (-4) Thus, the angular dependene of the ritial field for an ultra thin superonduting film an be written as 1 os ) ( sin ) ( // = + H H H H θ θ θ θ (-5) where ) ( 3 // T d H ξ π Φ = and ) ( T H ξ π Φ =, as given in Eq.-8 and Eq.-17 are the ritial fields in parallel and perpendiular diretions, respetively.

25 13 It has to be notied that Eq.-5 is only satisfied in the ultra thin film limit. Sine the only harateristi length in Eq.-19, whih is the base of this model, is ξ (T ), Tinkham assumed d << ξ (T ) in his original paper [38]. Another interesting ase is the superonduting thin strip, whih is very similar to the thin film exept the other onfinement is introdued in width, i.e. w < ξ,λ. Figure - shows the oordinate system hosen for a thin strip, where the sample lies along the x axis and is onfined between y w and d z. Here w and d are the width and thikness of the strip. z H H θ H // y x d w Figure -. Coordinate system of a superonduting strip in a magneti field. For onveniene, the parallel and perpendiular fields are defined along the y and z axis, respetively. In both diretions, the ritial fields are expeted to

26 14 have the same expression as a thin film in the parallel field (see Eq.-8) beause the area exposed to the magneti field has the same geometry. Using the orresponding thikness, w and d, the ritial fields in parallel and perpendiular diretions for a thin strip are H H // 3 Φ = π d ξ ( T ) 3 Φ = π w ξ ( T ) (-6) Obviously, anisotropy exists when the width w and thikness d are different. For a magneti field H v in the yz plane at an arbitrary angle θ from the z axis, as shown in Figure -, the ritial field an be obtained by diretly minimizing the Gibbs free energy beause both the parallel and perpendiular omponent have the same onfinement as a thin film in a parallel field. Choosing a vetor potential having only an x omponent, ( z sinθ osθ ) A x = H y, instead of = H z in Eq.-1 and doing integration A x with respet to both y and z, the Gibbs free energy per unit length of the strip an be obtained: w / d / G = f n + αψ + β ψ 4 + e H ( z sinθ y osθ) ψ H m * dydz w / d / 8π = w d f n + αψ + β ψ 4 H d + e H 8π m * ψ w d 3 sin θ + d w 3 os θ 1 1 = w d f n + αψ + β ψ 4 H d + e H d w ψ ( d sin θ + w os θ) 8π 6m * (-7)

27 Minimizing this expression with respet to ψ : 15 e H d w ( α + β ψ ) + ( d sin θ os θ ) = w d + w (-8) 1m gives the density of Cooper pairs: ψ ( d sin θ + w os θ ) e H α = 1m (-9) β Substituting the expression of α and β from Eq.-, the ritial field at an angleθ, H (θ ) is obtained when ψ : d 1 H ( θ ) sin θ w H ( θ ) os 4H λ 4H λ θ = (-3) Thus, the angular dependene of the ritial field in a strip is H ( θ ) sin H // θ H + ( θ ) os H θ = 1 (-31) where H // 3 Φ = and π d ξ ( T ) H 3 Φ = are the ritial fields in the π w ξ ( T ) parallel and perpendiular diretion, respetively. They are onsistent with the result in Eq.-6. Fluxoid Quantization and the Little-Parks Effet To analyze the state of a multiply onneted superondutor, for example a superondutor ontaining holes, in the presene of a magneti field, the onept

28 of the fluxoid Φ was introdued. It is assoiated with eah hole (or normal 16 region) and the definition is * 4π m Φ = Φ + λ J s ds = Φ + vs ds * e (-3) where Φ = = h ds A ds is the applied magneti flux through the path. Sine A = and J = in superonduting materials, Φ = is satisfied for any path s whih enloses no hole (see Path A in Figure -3) and Φ holds onstant value for any path around a given hole (see Path B in Figure -3). Path B Superonduting material Hole Path A Figure -3. A superondutor ontaining holes and different paths for fluxoids. Path A enloses no hole; Path B enloses a hole. Furthermore, applying the definition of the superurrent veloity introdued in GL theory,

29 * m v s 17 * * e A e A = ps = h ϕ (-33) it is found that Φ should be restrited to a disrete set, i.e. integer multiples of a h flux quantum Φ = =.7 1 e 7 G m, whih is shown as follows, * * e A h Φ = m v + = = n = n = n Φ * s ds h ϕ ds h π * * * e e e e (-34) The Little-Parks experiment [43, 44] suessfully demonstrated this effet. Let R be the radius of a thin-walled ylinder, i.e. the thikness d is less than the oherene length, whih means there is no magneti flux through the wall, and H be the applied field along the axis of the ylinder, as shown in Fig. -4. Then the magneti flux from the applied field is Φ = πr H. H d d < R oherene length ξ Figure -4. Experimental setup for the Little-Parks effet.

30 Putting the above result of Eq.-34 into Eq.-3, 18 n m e m e * * * Φ Φ = vs ds = vs R = * h the superurrent veloity is speified by Φ m R π v (-35) s v s = h m R * n Φ Φ (-36) where n is an integer number for whih v s is a minimum at a given H. In this way, the system remains superonduting at the lowest energy level. Obviously, v s is a periodi funtion of Φ, whih is shown in Fig. -5. Φ v s Φ Φ v s ΔT Φ Φ Figure -5. Shemati v s and - Δ T (suppression of T ) with respet to the applied field in the Little-Parks experiment.

31 To obtain the relation between the superurrent veloity v s and the 19 suppression of the ritial temperature Δ T = T T, Eq.-33 is applied to the Ginzburg-Landau equation (Eq.-9): 1 αψ + β ψ ψ + v ψ = * s m (-37) Thus, the order parameter an be written as α v = s α v = s ξ ( T ) ψ 1 1 * (-38) β m α β h The superonduting-normal transition ours when ψ = : v s h = h ξ ( T ) = h ξ () (1 T / T ) = ( ΔT ) (-39) T ξ () Hene a periodi T osillation appears. The shemati plot is also shown in Fig. -5. Here the problem has been idealized to the simplest ondition a single ring. A similar yet more ompliated effet in wire networks will be demonstrated later in this thesis. Commensurate Effet and Possible Origins The ommensurate effet is often observed in superondutors ontaining periodi defet-arrays [4-16]. As shown in Figure -6, this effet represents itself as minima in the field dependene of the resistane, R (H ), or maxima in the field dependene of the ritial urrent I (H ) at ertain partiular magneti fields,

32 H n = nh 1, where H 1 = Φ / A is the first mathing field, i.e. one flux quantum Φ per unit-ell area A ; and n is an integer whih orresponds to the flux quantum number in eah unit-ell. In some ases, the ommensurate effet an also be observed even when n is a frational number. For example, E. Rosseel et al. [45] found these frational mathings at n = 1/ 4, 1 /, and 3 / 4 during transport measurements on W.67 Ge.33 films with square hole-arrays. However, the minima are muh less pronouned at the frational mathing fields. In this thesis, I will onentrate on the integer ones. n Figure -6. Field dependene of the resistane and ritial urrent for a Nb film with a square array of holes [46].

33 There are several different mehanisms to interpret the observed 1 ommensurate effet [47-61]. The widely used one is the pinning enhanement model [16-18, 57-6]. In the mixed state of type-ii superondutors, Abrikosov vorties arrange in various phases, from the ordered triangular Abrikosov lattie to totally disordered phases [61, 6] depending on the temperature and the magneti field. The resistane of the superondutor is aused by the vortex motion aross the sample. As a simple model [49], the resistane is determined by the density of free vorties n f (T ) and their mobility μ (T ), whih is given by R( T ) = Φ n f ( T ) μ( T ) (-4) where is a onstant. When periodi artifiial pinning enters, suh as submiron holes (antidots) and magneti or nonmagneti dots, are introdued into the superondutors, the well defined periodi pinning potential is formed and interats with the vorties [63, 64]. At zero magneti field, free vorties still might be reated by thermal exitation or by urrent-indued vortex pair unbinding. There is always a probability for a vortex to ross the sample. It is reasonable to use the Maxwell-Boltzmann distribution for thermal ativation to estimate the vortex mobility as follows, Ubar K BT = 1 e = 1 U kin + U wire K BT μ ( T ) e (-41) U bar is the energy barrier for a vortex to ross from one ell to the next, whih inludes two parts: U kin is the kineti energy barrier for a vortex to ross a unit

34 distane without pinning influene and U wire is the energy hange assoiated with phase differenes between adjaent ells. When a perpendiular magneti field is applied, the field-indued vorties enter the sample and U kin is dramatially lower than the zero field value [65]. The resistane inreases signifiantly. However, at ertain speial applied fields, the period of vortex lattie oinides with the hole-array, resulting in a very stable state, and all vorties are pinned by the holes. Both the density and mobility of free vorties drop dramatially to a level omparable to the zero field situation. With this pinning enhanement, a resistane dip appears at eah mathing field. In the temperature dependene of the resistane, R (T ), at various fixed magneti fields, a distint rossover is observed at the mathing field [] shown in Figure -7. The arrows point out the rossover temperature between the integer and half integer mathing field. This phenomenon an be explained by the pinning enhanement model. Sine all fluxions are pinned at a mathing field, small hange of the temperature in the transition region an signifiantly affet the density of free vorties n f (T ) and their mobility μ (T ). Therefore, the transition of the R (T ) urve is very sharp. On the other hand, at nonmathing fields, there always exit field-indued free vorties. Thus, the hange of the resistane is not as dramati as that at the mathing fields, whih results in a muh broader transition. As a result, rossovers of the R (T ) urves at different fields appear.

