1
Melting of vortex lattice in 2D MoGe thin film Surajit Dutta 1 , Indranil Roy 1 , Aditya N. Roy Choudhury 1 , Somak Basistha 1 , IIaria Maccari 2 , Soumyajit Mandal, John Jesudasan 1 , Vivas Bagwe 1 , Claudio Castellani 2 , Lara Benfatto 2 and Pratap Raychaudhuri 1 1 Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai- 400005 2 ISC-CNR and Department of physics, Sapienza university of Rome, P.Ie A. Moro, 00185, Rome, Italy Sample preparation Thin film of MoGe is deposited on thermally oxidized Si substrate using pulse laser deposition technique (PLD) at room temperature and base pressure ~ 1 − 6 mbr. Measurement Techniques 1. Ac Screening Response : Sample is placed between primary coil (quadrupolar) and secondary coil (dipolar). Ac screening response of thin film superconductor is determined using probing field 3.5 mOe by passing small ac drive current (I d ) 0.5 mA (frequency, f = 31 kHz) through primary coil. Mutual Inductance between primary and secondary coil is defined as, = + = 2 M R & M I are real part and imaginary part of mutual inductance. Mutual inductance between two coil depends on complex screening length of the sample. Therefore complex screening length can be computed numerically using experimental value of mutual inductance. Expression of complex screening length is given by, λ −2 = 2 0 =λ −2 + −2 λ is penetration depth and δ is skin depth, σ is complex conductivity of the material, 0 is free space permeability. 2. Magneto-transport : Resistivity measurements are performed in 4 probe geometry configuration by passing 50 μA dc current as function of temperature and magnetic field. Dc nano voltmeter is used to measure the voltage. 3. Scanning Tunneling Spectroscopy (STS) : Scanning tunneling spectroscopy is done by using low temperature home made scanning tunneling microscope (STM). Lowest temperature of the system is 450 mK and Maximum magnetic field can be applied to 9 T. STM is operated at constant current mode and tip of ourSTM system is metallic (Pt-Ir). Expression of tunneling current between normal metal and superconductor is given by, = −∞ ∞ () (0) [ − ( + )] And tunneling conductance can be written as, ∝ 1 න −∞ ∞ ()[− ( + ) () ] Experimentally tunneling conductance is obtained by applying voltage modulation technique. + ≅ + ( ()) Superconducting and normal region can be easily distinguished base on tunneling conductance. Therefore vortex images are taken using this technique. V A Introduction We report melting of vortex lattice with increasing magnetic field in amorphous MoGe thin films of thickness 21 nm which is much less than the bending length of the vortex lines. Therefore, vortex lattice of this film is 2 dimensional. Here melting process follows 2-step BKTHNY melting : (1) solid phase- hexatic fluid phase (appearance of dislocation pairs) and (2) hexatic fluid phase- isotropic liquid phase (breaking of dislocation pairs into isolated dislocations: disclination). This intermediate hexatic fluid phase has quasi long range orientational order and exponentially decaying translation order. It has zero share modulus. Therefore no finite critical current can exist in hexatic fluid phase. we show this sequence of phase transition of vortex lattice driven by magnetic field at temperature 2K in this Amorphous MoGe thin film by combining real space imaging of vortex lattice using Scanning Tunneling Spectroscopy (STS), magneto-transport and ac screening response (low frequency penetration depth) measurements. Characterization of Solid phase & liquid phase based on experimentally measurable quantity Thermally activated flux flow resistance : In the solid phase, underlying potential of the vortex lattice is given by [3] , U I =U 0 ( I c I ) α Thermally activated flux flow resistance, R TAFF =R FF Exp − U K B T R FF is flux flow resistance which is defined by usual Bardeen – stephen relation, V=R FF (I − I c ) and I C is critical current. Therefore, R TAFF → 0 when I → 0 In liquid phase, underlying potential of the vortex lattice is U L which is independent of current, I. Thermally activated flux flow resistance can be written as, R TAFF =R FF Exp(− U L K B T ) So, R TAFF ≠ 0 when I → 0 Complex ac screening length [2,3] : In presence of ac magnetic field, complex screening length is defined as in liquid state, λ −2 = 0 2 [η + 0 ] Where as in solid phase, λ −2 = 0 2 Experimental results Summary ➢ Thickness of the sample is measured using