Vortices in Classical Systems

4 He-II vortices: Vortices in Quantum Systems STM of NbSe 2 vortices: G. A. Williams, R. E. Packard, Hess PRL (1989). Phys. Rev. Lett. 33, 280 (1974) Pan, Hudson, Davis, RSI (1999) Hall probe image of YBCO film at T = 40 K: Checkerboards in Bi 2 Sr 2 CaCu 2 O 8+x vortices: Gauss s 3.8 0 Hoffman et al, Science (2002) 20 um B applied = 1 Gauss

Single Vortex Manipulation in Superconducting Nb Vortices in Nb Film Cooled to 5.3K in 100G Eric Straver Jenny Hoffman Dan Rugar Kathryn Moler Δf=2.01Hz 1μm Stanford University Funded by Packard and AFOSR

Goals Part I: eliminate i uncontrolled vortex motion bigger magnets efficient power lines quieter sensors & circuits Part II: controlled single-vortex manipulation model for soft condensed matter vortex entanglement t vortex ratchet automaton Luttinger liquid FermiLab

Part I: eliminate uncontrolled vortex motion

Motivation: eliminate uncontrolled vortex motion 40 30 H c2 H* (where vortices move) Field (T) 20 YBCO 10 0 0 30 60 90 120 Temperature (K) Larbalestier, Nature 414, 368 (2001)

Motivation: eliminate uncontrolled vortex motion 40 30 H c2 H* (where vortices move) Field (T) 20 YBCO 10 0 0 30 60 90 120 Temperature (K) Larbalestier, Nature 414, 368 (2001) e- e- e- e-

Motivation: eliminate uncontrolled vortex motion 40 30 H c2 H* (where vortices move) Field (T) 20 YBCO 10 0 0 30 60 90 120 Temperature (K) Larbalestier, Nature 414, 368 (2001) Apply current I: Cooper pairs flow without dissipation e- e- e- e-

Vortex Pinning Measurements: Bulk Transport 10 GB, B = 0 Tesla 4 GB, B = 1 Tesla intra-grain (upper scale) grain boundary Redwing et al, APL 75, 3171 (1999). Hogg et al, APL 78, 1433 (2001).

Vortex Pinning Measurements: Bulk Transport 10 GB, B = 0 Tesla 4 GB, B = 1 Tesla intra-grain (upper scale) grain boundary Redwing et al, APL 75, 3171 (1999). Hogg et al, APL 78, 1433 (2001). What s really going on here? First detectable Voltage = many, many vortices

Examples of Previous Single Vortex Depinning Force Measurements with Transport Current Finnemore and coworkers ongoing work (1988-present) Cabrera and coworkers 1992 + several other groups

Examples of Previous Single Vortex Depinning Force Measurements with Transport Current Finnemore and coworkers ongoing work (1988-present) Cabrera and coworkers 1992 But we d like a more reliable way to know the force at the vortex location; and to detect vortex motion with finer spatial resolution on order ξ ~ 10-20 nm + several other groups

Vortex Pinning Measurements: STM Imaging g reduce field from 3Teslato15Tesla 1.5 estimate pinning force from flux density gradient across twin boundary Maggio-Aprile et al, Nature 390, 487 (1997)

Vortex Pinning Measurements: STM Imaging g reduce field from 3Teslato15Tesla 1.5 estimate pinning force from flux density gradient across twin boundary Maggio-Aprile et al, Nature 390, 487 (1997) But we d like a more versatile way to directly measure pinning forces at arbitrary defects.

Magnetic Force Microscopy Pros and Cons of MFM Force between tip and sample: F = ( m B) Other signals Image cantilever resonant frequency Δff 0 ~ df z /dz better signal-to-noise Vertical force gradient imaging Horizontal force manipulation Tip Geometry Con: Imperfectly known Pro: Up to 20 nm spatial resolution Signal to Noise Con: Not as good as SQUIDs, Hall probes Pro: Good enough to see vortices Con: See atomic forces too Pro: Simultaneous topography Invasiveness Con: Tip exerts force on vortex Pro: Tip exerts force on vortex

Vortex Depinning in Nb T: Decreasing Pinning 5.2K 5.5K 6.0K 6.5K 7.0K 7.5K 4 μm Colormap adjusted separately for each image.