35 3 Figure -7. Resistane as a funtion of the temperature under various fixed magneti fields for a Nb film with a square array of holes []. ( H = 1 9 G) Another important model is based on the loalization transition effet [6, 66]; i.e., all vorties drop inside holes when the width of the superonduting strips irumventing holes is omparable to the oherene length. Sine the 1 superonduting oherene length ( ) ξ 1 T / T is temperature dependent, at temperatures lose to T, a superonduting film ontaining a periodi hole-array is analogous to a well studied system, the superonduting wire network [67]. Early researh [47-5] shows a periodi ritial temperature osillation in the superonduting wire network system due to the additional suppression of T from the fluxoid quantization effet, whih is similar to the Little-Parks effet.

36 4 However, there is a phase shift beause of the orrelation between superurrent loops in the wire networks. I will briefly introdue this problem in the following setion. First, let s onsider a simplified problem: the infinite film with only one hole of radius r at the origin with a uniform magneti field perpendiular to the film, whih is governed by the linearized GL equation: 1 m and the boundary ondition: e e r h ih A ψ = ψ (-4) m ξ ( T ) e 1 e n ih A ψ h ρ =r = (-43) where n is the unit vetor perpendiular to the boundary of the superondutor. This problem is omparable to the Shrodinger equation for a single partile with mass ondition: m e and harge e in the same magneti field and with the same boundary e e i i i h A ψ = E ψ (-44) 1 r i m The energy 36): i E has the same form as the result of Little-Parks experiment (Eq.- E n 1 h = m = evs n mer Φ Φ (-45)

37 5 Hene the ritial temperature of the superondutor governed by Eq.-4 an be speified by applying the ground state energy E = min( E i ) from Eq.-45 [68]: E = h h = m ξ ( T ) m [ ξ () /(1 T / T )] e e (-46) thus, T T m () eξ 1 E h = (-47) Having got this result, it s time to solve the problem in a superonduting network, whih is roughly onsidered as an array of loops with the same relation for the energy (Eq.-45). Due to the phase orrelation between loops, it is not possible for all of them to be at the lowest energy level. In general, when Φ n < < n + 1, there are two possible states for any unit ell: E n and E n+ 1. So Φ the average energy per unit ell an be approximately written as E Φ Φ E Φ h 1 Φ 1 + ( 1 ) E = n n Φ m r 4 Φ e = n+ 1 (-48) Putting it bak to Eq.-47, the magneti field dependene of the normalized ritial temperature is found, T T m eξ () ξ () 1 Φ 1 = 1 E = 1 n h r 4 Φ (-49) A typial ase when ξ ( ) / r = 1 is plotted in Figure -8. Comparing the result of the Little-Parks effet (see Figure -5), a half period phase shift shows

38 6 up. Thus, the periodi T suppression is learly observed and its period is exatly following the mathing field. The resistane minima our at the integer flux quantum mathing fields. Figure -8. Magneti field dependene of the redued ritial temperature in a wire network. ( ξ ( ) / r = 1) Both of the mehanisms seem to work at this point, i.e., qualitatively explaining the ommensurate effet. However, the pinning enhanement model

39 7 onentrates more on the vortex motion behavior. In ontrast, similar to the Little- Parks effet, the wire network model studies the total fluxoid through eah hole in the film. No methods exist to distinguish these two mehanisms. That is, the ommensurate effet observed in superonduting films ontaining periodi holearrays an originate from hole-indued pinning enhanement at the integer flux quantum mathing fields or the T suppression due to noninterger flux quantum fields or even both. In the later part of this thesis, I will present experimental approahes to reveal the origin of the ommensurate effet observed in superonduting niobium at temperatures lose to T.

40 CHAPTER 3 METHODOLOGY Sample Fabriation In order to observe the ommensurate effet, the size and the separation of the artifiial defets have to be omparable with the relevant length sales of the superondutor suh as the Ginzburg-Landau oherene length ξ (T ), whih is typially in the submiron range. For example, the BCS oherene length of pure niobium is ξ 38 nm [69]. A niobium film is typially in the dirty limit, i.e., the = eletron mean free path l is muh smaller than the oherene length. Its zerotemperature Ginzburg-Landau oherene length ξ () depends on l and an be expressed as ξ ( ) =.855( ξ l ) 1/. Sine ξ(t) = ξ() (1 T /T ) 1/ is temperature dependent, smaller features will enable larger regions to observe the ommensurate effet. The modern nanoengineering tehnologies, e.g. eletron beam (E-beam) lithography [7-73], laser interferometri lithography [74], and self-assembled templates like anodi aluminum oxide (AAO) [4, 11, 17], make it possible to fabriate nanosale hole-arrays. However, most of the lithography tehniques are

41 9 restrited to the submiron limit and the ommensurate effet only takes plae at relatively low mathing fields (~ G) and a narrow temperature range very lose to T, the zero-field ritial temperature. On the other hand, the self-assembled anodi aluminum oxide (AAO) membrane [4] an provide a triangular lattie with the hole separation down to ~ 1 nm and hole diameter as small as ~ 45 nm, whih is the smallest size ever reported. But the hole lattie is not perfetly ordered throughout the whole sample. Instead, the perfet hole lattie is onstrained within several mirometer size domains and the surfae roughness of the film is also very diffiult to ontrol. In this thesis, niobium thin films ontaining periodi hole-arrays are fabriated with E-beam lithography followed by foused ion beam (FIB) milling. Detailed proedures are shown in Figure 3-1. First, a ~3 nm thik E-beam resist layer, polymethyl metharylate (PMMA-95-A4), is spin-oated on top of the silion wafer, whih is overed by a 1-nm-thik SiO insulator layer. Then the sample is baked on a hot plate at 18 C for 9 seonds. A 5 µm by 5 µm bridge with standard four-probe transport measurement pads is exposed by the Raith 15 E-beam lithography tool. The pattern is developed in E-beam resist developer, MIBK: IPA (1:3) solution, followed by Isopropanol for 75 seonds eah. Thus, the pattern is defined on the wafer. Next, a niobium film with desired thikness is sputtered on the pattern with an AJA ATC4 sputtering system. The base vauum of the system is better

42 than Torr. The pressure of the working argon gas is maintained at mtorr during the sputtering. The growing rate is ~ 3.1Å/seond. After the sputtering, a standard liftoff proedure with aetone is followed. Only the niobium bridge struture is left on the wafer. PMMA SiO Si 1. Coat and bake E-beam resist 4. Nb Deposition (sputtering) E-beam. Expose with E-beam 5. Lift-off Ga beam 3. Develop 6. Milling hole-array with FIB Figure 3-1. Proedures of sample preparation. Finally, a well defined hole-array is diretly milled out in the niobium bridge with a FEI Nova 6 FIB mahine. A gallium soure is used for produing the ion beam. The aelerated voltage is set at 3 kv and the beam urrent is

43 31 fixed at 98 pa. In this ondition, the approximate beam spot size of the ion beam is ~ nm. The hole diameter ontrolled by the dwell time is about 3 nm for the dwell time of milliseonds. The SEM images (top view and side view) of the niobium film ontaining the hole-array with 15 nm hole to hole distane are shown in Figure 3-. Aording to the 45 side view image at the edge of the film, the holes are learly milled throughout the niobium film to the SiO insulate layer. Surrounding the holes, small regions with lighter ontrast are observed in the SEM image. It is due to the niobium redeposition and gallium ontamination during the FIB milling. However, the quality of the sample will not be affeted too muh beause these ontaminations mainly happen on the top layer of the sample surfae. Although the gallium ions might penetrate into the film and destroy the superondutivity in that part, this region is onstrained very lose to the edge of the hole and an be onsidered, as the effetive radius of the hole is a little bit larger than what is observed.

44 3 Figure 3-. SEM images of a triangular hole-array in a niobium film. (left) Top view; (right) Side view. There are many advantages using this new method. First, unlike the AAO template in whih the hole-array is perfetly ordered only in mirometer size domains, the hole-array fabriated by FIB milling uniformly overs the whole sample (the bridge between the two voltage leads). The lattie parameters of AAO templates, suh as the hole diameter, the hole shape, the lattie type and period, are very diffiult to adjust, whereas FIB pattering an easily ontrol them. Seond, ompared with diret E-beam pattering, the ontamination due to the resist is limited near the edges. Thus, the quality of the niobium film is muh better than those samples diretly patterned with E-beam lithography, in whih the resist ontaminates the whole sample sine the resist exists at all hole loations. Third, the roughness of the sample surfae is almost as good as the ontinuous film, whih redues unwanted topology effets. Five different samples, Sample A through E, were fabriated with the above method. Sample A is the referene niobium film bridge without hole-arrays. The optial mirosope image is shown in Figure 3-3. A niobium bridge is defined within a 5 µm (long) by 5 µm (wide) retangular area between the two voltage ontat leads. The two rosses are the referenes for the oordinate alibration.