Ambios XP2 stylus profilometer Variation of resistance with temperature at zero magnetic field 1 2 3 4 5 6 7 8 0 50 100 150 200 250 R () T (K) Low frequency magnetic shielding response 20 40 60 80 2 3 4 5 6 7 8 -40 -20 0 M R (nH) 0 Oe 3 Oe 4 Oe 5 Oe 7 Oe 12 Oe 20 Oe 50 Oe 0.1 kOe 0.2 kOe 0.3 kOe 1 kOe 5 kOe M I (nH) T(K) Real part & imaginary part of the mutual inductance Basic characterization of the sample Tunneling Spectra and corresponding temperature dependence of energy gap ( Δ) which is well fitted with conventional BCS relation, ∆(0) K B T c = 2.17 0 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0 1.2 , (meV) T (K) BCS fit -4 -2 0 2 4 0.3 0.6 0.9 1.2 1.5 G [V] V (mV) 0.45 K 0.97 K 1.55 K 2.15 K 2.65 K 3.75 K 4.45 K 5.30 K 6.60 K 7.60 K Tunneling Spectra ( tunneling conductance, G [V]) is fitted with usual BCS relation with modifying density of state of superconductor, N s E = Re{ E+iΓ |E|+iΓ 2 −∆ 2 } . Γ is phenomenological temperature broadening parameter. Critical temperature is 6.95 K Real part of ac screening length (λ −2 ) goes to zero for magnetic field 5 kOe (> H c1 = 1.8 Oe ). Therefore, ac screening length is completely imaginary and it reveals existence of vortex liquid state around this field. Here we have used ac screening response measurement technique in transmission geometry. Circuit diagram Finding of melting field at 2K from Magneto-Transport measurements I-V characteristics 0 2 4 6 0 15 30 45 V (mV) I (mA) 0.3 kOe 0.9 kOe 1.5 kOe 1.8 kOe 1.9 kOe 2.2 kOe 2.5 kOe 3.1 kOe 4.3 kOe 5.2 kOe 8.0 kOe 20 kOe 45 kOe 65 kOe 80 kOe 100 kOe Notional Critical current at low field regime 0 2 4 6 0 10 20 30 40 50 V (mV) I (mA) I c 1 2 3 4 5 0.0 0.3 0.6 0.9 1.2 1.5 1.8 I c (mA) H (kOe) When Lorentz force on vortices is more than the pinning force then vortices are in flux flow regime (large current limit) and I-V Characteristics follow usual Bardeen-Stephen relation. I-V characteristics for I << I c In this regime (I ≤ 100 uA ≪ I c ), vortex motion is driven by thermal energy and it is called thermally activated flux flow regime. 0 20 40 60 80 100 0 1 2 3 4 5 V (V) I (A) 0.3 kOe 0.9 kOe 1.9 kOe 2.2 kOe 2.5 kOe 3.1 kOe 3.4 kOe 3.7 kOe 4.0 kOe 4.1 kOe 4.6 kOe 4.9 kOe 5.2 kOe 0 1 2 3 4 5 0.0 0.1 0.2 0.3 TAFF (n - m) H (kOe) Nature of I-V Characteristics 0 5 10 15 0.0 0.2 0.4 0.6 0 5 10 15 0.0 0.2 0.4 V(V) 1.5 kOe V (V) I (mA) 1.8 kOe 2.2 kOe I(mA) 2.5kOe ➢ Below 1.9 kOe, I-V curves are well fitted with expected form (red lines) for vortex soild with =1 which is matching with predicted theoretical value. ➢ Above 1.9 kOe, I-V curves can not be fitted with the same form of vortex solid. Rather, curves are linear for I ≤ 250 uA (green lines) as expected for vortex liquid. Evolution of vortex lattice with magnetic field 10 kOe 55 kOe 70 kOe 85 kOe Vortex images are taken using our home made low temperature STM. Observation: ➢ Orientational order of vortex lattice persists up to 70 kOe with the presence of dislocation pairs. ➢ Above 70 kOe, six distinct spot in FFT becomes isotropic ring. Characterization of vortex liquid phase by real space vortex imaging Temporal dynamics of vortex liquid phase 10 kOe 25 kOe 55 kOe t = 0 t = 1.5 ho urs t = 3 ho ur s Direction of movement of vortices 85 kOe 55 kOe 25 kOe ➢ 3 consecutive images are taken in 1.5 hours interval to observe temporal dynamics of vortices . ➢ To capture the movement of vortices, 12 consecutive images are taken in 15 minutes interval over same area. Observation 1. Temporal dynamics shows that dislocations are appeared randomly in pair over time up to field 70 kOe and isolated dislocations points are observed above of this field. 2. Below 70 kOe, motion of vortices are along principle axis. 3. Above 70 kOe, motion of vortices become completely random like isotropic liquid. Therefore, there is another phase transition at 70 kOe. Time Distinction between Hexatic fluid and isotropic liquid based on spatial variation of Orientational order Transition point of hexatic fluid phase to isotropic liquid phase 0 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8 H (kOe) Hexatic Fluid Isotropic Liquid 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 TAFF ( m) Six-fold orientational order parameter is defined as, ψ 6 = 1 <σ , 6( − ) > For perfect hexagonal lattice, ψ 6 =1 Above 70 kOe, orientational order parameter goes to zero and thermally activated resistivity (ρ TAFF ) increases rapidly