45 Sample A 33 V + V - I + I - Figure 3-3. Optial mirosope image of the referene niobium film. For omparison, a niobium thin strip, Sample B, was also fabriated. It is defined in the niobium bridge with 1-nm-wide groves, whih is ut out of the niobium film with FIB. The width of the strip is ~15 nm and the distane between the enters of the two voltage ontat leads is µm. In order to ahieve the best quality, the struture is plaed near the middle of the film bridge (see Figure 3-4).

46 34 Sample B V + V - I + I - Figure 3-4. SEM image of the niobium thin strip. Samples C through E are niobium films ontaining hole-arrays with different film thiknesses and lattie types for investigating the effets of these parameters. The base film used for FIB milling is a 5 µm by 5 µm retangular niobium bridge whih is idential to the referene film struture, Sample A. However, the thikness varies; Samples C and D are 6 nm thik, whereas the thikness of Sample E is 1 nm. The sizes of the holes are fixed, ~3 nm, by using the same parameters during FIB milling. The SEM images are presented in

47 35 Figure 3-5. The designed dimensions of the hole-array patterns are µm larger than the niobium bridge in both the length and width diretions to ensure that the whole bridge is overed by the hole-arrays (Insets in Figure 3-5). The hole-arrays in Sample C and D are triangular latties. The distanes between two adjaent holes are 15 nm and 3 nm, respetively. Sample E ontains a square lattie of holes, in whih the hole to hole distane is 15 nm. The main parameters of all five samples are summarized in Table 1. Table 1 Niobium thin films ontaining different types of strutures Hole Separation Thikness T ξ () Sample Strutures (nm) (nm) (K) (nm) A None N/A B Strip N/A C Triangular hole-array D Triangular hole-array E Square hole-array Note: ξ () is obtained by fitting the linear part of the H (T ) phase diagram.

48 Sample C 36 Sample D Sample E Figure 3-5. SEM images of niobium films ontaining hole-arrays with different types of latties. The insets show the images of the entire patterned bridges.

49 PPMS Measurement System 37 Transport measurements were arried out in a Physis Property Measurment System (PPMS) ( from Quantum Design) by using a standard four-probe DC setup. The samples are eletrially onneted by 5-µm-thik gold wires with indium diretly pressed on the ontating pads. The applied DC urrent is 5 µa. For eah data point, averaging of three times is applied to redue the measurement noise. The magneti field is always applied in the vertial diretion and an be up to 9 Tesla. With the horizontal rotator option, the angle between the surfae of the sample and the magneti field an be preisely ontrolled (see Figure 3-6). The range of the rotation is from -1 to 37 with the resolution as high as.53. Instead of the system thermometer, whih is loated at the bottom of the sample hamber, a user thermometer on the rotator is used for temperature ontrol. Sine it is very lose to the sample, superior temperature ontrolling is ahieved. The temperature is stablized within ± mk during the measurement.

50 38 H Magneti field I Figure 3-6. Sample rod with the horizontal rotator.

51 CHAPTER 4 ANISOTROPY IN D AND 1D SUPERCONDUCTORS Critial Field of a D Niobium Superonduting Thin Film As mentioned in the previous hapter, hole-arrays are ahieved in niobium thin films through FIB milling. Thus, the quality of the base film is very ritial to obtain high quality patterned films. It is neessary to investigate basi properties of the base film first. Besides, omparing the results of the FIB patterned samples with the referene ontinuous film an also help to better understand the effets indued by a periodi hole-array. A 6-nm-thik niobium film was well defined on the SiO /Si substrate by E- beam lithography tehniques. The bridge between voltage leads is a 5 µm (long) by 5 µm (wide) retangular area (see Figure 3-3). The film thikness was ontrolled by a alibrated quartz-rystal thikness monitor during the sputtering. The applied urrent was fixed at 5 µa for standard four-probe transport measurements at various magneti fields and different angles between the normal diretion to the surfae of the film and the applied field.

52 Figure 4-1 shows the zero-field resistive transition of the D ontinuous 4 niobium film. The zero-field ritial temperaturet is ~ 8.65 K by hoosing the resistive midpoint as the riterion. The transition is sharp and displays no strutures, indiating the high quality of the film. The transition width, i.e., the differene of the temperatures at 1% and 9% of the normal resistane, is only ~ mk. Figure 4-1. Resistive transition of the niobium referene thin film at zero-field.

53 41 This ritial temperature is a little bit lower than that of the niobium bulk sample, whih is well known as 9.46 K [75, 76]. There are several possible reasons. First, the sputtering system used for fabriation is not an ultra high vauum (UHV) system. The system vauum is restrited at ~1-8 Torr. Seond, it is not possible to avoid exposure of the film to air and prevent formation of a thin metalli oxide on top of the niobium film. Third, the thikness of the film is in the intermediate range ompared to the thik films. It also redues the ritial temperature [77, 78]. The superonduting phase diagrams (see Figure 4-) with the magneti field along both the parallel and perpendiular diretions have been obtained by measuring the temperature shift of the midpoint of the normal-to-superonduting resistive transition at different fields (some urves obtained in parallel fields are shown in the inset of Figure 4-). In these phase diagrams, the temperature is plotted in terms of its redued value, t = T / T. As expeted, when the magneti field is applied perpendiular to the surfae of the film, the phase boundary has a linear temperature dependene, whih follows the expression of Eq.-17. For onveniene, it is rewritten in the temperature dependent form: H ( T ) = Φ π ξ ( T ) = Φ (1 π ξ () T T ) (4-1) where ξ (4-) ( ) () (1 / ) 1/ T = ξ T T is the Ginzburg-Landau oherene length.

54 4 4G G H // H Figure 4-. Phase diagrams of the niobium referene film in parallel and perpendiular fields. ( T =8.65 K). The inset shows resistive transition on field ooling in seleted fields along parallel diretion used to determine T (H ).

55 43 On the other hand, the phase diagram in the parallel field is a little bit more ompliated. In the temperature region lose to T, it follows the thin film limit of the Ginzburg-Landau theory, the square-root behavior, H ( T ) = 1 Φ π d ξ ( T ) = 1 Φ 1 () π d ξ T T 1 (4-3) where d is the thikness of the film. But at around.9t (~7.78 K) a rossover into linear temperature dependene is observed. The physis is lear here: when the temperature is lose to T, the oherene length beomes larger than the thikness of the film and the superondutivity is onfined to one-dimension (1D) along this diretion, whih gives the paraboli behavior of the superondutivity phase boundary; however, at temperatures far away from T, the oherene length beomes muh smaller than the thikness and the thin film onfinement doesn t work any more. H. J. Fink [79] first studied this dimensional transition for superonduting thin films and onluded that the rossover point appears at d 1.84 ξ ( T ), where vorties start to enter the materials. As shown in Figure 4-3, the linear and nonlinear parts of the phase diagram in parallel fields (red open irles) are niely fitted with Eq.4-1 (blue dashed line) and Eq.4-3 (green dashed line), respetively, whih yields ξ ( ) 9. 8 nm. Using this result, the alulated temperature dependene of the Ginzburg-Landau oherene length (Eq.4-) is also plotted as a blak dashed line. At the rossover

56 44 point, ~.9T, the alulated value of the oherene length is ~ 31. nm, whih is very lose to the Fink s limit, ξ = d rossover 3. 6 nm Linear region Paraboli region ξ ( T / T ) 31. Figure 4-3. Analysis of the phase diagram in the parallel field for the niobium referene film. ( T =8.65 K). Open irles represent the measured phase diagram. The dashed lines are the fitting urves.

57 The fitting result, ξ ( ) 9. 8 nm, is signifiantly smaller than the BCS 45 oherene length of niobium, ξ 38 nm [8]. It indiates that the film is in the = dirty limit and the eletron mean free path an be estimated by the dirty limit equation [37]: ξ ( ) =.855 ( ξ l (4-4) 1/ ) whih gives l nm. Finally, the angular dependene of the ritial field at fixed temperatures is studied in the region lose to T (paraboli region). The ritial fields are obtained from the magneti field dependene of resistane urves. The midpoint of the normal resistane is still used as the riterion. Shown in Figure 4-4 with red solid irles are the ritial fields at the orresponding angles θ (see Figure -1 for the definition). The temperature is fixed at ~.98T (8.5K) and the ritial fields are normalized with H ( θ = ), i.e., the perpendiular ritial field. The measured data are niely fitted by Tinkham s thin film angular dependene formula for the ritial field (Eq. -5). The fitting results are ompared at different fixed temperatures,.98t,.95t and.9t, as shown in the inset of Figure 4-4. Clearly, Tinkham s model works better at temperatures loser to T. At.9T, the fitting urve starts to diverge from the measured data. It is beause this temperature is on the edge between the paraboli region and the linear region, and the vorties start to appear.

58 46.98 T Figure 4-4. Angular dependene of the ritial field for the niobium referene film at a fixed temperature. ( T = 8.65 K). The inset ompares the results at different fixed temperatures. The dashed lines are the fitting urves with Tinkham s formula. In summary, high quality niobium films an be fabriated, whih offers a good base for future researh. The estimation of the oherene length shows the films are in the dirty limit. The angular dependene of the ritial field at a temperature near T follows the Tinkham model.

59 Resistive Anisotropy of a 1D Niobium Strip 47 When the thikness of a superonduting film is omparable to the oherene length, vorties do not appear inside the sample in parallel fields, while they an form in perpendiular fields. Unlike a thin film, a superonduting strip should have similar behavior when the magneti field is applied either along parallel or perpendiular diretions. Obviously, the angular dependene of the ritial field should also be quite different with a thin film. This is a very interesting topi to study. Furthermore, sine strips are the fundamental struture in a superonduting wire network, studies on the properties of a strip will shed light on the properties of wire networks. For a better omparison with the niobium films ontaining hole-arrays, whih will be investigated later, strips studied in this thesis are also ut out of high quality films so that they have the same mirostrutures. Thus, results obtained in the thin strips an be diretly applied to the analysis of the data obtained in patterned films. As shown in Figure 3-4, a 15-nm-wide niobium thin strip with four transport measurement ontats is defined by FIB in the referene film, whose thikness is 1 nm. The distane between the two voltage leads is µm. The zero field ritial temperature T is ~ 7.4 K, whih is signifiantly lower than the referene film. The main reason is the gallium ontamination during the FIB milling, that has also been disussed in that hapter.

60 The superonduting-normal phase diagrams in the parallel and 48 perpendiular magneti fields (see Figure - for the definition of the oordinates) are presented in Figure 4-5. Unlike the referene film, the paraboli behavior appears in both urves at temperatures lose to T. It is reasonable beause in eah ase the area being exposed to the magneti field has the same geometry as that of the referene film in parallel fields. To fit the experimental data in the paraboli region, the theoretial ritial field formula for a thin strip in both parallel and perpendiular diretions, Eq.-6, are rewritten in a temperature dependent form as below, H H // 1 Φ 1 Φ ( T ) = = 1 ( ) () π d ξ T π d ξ 1 Φ 1 Φ ( T ) = = 1 ( ) () π w ξ T π w ξ T T T T 1 1 (4-5) where d = 1 nm and w = 15 nm are the thikness and the width of the strip, respetively. When ξ ( ) = 7. 6 nm is used in the above equation, the theoreti H (T ) values for both perpendiular and parallel magneti fields (see blue dashed lines in Figure 4-5) are in good agreement with the experimental data. This fitting result indiates that the strip is also in the dirty limit and the eletron mean free path is l. 8 nm.

61 49 H // H H // H Figure 4-5. Phase diagrams of the niobium referene strip in parallel and perpendiular fields. ( T =7.4 K). The dashed lines in the main panel are the fitting urves. The inset shows the saling relation between these two phase diagrams. The solid squares in the inset represent the saling result after multiplying a fator on the field of the phase diagram in parallel fields.

62 Eq.4-5 also indiates a relation for the ritial fields in parallel and 5 perpendiular diretions: d H ( T ) = H // ( T ) (4-6) w This implies that the superonduting phase diagram in perpendiular magneti fields an be reonstruted by multiplying a oeffiient, d α = w, to the magneti fields of H (T ) phase diagram in parallel fields. The result is shown in the inset of Figure 4-5. The saling fator, α =. 667, almost perfetly follows the theoretial value. This saling method will be of great help in identifying the base line of the phase diagram for a system whih has muh more ompliated phase boundary, for example, a superonduting wire network. Next, the angular dependene of the ritial field is investigated. Choosing a temperature lose to the zero-field ritial temperature,.98t (6.9 K), the H (θ ) is determined by measuring R (H ) urves at various angles and is presented in Figure 4-6 with red open irles with θ varying from 18 to. In the PPMS system, the magneti field is fixed along the vertial diretion. The sample an freely rotate through a horizontal axis. o θ = 18 represents the o situation in whih the sample horizontally faes down to the ground and θ = is the opposite way. In both ases the magneti field is perpendiular to the width diretion. The fitting urves of the strip model (blue dashed line) and Tinkham model (green dashed line) are obtained from Eq.-31 and Eq.-5, respetively.

63 51 Figure 4-6. Angular dependene of the ritial field for the niobium referene strip at a fixed temperature. ( T =8.65 K). The inset ompares the results at different fixed temperatures. The dashed lines are the fitting urves. It is not surprising that the Tinkham fitting doesn t work well in this ase. Roughly speaking, in the Tinkham model with the magneti field perpendiular to the film, the magneti fluxes penetrate the film and vorties form, whereas in parallel fields, vorties an not exist due to the onfinement along the thikness diretion. Hene a usp-like peak appears at θ = 9. However, a strip remains 1D

64 5 at temperatures lose to T during the sample rotation, i.e., both d and w are omparable to the oherene length and no vorties or sreening effets exist in the strip. Thus, the anisotropy purely depends on the geometri differene, whih has been desribed by the thin strip model in Chapter. Sine the mehanism is the same in both diretions, the hange of the ritial field at around 9 annot be as dramati as that of a thin film and a sinusoidal type peak shows up. The fitting results of the thin strip model are ompared at different fixed temperatures,.98t,.96t, and.94t, and given in the inset of Figure 4-6. Beause of the symmetry, the data are provided from to 9 only. Just like the Tinkham model for the thin film, the thin strip model works better at temperatures loser to T. Signifiant deviation between the experimental data and the fitting urve an be seen for the one at.94t beause the strip has reahed the transition between paraboli and linear regions. In the previous disussion, the ritial field is always defined with 5% of the normal resistane riterion. Naturally, one might ask whether this thin strip angular dependene relation still works or not if the riterion has hanged to different perentages. The best way to answer this question is to sale R (H ) urves at different angles by multiplying the applied magneti field with the orresponding fators obtained from the angular dependene relation of the ritial field, Eq.-31, H 1 A ( θ ) = = os θ + sin θ (4-7) H ( θ ) γ

65 53 H // w where γ = =. If all of R (H ) urves an be merged to the one at, we H d an safely onlude that this relation does not depend on the riterion used to determine the ritial field. Ten R (H ) urves from to 9 with a step size of 1 at the fixed temperature.98t (6.9K) are hosen (see the inset of Figure 4-7). The results after saling are given in the main panel of Figure 4-7. All the onverted urves overlay on the measured R (H ) urve at. It indiates that all the information behind the angular dependene of the magnetoresistane R (θ ) is ontained in one base urve, R (H ) at, plus the orresponding angular dependene of the ritial field. Hene, the R (θ ) and R (H ) urves at the fixed temperature an be onverted to eah other.

66 54 9 Figure 4-7. Saling behavior of R (H ) urves at various field diretions for the niobium referene strip at a fixed temperature. ( T =.98T or 6.9K). The inset provides the measured R (H ) urves at different angles. The main panel represents the data after saling by the fator 1 A ( θ ) = os θ + sin θ with γ γ = 1.5.

67 The method to reonstrut the R (H ) at with R (θ ) urves at different 55 fields is pretty straightforward. Sine the temperature is fixed during the measurement, the resistane of the sample depends on two variables only, the magneti field H and the angle θ. For eah point of the R (θ ) urves, the orresponding point with the same value of ( H, θ ) in R (H ) urves an always be found. Their resistane should be idential if the measurements are onsistent. Using the onlusion obtained from the previous setion, this point an be mapped into the base urve, R (H ) at, with the saling fator, A (θ ). A shemati drawing is given in Figure 4-8. The formula to alulate the orresponding H for eah R (θ ) urve at fixed field H app is 1 H = H app A( θ ) = H app os θ + sin θ (4-8) γ θ = α ( θ 1 ) R H 1, θ ) ( 1 R H 1, θ ) ( 1 Figure 4-8. Shemati outline of R (θ ) to R (H ) onversion.

68 56 Shown in Figure 4-9 is the result at temperature.98t. The data onverted from R (θ ) urves at different applied fields (thin lines with different olors) almost perfetly reonstrut the measured R (H ) urve at (red open irles). The originally measured R (θ ) urves at different fields are also provided in the inset. Figure 4-9. Reonstrution of R (H ) at from R (θ ) urves. The inset provides the measured R (θ ) urves in various fixed fields. ( T =.98T or 6.9K)

69 57 Meanwhile, the inverse onversion should also be valid based on the same strategy, i.e., the R (θ ) urves at fixed applied field H app an be derived with one basi urve, R (H ) at. The only differene is for eah data point, θ has to be speified orresponding to the applied field. The relation is given as follows, θ = arsin H H app 1 γ 1 1 1/ (4-9) The fitting for R (θ ) urves at different temperatures with the applied field fixed at 4 G is shown in Figure 4-1. Within the measurement error, the values derived from R (H ) at (open symbols) niely fit the diretly measured data (thin lines). As a onlusion, the niobium thin strip demonstrates a variety of interesting anisotropi properties. Beause of seond onfinement along width, the angular dependene of the ritial field follows the thin strip formula developed in Chapter rather than the Tinkham model, whih works for the thin film. A saling relation has also been found for the magnetoresistane R (H ) at different angles. This result demonstrates that R (H ) and R (θ ) urves an be onverted to eah other based on the angular dependene of the ritial field.

70 58.98 T.975 T.96 T.95 T.945 T.97 T.94 T.965 T.93 T Figure 4-1. R (θ ) fitting for the referene strip at different temperatures in a fixed field (H = 4 G). The thin lines are the measured data and the open symbols are the fitting results.