above the same field. Therefore hexatic to isotropic liquid transition occurs around field 70 kOe at 2K. ρ TAFF at different temperatures ➢ Solid to hexatic fluid phase transition are identified from the point above which I-V curves do not fit with expected form of soild phase as well as ρ TAFF becomes finite as I goes to zero. ➢ Hexatic to isotropic liquid transition is obtained from the point where ρ TAFF starts increasing rapidly. ➢ Upper critical field is found from the point above which ρ TAFF reaches its normal state value. Phase diagram 2 3 4 5 6 7 20 40 60 80 100 H c2 H (kOe) T (K) Hexatic fluid Vortex liquid Vortex Solid Normal state x10 Low temperature magneto-transport measurements down to 300 mK Solid Liquid 2 3 4 5 6 7 8 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 -2 , -2 (m -2 ) T (K) BCS fit BCS fit of penetration depth in zero magnetic field, λ −2 () λ −2 (0) = ∆ ∆ 0 ℎ ∆ λ 0 = 534 Observation at 2K ➢ Ac screening response measurement and Tunneling spectroscopy reveals that Our system is well behaved BCS like type II superconductor. ➢ All the phase transition points are matched properly with these three different set of measurements. ➢ Existence of hexatic fluid phase is conformed by finding decay length scale of orientational order in real space of vortex lattice and motion of vortices as well as appearance of finite value of ρ TAFF in zero current current limit. ➢ Similar kind of behavior in resistivity (at 2K) is also observed in low temperature regime 300 mK. ➢ Appearance of finite R TAFF (in low current limit) at 300 mK suggests hexatic fluid is quantum fluid and we have quantum phase transition. Future work ➢ To verify the phase transition at low temperature (300 mk), we need to observe temporal dynamics of vortex lattice. ➢ Using classical BKTHNY theory we can not explain low temperature behavior. So, we need Quantum analog of this model. References 1. Indranil Roy, Surajit Dutta, Aditya N. Roy Choudhury, Somak Basistha, IIaria Maccari, Soumyajit Mandal, John Jesudasan, Vivas Bagwe, Claudio Castellani, Lara Benfatto, Pratap Raychaudhuri, Melting of the vortex lattice through intermediate hexatic fluid in a- MoGe thin flim, arxiv:1805.05193 (2018). 2. Enrst Helmut Brandt, Penetration of Magnetic ac Fields into TypeII Superconductor, Phys. Rev.Lett. 67,16 (1991). 3. C.J. van der Beek, V.B Geshkenbein, V.M Vinokur, linear and nonlinear ac response in superconducting mixed stated, Phys. Rev.B 48,5 (1993 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.1 0.2 0.3 R TAFF () T (K) Finite thermally activated Resistance is observed at low temperature 300 mK. Therefore this hexatic fluid is quantum fluid. 0 20 40 60 80 100 1E-6 1E-5 1E-4 1E-3 0.01 0.1 1 10 100 1000 R TAFF () H (kOe) 0.3 K 0.45 K 0.79 K 1.1 K 1.45 K 2.03 K 3.5 K 4.5 K 5.5 K 6.5 K Low temperature vortex lattice (450 mK) 10 kOe 55 kOe 70 kOe Real space orientation order is defined as, G 6 r =< Cos 6 θ r −θ 0 > θ r : Angle between an arbitrary fixed axis and the line connecting between two nearest neighbor located at position r. For a perfect hexagonal lattice, G 6 r =1 Hexatic fluid has quasi long range oriententional order. So, G 6 r decays as power law with r (10 kOe, 25 kOe, 55 koe). For, isotropic liquid G 6 r decays like exponentially (no orientational order; 70 kOe and 85 kOe). Sudden increment in R TAFF is also appeared around 80 kOe at low temperature. Therefore there is hexatic to isotropic liquid transition around this point. (η : viscous drag coefficient and 0 is intrinsic vortex lattice frequency, is average value of labush parameter) In field, complex ac screening length is defined as, λ 2 =λ 2 +λ 2 λ is London penetration depth and it’s contribution can be neglected for such low field (λ −2 ∝ (1 − 2 )). Therefore, λ 2 ≈λ 2 = 2 0 λ is campbell penetration depth and in the linear response regime, it is modified as, → + η In Low temperature, Orientational order also persists up to 70 kOe with random dislocation pairs. 0 20 40 60 80 100 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 TAFF (-m) H (kOe) 1.45 K 2.0 K 3.5 K 4.5 K 5.5 K 6.5 K 10 100 1000 10000 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 -2 , -2 (m -2 ) H (Oe) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -2 / -2 0 1 2 3 4 5 6 7 8 0 50 100 150 200 250 R TAFF () T (K) 0 kOe 0.5 kOe 2 kOe 5 kOe 10 kOe 15 kOe 20 kOe 25 kOe 30 kOe 40 kOe 50 kOe 60 kOe 70 kOe 80 kOe 90 kOe 100 kOe