71 CHAPTER 5 ANISOTROPIC PROPERITIES IN SUPERCONDUCTING FILMS WITH PERIODIC HOLE-ARRAYS Superonduting niobium films ontaining periodi hole-arrays exhibit rih novel vortex phenomena and have attrated muh interest in reent years. The ommensurate effet is one of the hot topis in this area [4-17]. Exploring the mehanism behind this effet an greatly help us to understand the omplex vortex dynamis in this kind of system. At temperatures lose to T, different mehanisms an aount for this effet observed in transport measurements. However, no effetive methods exist to distinguish them. In this hapter, I present experimental approahes to investigate the origin of the ommensurate effet. This work also reveals that a hole-array affets the anisotropi properties of a superonduting film. For example, the angular dependenes of the ritial temperature T (θ ) and magnetoresistane R (θ ) of patterned films are different from those of a ontinuous film. Suh data an provide more onlusive information on the origin of the ommensurate effet in this system. The experiments are arried out on three niobium films with different types of hole-arrays (Sample C through Sample E). All of them are fabriated with the

72 6 same method, E-beam lithography followed by FIB milling. The hole diameter is fixed at ~3 nm sine the same parameters are used during FIB patterning. The hole-arrays in Sample C and Sample D are triangular latties. The thikness of the niobium film is 6 nm for both samples. To ompare the influene of the neighboring hole separation on the temperature range in whih the ommensurate effet an be observed, the hole-to-hole distane is set to 15 nm and 3 nm for Sample C and Sample D, respetively. Sine the symmetry of the hole-array an also influene the ommensurate effet, a hole-array with a square lattie is fabriated into Sample E. The distane between two adjaent holes is 15 nm. The thikness of the niobium film inreases to 1 nm, whih will largely hange the oherene length and therefore the temperature range of the ommensurate effet. The main parameters of these three samples are summarized in Table 1 and more details on sample haraterizations an be found in Chapter 3. A standard four-probe d transport measurement is applied using onstant urrent mode. A riterion of.5 R N is used to define the ritial temperature T and the ritial field H, where R N is the normal state resistane. Relation of the Properties in Parallel and Perpendiular Fields At temperatures near T, transport measurements on superonduting films ontaining periodi hole-arrays show an intriguing ommensurate effet whih appears as minima in the magneti field dependene of the resistane, R (H ), or

73 maxima in the field dependene of the ritial urrent I (H ) when an integer 61 number of flux quantum, n Φ, is ommensurate with unit ell of the hole-array. In the following investigation, I will fous on the resistane, whih an also be translated to the ritial urrent I defined as I = V R. / Two mehanisms an aount for this effet [47-6]. The first one is based on the pinning enhanement model, in whih the ommensurate effet is attributed to ommensurate pinning of the entire vortex lattie by the hole lattie at mathing fields. In this state, the resistane of the system is signifiantly redued at the mathing fields due to the large derease of the mobility of the vortex. The other one is the wire network model. Sine the superonduting T oherene length, ξ ( T ) ~ 1 T 1, is temperature dependent, superonduting films ontaining hole-arrays should resemble superonduting wire networks at temperatures lose to T, where the width of the superonduting strips irumvented with holes is omparable to ξ (T ). The system resistane inreases beause of the additional suppression of the superonduting ritial temperature T by magneti field at noninteger flux quantum values arising from the fluxoid quantization effet (Little-Parks effet) exept at integer flux quantum mathing fields where no T suppression ours. As a result, the resistane minima appear at the mathing fields.

74 To distinguish these two possible origins, an experimental approah is 6 designed. The shemati outline is displayed in Figure 5-1. The blak solid irles represent a typial R (H ) urve obtained in a superonduting film with a periodi hole-array at a fixed temperature near T. Resistane minima appear at mathing fields H n, where the magneti flux threading eah unit ell is integer times the flux quantum, n Φ. This measured urve an be onsidered as a superposition of a base line without the influene of the hole-array and the resistane hange indued by the hole-array. If the minima are due to pinning enhanement, the resistanes of the base line without the influene of the holearray (blue dashed line) should be always larger than or equal to the measured values. On the other hand, additional T suppression from the flux quantization effet results in exess resistane. Consequently, the R (H ) urve for a wire network in the absene of the hole-indued Little-Parks effet (red dashed line) should lie below the measured urves exept at the integer flux quantum mathing fields where they have the same values. Thus, developing a way to determine the base lines in the absene of the influene of the hole-array is ruial for understanding the origin of the ommensurate effet.

75 63 Baseline for pinning enhanement Typial measured R (H ) urve Baseline for a wire network Figure 5-1. Shemati outline to identify the origin of the ommensurate effet. In order to demonstrate the influene indued by the hole-array in a superonduting film, one usually ompares a patterned film with a referene ontinuous film. However, it is diffiult to reah a reliable onlusion sine T and the transition widths of the patterned films often differ from those of the referene film. Furthermore, by taking a loser look at a typial experimental H (T ) phase boundary near T for a patterned film (See Figure 5-, part A), it follows a paraboli bakground augmented by T osillation rather than the linear behavior of the D ontinuous thin film, T H ~ 1 (See Figure 5-, part B). T

76 It an not be understood with the pinning enhanement model, whih assumes 64 that the hole-array does not alter the properties of a ontinuous film exept induing the ommensurate pinning. In fat, it indiates that the patterned film has beome a new system whih is different from a ontinuous film [3, 4, 15, 6]. On the other hand, in the wire network senario, the sample is treated as a D multiply onneted superonduting strip system. Thus, at temperatures lose to T, the H (T ) phase diagram of a film with a periodi hole-array should have a bakground whih is similar to that of a superonduting strip. Aording to previous disussions, under the perpendiular magneti field, the superonduting strip also demonstrates a paraboli H (T ) phase boundary near T (see Figure 5-, part C). From this point of view, the wire network model is more suitable in this ase and a thin strip should be used to ahieve the baseline without the influene of the hole-arrays shown in Figure 5-1. However, fabriating a superonduting thin strip with exatly the same quality ( T and transition width) as that of a film with a hole-array is very diffiult due to potential damage or degradation aused by the patterning proess. An alternative approah has to be developed.

77 65 A B C Figure 5-. H (T ) phase diagrams of niobium film with different strutures in the perpendiular field. (A) with a hole-array; (B) ontinuous film; and (C) thin strip. In Chapter 4, I demonstrated that the Tinkham formula for the ritial field of a thin film with a thikness d in parallel fields (see Eq.4-3) an also be applied to a superonduting thin strip (its width w and the thikness d are both less than 1.84ξ ) with the magneti field applied either in perpendiular or parallel diretions sine the area exposed to the magneti field has the same geometry. The only differene is that we have to replae d with w in perpendiular fields d (see Eq.4-5). Consequently, with a saling fator, α =, the H (T ) phase w diagram and the magneti field dependene of the resistane R (H ) in

78 66 perpendiular fields an be reonstruted from the orresponding data in parallel fields. Both theory [81] and experiments [81, 8] have shown that there is no Little-Parks effet and ommensurate effet when the magneti field is applied parallel to the superonduting wire network system, whih implies that if a film ontaining a hole-array behaves like a wire network, R (H ) urves in parallel fields an be used to derive the baselines for those in perpendiular fields. As demonstrated below, by using this approah the origin for the ommensurate effet in niobium films ontaining periodi hole-arrays an be revealed. The red open irles in Figure 5-3 represent the R (H ) urve at.99t (5.9 K) in perpendiular fields for Sample C with T =5.96 K. Resistane dips our at mathing fields with the period 17 G. This value is onsistent with the alulated first mathing field for a triangular hole lattie, H 1 Φ = =16 G 3 a where a = 15 nm is the separation of neighboring holes. In order to obtain the base line for the sample, as suggested in Figure 5-1, the related R (H ) urve at the same temperature but in parallel fields is also measured (blue solid squares in Figure 5-3). If the resistane minima were indued by pinning enhanement at the mathing fields, the blue dashed baseline suggested in Figure 5-1 ould have been reonstruted by saling the R (H ) urve in parallel fields with a fator of α = H / H //, whih represents the anisotropy of the film geometry. The blak

79 67 dashed urve in Figure 5-3 is obtained in this way, where a saling fator of.41 is used. It is lear that this saling approah does not produe the desired baseline. x.41 x.587 x.587 Figure 5-3. Baseline reonstrution for the R (H ) urve ( T =.99T or 5.9 K) and H (T ) phase diagram (inset). The solid and open symbols are experimental data obtained in parallel and perpendiular fields, respetively. The blue dashed and solid urves represent the baselines derived based on saling the measured urved in parallel fields. The blak dashed urve is the saling result by using a different fator for pinning enhanement model.

80 68 On the other hand, in a wire network, superurrent irulating the holes ontributes an additional T suppression mehanism in the presene of a magneti field along the perpendiular diretion exept the mathing fields, at whih the fluxoid through eah hole is integer flux quanta and no superurrents exist. Hene, the H (T ) phase diagrams for both perpendiular and parallel fields an be used to get the proper saling fator. By multiplying a fator, the phase diagram ( ) H // T in the parallel field should transform to a baseline whih forms an upper bound to the phase diagram of the perpendiular field. The inset of Figure 5-3 gives the result of this saling for Sample C. The red open irles and blue solid squares represent the H (T ) phase diagrams in perpendiular and parallel fields, respetively. By applying the saling with a fator of.587, a baseline (blue dashed line) is obtained whih remarkably overlaid with those points at mathing fields in the phase diagram of the perpendiular magneti fields. Sine this saling fator depends only on the geometri anisotropy, i.e. the thikness d and the width w of the superonduting segment between holes, it should also be appliable to derive the baseline for the R (H ) urves in perpendiular magneti fields (the blue solid line in Figure 5-3). After multiplying the magneti fields of the R (H ) urve in parallel fields with the same fator.587, a baseline is obtained lying below the measured urve exept at the integer flux quantum mathing fields where they have the same values. It is exatly what is expeted aording to the shemati outline in Figure 5-1, whih implies that

81 lose to T, a film ontaining a hole-array behaves like a wire network. 69 This saling relation should be valid at different temperatures, too. In Figure 5-4, baselines of R (H ) in perpendiular fields are investigated at various temperatures for Sample C. Two temperatures are hosen,.98t (5.85 K) (left) and.998t (5.95 K) (right). The measured R (H ) urves in perpendiular fields and parallel fields are represented by red open irles and blue solid squares, respetively. Multiplying the same saling fator,.587, on the fields H of the urves in parallel fields, baselines for the R (H ) urves in perpendiular fields are reonstruted. It is lear that the saling fator is independent on the temperature, whih is onsistent with the theory. T=5.85 K T=5.95 K x.587 x.587 Figure 5-4. Baseline reonstrution for the R (H ) urve at different temperatures. The solid and open symbols are experimental data. The blue dashed urves represent the baselines derived by saling the measured urved in parallel fields.

82 For a thin strip the saling fator, d α = w 7, is deided by the thikness d and the width w. Comparably, the wire network system should also follow this relation. For Sample C, the width w of the strips is the width (~ 14 nm) of the superonduting setion between two neighboring holes, i.e., the differene between the lattie period and the hole diameter (refer to the SEM image in Figure 3-). With the designed thikness 6 nm, the alulated ratio between the baseline of H and H // is ~.57. It is very lose to the experimental value of.587. The disrepany an be aused by surfae oxidation of the niobium film (usually a few nanometers), whih redues the atual thikness of the film. It is also diffiult to preisely ontrol the thikness during the sputtering beause the rystalline thikness monitor is a little bit away from the samples. Another reason is that the strips in niobium films ontaining hole-arrays are not as simple as an individual strip. It is irumsribed by holes, whih makes it diffiult to preisely define their edges and the real width. To verify the generality of the wire network behaviors in superonduting films ontaining hole-arrays, samples with different lattie parameters (hole separation and lattie type) are also investigated. Exept for the period of the hole-lattie hanges to 3 nm, Sample D is idential with Sample C. More details inluding SEM images refer to Chapter 3 and Table 1. The zero field ritial temperature T of Sample D is ~ 6.95 K, whih is muh higher than that of Sample C, ~ 5.96 K. It is beause of the smaller density of the holes in Sample

83 D. Less gallium ontamination is introdued during the FIB milling and the 71 integrality of the superondutor keeps better. Sample E ontains another type of hole-arrays, square lattie of holes. Both hole diameter and separation distane are idential to Sample C. But the thikness of the film hanges to 1 nm, whih is the thikest one among all three samples. Its zero field ritial temperature, ~ 7.65 K, is also the highest one. The main reason is that for a superonduting film T depends on the thikness of the film [77, 78]. Thiker film behaves more like the bulk sample, whih has a higher ritial temperature. Also, for a thiker film, the ratio of the ontaminated part to the unaffeted part, both from oxidization and gallium ontamination, is signifiantly redued. Thus, better superondutivity is ahieved in it. Figure 5-5 shows the baselines of R (H ) urves in perpendiular fields obtained from saling the orresponding R (H ) urves with the magneti field in the parallel diretion for Sample D (left) and Sample E (right). The red open irles represent the R (H ) urves at.998t (6.94 K) and.995t (7.61 K) in the perpendiular applied field for Samples D and E, respetively. The first mathing fields are 7 G for Sample D and 95 G for the square lattie. These results are very lose to the alulated ones, 65 G and 9 G. A frational mathing learly appears at half mathing field for the film with a square holearray. This phenomenon hasn t been observed in the two samples with triangular latties of holes. Simulations in the wire network [83] show that at frational mathing fields the vorties distribution in the square lattie are more stable than

84 7 those in the triangular lattie and T suppression is signifiantly redued. That is the reason why the frational mathing is observed in Sample E only. Compared with those at integer flux quanta, this frational mathing is extremely weak. Thus, I will onentrate on the resistane minima at integer number flux quanta only. The H (T ) phase diagrams are also inluded in the insets. T osillation with the same field periods as that in the observed R (H ) urves in perpendiular fields an be learly identified for eah sample (red open irles in the insets). By multiplying H // with a saling fator of.33 (Sample D) and.64 (Sample E), baselines (blue dashed lines) for both R (H ) urves and H (T ) phase diagrams in the perpendiular magneti fields are obtained. All of the baselines almost perfetly pass through the data points at the mathing fields and form lower (upper) boundaries to the R (H ) urves ( H (T ) phase diagrams), whih are exatly expeted in the wire network model. Thus, it is lear that all niobium films ontaining hole-arrays behave as superonduting wire networks at temperatures lose to T. The method for obtaining baselines for the R (H ) urves or H (T ) phase diagrams in perpendiular fields by saling the orresponding ones in parallel fields are also universally appliable to other superonduting films with periodi hole-arrays.

85 73 A B X.33 X.64 X.33 X.64 Figure 5-5. Baseline reonstrution of the R (H ) urve and H (T ) phase diagram (inset) for different samples. (A) for Sample D (triangular hole array with 3 nm hole separation; T =.998T or 6.94 K); (B) for Sample E (square hole array with 15 nm hole separation; T =.995T or 7.61 K). The solid and open symbols are experimental data obtained in parallel and perpendiular fields, respetively. The blue dashed and solid urves represent the baselines derived based on saling the measured urved in parallel fields. Another problem whih is diffiult to be understood by the pinning enhanement mehanism is the highest mathing fields that an be observed. Normally, the saturation number, n s D = with D as the hole diameter, is 4 ξ ( T ) used to estimate the maximum number of flux quanta that an be trapped in a single hole [6, 14, 17, 84, 85]. Take Sample C as an example. Fitting the linear part of the H (T ) phase diagram with fields applied along the perpendiular

86 diretion (see Figure 5-6) with the bulk expression, H ( T ) = Φ π ξ ( T ), yields 74 ξ ( ) = 8.5 nm. Thus, in our experimental temperature range the saturation number is n 1. However, the R (H ) urves in perpendiular field at different s temperatures (see the inset of Figure 5-6) show the ommensurate effet an be observed at the field as high as H = 85 G 8H 1. Usually it is explained by interstitial vorties whih experiene a aging potential originated from the pinned vorties in the holes. But aording to the H (T ) phase diagram the temperature is still very lose to T (~.99T ) even at fields higher than the seond mathing field. The paraboli bakground also indiates it is still in the wire network region and no vorties an enter the superonduting segments between the holes. It ontradits with the interstitial vorties explanation. On the other hand, if the ommensurate effet originates from the hole-indued T suppression, everything is straightforward. The region, either in temperature or in field, in whih the ommensurate effet an be observed, should be oinident with the T osillation region in the H (T ) phase diagram in the perpendiular field. As indiated in Figure 5-6, the T osillation in Sample C disappears exatly at H = 85 G.

87 75 ~85 G ~85 G Figure 5-6. Magneti field region of the ommensurate effet. For Sample C (triangular hole array with 15 nm hole separation). The main panel is the phase diagram in perpendiular fields. The inset provides the R (H ) urves measured at different fixed temperatures in perpendiular fields. Sine the T osillations are onstrained in the wire network region (nonlinear region) of the H (T ) phase diagram, there are several methods that an be used to adjust the temperature region in whih the ommensurate effet an be

88 observed: for instane, hanging the hole-hole separation or diameter; and 76 hanging the oherene length. Shown in Figure 5-7 is the omparison of phase diagrams in perpendiular fields for all three niobium films ontaining hole-arrays. Sample D is idential to Sample C exept the hole separation is inreased to 3 nm, whih signifiantly shrinks the nonlinear temperature region sine it is roughly deided by the wire network riterion, 1.84ξ () w < 1.84 ξ ( T ) =, 1 T / T where w is the distane between the edges of two neighboring holes. The maximum number of flux quanta of the mathing field also redues to ~ 4. On the other side, Sample E has the same hole separation with Sample C, whih produes almost the same temperature region of nonlinear part in the phase diagram. But the different thikness, 1 nm in Sample E ompared with 6 nm in Sample C, gives larger oherene length. Fitting the linear part of the H (T ) phase diagram of Sample E yields ξ ( ) 1 nm, whih is ~ 17.6% larger than the one in Sample C. Sine the inreasing rate of the paraboli bakground in the phase diagram is inversely proportional to ξ () (see Eq.4-3), the magneti field on the edge of the nonlinear region of Sample E is muh lower than that of Sample C. Thus, the maximum number of the observed mathing fields is also redued.

89 77 Figure 5-7. Comparison of T osillation region of the phase diagram in perpendiular fields for niobium films with different hole-arrays. See Table 1 for more detail of the sample parameters. In onlusion, an experimental approah has been developed to determine the origin of the ommensurate effet in superonduting niobium films ontaining periodi hole-arrays where minima and maxima appear in the magneti field dependene of the resistane and the ritial urrent, respetively. By saling the field dependene of the resistane and the ritial temperature with magneti fields along the parallel diretion, the orresponding baselines in perpendiular fields are reonstruted. It is found that in superonduting niobium films with both triangular and square hole-arrays the ommensurate effet originates from hole-indued T suppression at noninteger flux quantum fields. This onlusion is supported by omparing the maximum mathing field and the T osillation

90 78 region of the H (T ) phase diagram in perpendiular fields for patterned films with various hole-hole separations and thikness. Angular Dependene of the Mathing Field In the previous setion a onlusion was reahed that the ommensurate effet originates from hole-indued T suppression at noninteger flux quantum fields. The baselines of R (H ) urves in perpendiular fields, whih exlude the influene indued by hole-arrays, an be reonstruted by the field dependene of the resistane with the magneti field along the parallel diretion. It implies that if the sample is tilted with an angle to the applied magneti field only the perpendiular magneti field omponent ontributes to the ommensurate effet. A shemati diagram is shown in Figure 5-8. Considering the horizontal surfae as the referene, whih is marked with light blue olor, the magneti field is always applied along the vertial diretion. The sample is tilted with an angle θ to the referene surfae. The magneti field an be analyzed by two independent omponents with respet to the sample surfae, H // and perpendiular magneti field omponent H. Beause the H is the only relevant quantity, the angular dependene of the mathing field should be desribed by H θ ) = H () / osθ (5-1) n ( n

91 79 H applied H // θ H θ I Figure 5-8. Shemati outline for the angular dependene of the mathing field. The experimental method to verify this relation is straightforward, measuring the magneti field dependene of the resistane R (H ) at different angles and extrating the orresponding mathing fields at the resistane minima. In Figure 5-9, the data for Sample C are given. The R (H ) urves at different angles, presented in the inset A, are obtained at the fixed temperature.99t o (5.9 K). For the urve at θ = (without any tilt), the first mininum appears at H () 173 G ± 5 G. The red solid irles in the main panel represent all 1 = mathing fields extrated from the orresponding urves. It should be notied that the resistane minima in R (H ) urves at angles higher than 7 are hard to resolve. The reason is that the minima beome smaller at high dissipation levels.

92 8 A B Figure 5-9. Angular dependene of the first mathing field for Sample C. (A) R (H ) urves in a magneti field along various diretions. (B) R (θ ) urves in various fixed fields. Temperature is fixed at.99t (5.9 K). The solid and open symbols are the first mathing fields in different angles obtained from inset A and B, respetively. The dashed line is the fitting urve with H ( θ ) 19 / osθ (G). 1 =

93 A more effetive and onvenient way to determine the loations of the 81 minima is to measure the angular dependene of the resistane R (θ ) at various fields. Shown in the inset B is the measured R (θ ) urves at different applied magneti fields with the temperature fixed at.99t (5.9 K). Extrating the orresponding angles at the resistane minima, the results are plotted with blue open squares in the main panel. Combining all data from both methods, a omplete H ( ) urve is ahieved. Although they are from two independent measurements, the fitting urve (blak dashed line), obtained from Eq.5-1, fits both sets of the data niely. The fitting result for H ( 1 ) is 19 G, whih is still within the experimental errors. In fat, if normalized by the mathing field H ( 1 ) at θ =, all niobium films ontaining hole-arrays samples follow the same 1/ os( θ ) relation. In Figure 5-1, the angular dependene of the normalized first mathing fields for all three samples, Samples C through E, are plotted with red, blue and green open irles, respetively. The hole-arrays in these samples are different not only in the holehole separation but also in the lattie type (triangular and square arrays). The orresponding first mathing fields along the perpendiular diretion (at ) are at 19 G, 9 G, and 95 G, respetively. Yet all the data niely follow the 1/ os( θ ) urve. Thus, it onfirms that only the perpendiular magneti field omponent ontributes to the ommensurate effet. 1 θ

94 8 Figure 5-1. Angular dependene of the normalized first mathing field for niobium films ontaining different hole-arrays. The open symbols are the normalized first mathing fields at different angles for Sample C through E. The dashed line is the fitting urve with H θ ) / H () 1/ osθ. 1 ( 1 =

95 Angular Dependene of the Critial Temperature 83 For a superonduting strip, the H (T ) phase diagrams at different angles between the sample and the applied magneti field all exhibit a paraboli behavior. They follow the saling behavior, i.e. by multiplying a saling fator on H, the phase diagram at any angle an be onverted to the baseline urve, the one in the parallel or perpendiular applied fields. However, for a superonduting film ontaining a hole-array, things beome muh more ompliated. Shown in Figure 5-11 is the phase diagrams of Sample C at different angles ( θ = 9, 75, 6, 45, 3, 15 and ). Although still based on the paraboli bakground, phase boundaries are greatly modified beause of the T osillation indued by the holearray exept the one in the parallel field ( θ = 9 ), whih still holds a pure paraboli behavior. In order to explore the anisotropi property of the superonduting-normal phase diagram, phase boundaries at all other angles also have to be measured. However it would take too muh work. The angular dependene of the ritial temperature at a fixed magneti field is an alternative way to explore this ompliated phenomenon. In this way, the tangled ritial temperatures are strethed out with respet to the angle.

96 G Figure Phase diagram at different angles for Sample C. The inset is the angular dependene of the ritial temperature at a fixed field (H=5 G). Like the H (T ) phase diagram, the angular dependene of the ritial temperature also shows T osillations. It is beause the T suppression only depends on the field omponent perpendiular to the sample. With the angle between the sample and the applied field varying, this perpendiular field omponent, H = H app osθ, also varies from ( H along parallel diretion) to the total value of the applied field ( H along perpendiular diretion). The inset of

97 85 Figure 5-11 provides the one with a field fixed at 5 G. In this urve, two peaks appear at 64 ±.5 and 31 ±.5, respetively. They orrespond to the first and seond mathing fields in the H (T ) phase diagram at θ =, and the alulated angles, aros n H1 H app, should be 64.6 and Furthermore, sine the T osillation is largely extended with the angle, some subtle strutures, like the frational mathing, whih have been buried in the phase diagrams, might be observed now. Beause of its omplexity, it is very diffiult to fit the angular dependene of T through theoretial simulations. However, if a superonduting film ontaining a hole-array behaves like a wire network at temperatures lose to T, as analyzed in the early part of this hapter, T an be treated as the superposition of a thin strip bakground and the periodi T osillation indued by holes. Thus, the angular dependene of the ritial temperature an be written as Strip T ( θ, H app ) = T ( θ, H app ) + ΔT ( θ, H app ) (5-) where H is the fixed applied field and Δ T θ, H ) is the angular dependene app ( app of the T suppression due to the hole ontribution in that fixed field. An experimental method an be developed to fit the angular dependene of T. Take the data of Sample C with the field fixed at 5 G (see the inset of Figure 5-11) as an example. First, the baseline of the strip has to be reonstruted. Aording to the preeding results, in the magneti field parallel to the sample (i.e.

98 86 at θ = 9 ), the phase diagram of the superonduting film with a hole-array has the same phase diagram with the superonduting thin strip that omposes the orresponding wire network sine the geometry exposed to the magneti field is the same. The temperature dependene of the ritial field is desribed by the thin film limit of the Ginzburg-Landau theory, the square-root behavior, H 1 1 Φ 1 Φ T // ( ) = 1 = // () 1 ( ) () T = H π d ξ T π d ξ T T T 1 (5-3) where 1 Φ H // () = is the parallel ritial field at temperature T =. π d ξ () For a thin strip, the ritial fields at different angle θ obey the saling rule, i.e. T H (, ) ( ) // ( ) ( ) // () 1 θ T = A θ H T = A θ H (5-4) T The saling fator A (θ ) for a thin strip has been given in Eq.4-7. To experimentally verify this relation in a patterned film, A (θ ) an be obtained by saling the phase diagram in the parallel field to ahieve the baselines whih form the upper bounds to the phase diagrams at various angles, just like what have been done to the perpendiular phase diagrams in Figure 5-3 and Figure 5-5. Saling fators at seven different angles, 9, 75, 6, 45, 3, 15 and, are obtained in this way and plotted by red solid irles in the inset of Figure 5-1. They follow the angular dependene relation of the ritial field for the thin strip (the blue thin line) niely. 1

99 87 H = 5G Figure 5-1. Calulated strip bakground of the T (θ ) urve for Sample C at a fixed field (H=5 G). The open irles are the measured T (θ ) urve. The dashed line is the alulated strip bakground. The inset provides the angular dependene of the saling fator for the baselines of the phase diagrams with field along different angles. The thin line is the fitting urve with the angular dependene of the ritial field for a superonduting strip.

100 88 From Eq.5-4, the angular dependene of the ritial temperature for a thin strip at a fixed magneti field now an be derived, T Strip H app ( θ ) = T 1 (5-5) A( θ ) H // () Using the fitting result ξ ( ) 8. 5 nm and = 5. Strip T 96 K, the alulated T (θ ) baseline is plotted in Figure 5-1 (blue dashed line). It forms an upper boundary of the measured T (θ ) urve (red open irles) and almost perfetly passes through the peak points at the mathing angles, where fluxoid at eah unit ell is integer flux quanta and the hole-indued T suppression does not our. The next step is to identify the T suppression due to the perpendiular omponent of the magneti field at eah angle. Obviously, it is equivalent to the T hange between the phase diagram in perpendiular fields and the baseline exluding the influene of the hole-arrays, whih is shown in the inset of Figure The phase diagrams in the perpendiular and parallel magneti fields are represented with the red and blue open irles, respetively. The baseline (blue dashed line) is obtained by multiplying a saling fator,.587, to the field of the phase diagram in parallel fields. Then the T osillation after taking off the paraboli bakground an be ahieved by subtrating the baseline from the phase diagram in perpendiular fields. The results are plotted in Figure Cusps in Δ ourring periodially at multiples of the first mathing field H = 17 G, T 1 are observed. This is a very typial behavior of the Little-Parks effet in the wire

101 89 network as introdued in Chapter. To get the T suppression for a field applied at an angle θ, the osine relation between the applied fields and the perpendiular H omponent, θ = aros( ), needs to be applied. In this way, Δ T at eah H app angle (See Figure 5-14, blue open irles) is obtained. x.587 Figure Δ T osillation in various perpendiular fields for Sample C. The inset presents the outline to obtain the paraboli bakground of the phase diagram in perpendiular fields by saling the phase diagram in parallel fields.

102 9 Finally, the angular dependene of the ritial temperature is reonstruted by superimposing Δ T on the baseline. As shown in Figure 5-14, the fitting result is presented with the red solid irles, whih is remarkably onsistent with the measured data (the red open irles). The orresponding base line and also provided in the blue dashed line and open irles, respetively. Δ T are ΔT Base line H = 5G Figure Experimental method to fit the T (θ ) for Sample C at a fixed field (H=5 G). The open and solid irles represent the measured data and fitting results, respetively.

103 91 Using the same method, all the angular dependene urves for the ritial temperature at different magneti fields an be niely fitted. Shown in Figure 5-15 are the results for all three samples C through E. In these fittings, the positions of peaks in the T osillations are preisely onsistent with the measured data. It implies that the T osillations in the angular dependene of the ritial temperature totally originate from the Little-Parks effet indued by the perpendiular omponent of the applied magneti field. In eah sample, the data at the relatively lower magneti fields are better fitted than those in the higher fields. It is beause in the low field region, ritial temperatures are loser to the zero field ritial temperature, ausing the wire network behavior to dominate. In higher magneti fields, ritial temperatures are away from the zero field ritial temperature, whih results in the muh shorter oherene length and the interstitial vorties might reside in the superonduting segment between the holes. In this situation the wire network model is not suitable any more. The maximum number of the osillations that an be observed in the angular dependene of T in Sample D and E is muh less than that in Sample C. Atually, the osillations in the angular dependene of T are the diret refletion of the osillations that an be observed in the phase diagram in perpendiular fields, whih depends on the lattie parameters of the hole-arrays and the oherene length of the materials. A detailed disussion has been done in the early part of this hapter.

104 9 Sample C Sample D Sample E Figure Experimental fitting for T (θ ) urves at various fixed fields for niobium films with different hole-arrays. The open and solid symbols represent the measured and fitting data, respetively. See Table 1 for the sample parameters.

105 93 In onlusion, an experimental way to fit the angular dependene of T for superonduting films ontaining hole-arrays has been developed based on the assumption that these films behave like wire networks at temperatures lose to T. The fat that the fitting results are in good agreement with the experimental data verifies this assumption. The results also demonstrate that the observed T osillations originate from the Little-Parks effet indued by the perpendiular omponent of the applied magneti field. It further onfirms the onlusion obtained in the early part of this hapter. Angular Dependene of the Magnetoresistane The angular dependene of the magnetoresistane an diretly reveal the sample s anisotropi properties in the magneti field. With a hole-array in it, a superonduting film exhibits unique properties in the angular dependene of the resistane at different magneti fields. Shown in Figure 5-16 are R (θ ) urves of Sample C at various applied fields with the temperature fixed at.99t (5.9 K). Pronouned resistane minima are observed in these urves. The origin of these osillations is still the ommensurate effet. As disussed before, only the perpendiular magneti field omponent ontributes to this effet. In a fixed applied magneti field, first from H app, the perpendiular omponent of the field sweeps H app to and then from to - H app as the angle between the field and

106 94 the normal diretion of the film surfae varies from to 18. Resistane minima appear at those positions where the perpendiular omponent of the field reahes the mathing field, i.e. H app osθ = n H 1. Take the R (θ ) urve at 5 G as an example. Two minima appear between and 9 region beause the field for the third mathing already exeeds the applied field. The first and seond mathing loate at 64 and 3.5, whih are very lose to the alulated values of and for the first and seond mathing angles with a first mathing field of H1 = 17 G. The other important observation is the amplitude of the resistane minima with different dissipation levels. It dereases signifiantly at the high dissipations. More detailed disussion will be provided below. Figure R (θ ) urves in various fixed fields at T= 5.9 K for Sample C.

107 95 Similar to the angular dependene of T, R (θ ) urves an also be analyzed as the strip bakground superimposed by the resistane osillation indued by the T suppression indued by the Little-Park effet. In this way, R (θ ) urves an be fitted. First, the baseline of the thin strip bakground has to be reonstruted. It has already been shown that for a superonduting strip, the angular dependene of the resistane an be retrieved from the R (H ) urve in the perpendiular field by onverting the applied fields of the R (H ) urve to the orresponding angles through Eq.4-9. For a niobium film ontaining a hole-array, the base line of the strip in the perpendiular field an be obtained by saling R (H ) urve with field along the parallel diretion (see the inset of Figure 5-17), as disussed in the early part of this hapter. Thus, there is no diffiulty in reonstruting the baseline for the angular dependene of the magnetoresistane. Shown in Figure 5-17, the red open irles represent the measured R (θ ) urve at 5 G and the blue dashed line is the alulated baseline. Beause of the symmetry, only the region that θ varies from to 9 is presented here. The baseline preisely passes through those resistane minima at the mathing angles where no T suppression ours. The remarkable overlay at 9 of the two urves demonstrates the onsistene of the two independent measurements with respet to the fields and the angles. The baseline forms a lower boundary of the R (θ ) urve and also provides additional evidene that the ommensurate effet

108 96 originates from the T suppression indued by Little-Parks effet rather than the pinning enhanement..587 Figure Calulated strip bakground of the R (θ ) urve in a fixed field (H=5 G) at T= 5.9 K for Sample C. The open irles represent the measured R (θ ) urve. The dashed line is the alulated strip bakground. The inset provides the outline to obtain the baseline of R (H ) in perpendiular fields by saling the data in parallel fields.

109 97 Next is to figure out the resistane hange due to the T suppression indued by the perpendiular field omponent. The most straightforward method is as follows. Taking the R (T ) urve at zero-field as a referene, if there exists a T suppression, the whole R (T ) urve should be parallelly shifted by this temperature hange, Δ T. To simplify the problem, any effets whih might ause the slope hange in the R (T ) urve will not be onsidered here. Comparing these two urves at a fixed temperature an give the orresponding resistane hange. A shemati outline is shown in Figure Now, the problem beomes how to obtain the orresponding Δ T at eah angle. In the early part of this hapter, Δ (H ) in perpendiular fields has been ahieved (see Figure 5-13). For a fixed applied field T H app at an arbitrary angle θ, the perpendiular field omponent is H ( θ ) = H osθ. Aordingly, Δ (θ ) app T for a fixed applied field an be obtained from Δ (H ) in perpendiular fields. It has to be notied that using this method the amplitude of the alulated resistane hange largely depends on the dissipation level. The reason is that the slope at eah point of the R (T ) urve hanges. At high dissipation levels lose to the normal resistanes, the slopes are muh smaller than those in the middle of the transition. Thus, even with the same T Δ T, the hange of the resistane at high dissipation is also muh smaller, i.e. Δ R < ΔR1 (shown in Figure 5-18). It is quite onsistent with what is observed in Figure Thus, hoosing the right

110 98 dissipation level is very important in the fitting. Theoretially, one might divide the referene R (T ) urve at zero-field to as many dissipation levels as possible so that the orret one an be obtained for eah data point. However, it takes too muh alulation. In fat, it has been found out that the slope does not hange dramatially between the adjaent data points in the referene urve. Using this observation, we an roughly divide the urve into several main regions with different dissipation levels yet still get aeptable fitting results. ΔT ΔR ΔR 1 ΔT ΔT Referene line Figure Shemati outline of alulating the resistane hange due to T suppression. The inset is the T suppression in various perpendiular fields.

111 99 The final fitting data are obtained by superimposing the resistane hange to the strip baseline, whih is shown in Figure 5-19 (the green open irles). Only three different dissipation levels are used during the fitting. Also provided are those in the different seleted fields, 5 G, 15 G, 5 G, 35 G, and 5 G. In the figure, the open irles, dashed lines and solid thin lines represent the fitting urve, the baseline and the measured data, respetively. Obviously, it is not a perfet fitting. There are several fats affeting the results. First, the baselines are obtained from independent measurements in whih experimental errors an not be avoided. Seond, the resistane hange alulation model is a simplified model in whih all effets that might ause the slope hange in R (T ) urve are negleted. Finally, only three different dissipation levels are used, whih may also introdue some errors. Despite all of these inauraies, the fitting results are still quite deent. It demonstrates again the wire network behavior of the niobium thin films with periodi hole-arrays at temperatures lose to T.

112 1 Figure Experimental fitting for R (θ ) urves in various fixed fields at T=5.9 K for Sample C. The the thin lines are the measured data and open symbols are the fitting results. The dashed lines are the alulated strip bakgrounds.