Vortex Pinning vs. Height at T = 5.5 K A z=576nm B z=518nm C z=461nm D z=403nm E z=346nm F z=288nm G z=230nm H z=173nm 2μ m Δ f (Hz) Δ f=0.63hz I -0.8-1.0-1.2-1 0 1 x (μ m) x y Δ f=0.73hz J -0.6-0.8-1.0-1 0 1 x (μ m) Δ f=0.85hz -0.2 K -0.5-0.8-1 0 1 x (μ m) Δ f=0.97hz Δ f=1.22hz 0.4 L 1.0M 0.0-0.4-1 0 1 x (μ m) 0.5 0.0-1 0 1 x (μ m) Δ f=1.50hz N 1.5 1.0 0.5-1 0 1 x (μ m) Δ f=2.02hz 30O 3.0O 2.0 1.0-1 0 1 x (μ m) Δ f=2.79hz P 500nm vortex motion event Important parameters: k spring = 2.1±0.7 N/m f 0 = 71128.28 Hz λ Nb = 90 nm

Model: Monopole Tip Monopole Vortex +M d offset z λ B -M superconductor vortex model vortex as monopole distance λ beneath surface --J. Pearl, J. Appl. Phys. 37, 4139 (1966). df dz z = 2k spr df f = M zφ 2π 0 ( x x 0 ) 2 ( y 2 2 2 ( x x ) + ( y y ) + ( z + λ + d ) 5/ 2 0 0 0 offset ) y 0 ) 2 + 2( z + λ + d offset ) 2 )

Quantifying the Depinning Force at T = 5.5 K 10-4 df/dz (N/m m) z (nm) 100 200 300 400 600 900 B -1.00-1.25 17G vs. z 4G vs. z -1.47-1.63-1.72-1.85-1.93-1.97 10-5 100 200 300 400 600 900 z+λ+d (nm) min

Quantifying the Depinning Force at T = 5.5 K 10-4 df/dz (N/m m) z (nm) 100 200 300 400 600 900 B -2.77-2.95-3.00-3.03 z (nm) 100 200 300 400 500 600-2.97 17G vs. z -3.00 40-3.00 4G vs. z 30-2.95 17G vs. z+λ+d λ min 20 4G vs. z+λ+d min 10-5 100 200 300 400 600 900 z+λ+d (nm) min F z (pn) 10 4 Gauss 17 Gauss df z /dz ~ (z+λ+d) -3 5 400 500 600 700 800 900 z+λ+d min (nm)

Quantifying the Depinning Force at T = 5.5 K 10-4 df/dz (N/m m) z (nm) 100 200 300 400 600 900 B -2.77-2.95-3.00-3.03 z (nm) 100 200 300 400 500 600-2.97 17G vs. z -3.00 40-3.00 4G vs. z 30-2.95 17G vs. z+λ+d λ min 20 4G vs. z+λ+d min 10-5 100 200 300 400 600 900 z+λ+d (nm) min df z /dz ~ (z+λ+d) -4 F z (pn) 10 5 4 Gauss 17 Gauss fit bound 400 500 600 700 800 900 z+λ+d min (nm)

Quantifying the Depinning Force at T = 5.5 K 10-4 df/dz (N/m m) z (nm) 100 200 300 400 600 900 B -2.77-2.95-3.00-3.03-2.97-3.00-3.00-2.95 17G vs. z 4G vs. z 17G vs. z+λ+d λ min 4G vs. z+λ+d 10-5 min 100 200 300 400 600 900 z+λ+d (nm) min df z /dz ~ (z+λ+d) -4 F z (pn) z (nm) 100 200 300 400 500 600 40 30 20 10 4 Gauss 17 Gauss fit bound 5 k spring bound 400 500 600 700 800 900 z+λ+d min (nm)

Depinning Forces in Two Datasets at 5.5 K z=576nm z=518nm z=461nm z=403nm z=346nm z=288nm z=230nm z=173nm A B C D E F G H 2μ m Δ f=0.63hz Δ f=0.73hz Δ f=0.85hz Δ f=0.97hz Δ f=1.22hz Δ f=1.50hz Δ f=2.02hz Δ f=2.79hz 12 10 # vortices # vort tices 8 17 Gauss 6 4 Gauss 4 2 0 0 10 20 30 40 50 F z (pn) F lateral,max = 0.38*F z,max F depin ranges from 4 to 12 pn at 5.5 K

Nb Depinning Forces: Quantitative Comparison f p (pn/μ μm) 1000 100 10 1 0.1 001 0.01 Breitwisch (PRB 2000): T = 8.55 K; t = 400 nm c Park (PRL 1992): T c = 7.35 K; t = 20 nm Breitwisch Allen (PRB (PRB 1989): 2000): T c = 8.55 T K; t = 400 nm Park (PRL 1992): T c = 7.35 c = 8.91 K; t = 20 nm K; t = 20 nm Allen (PRB 1989): T Allen (PRB 1989): T c = 8.91 c = 5.86 K; t = 50 nm K; t = 20 nm Stoddart (SST 1995): T c = 9.20 K; t = 700 nm Allen (PRB 1989): T c = 5.86 K; t = 50 nm Stoddart (SST 1995): T c = 9.20 K; t = 700 nm 0.01 0.1 1 - T/T c F depin ranges from 15 pn/μm to 40 pn/μm # vort rtices 12 10 Limits on motion detection from bootstrapping vortex fits: limited by S/N, can be improved r(95%) 60 (nm)80 40 20 0 100200300400500600 z (nm) 8 17 Gauss 6 4 Gauss 4 2 0 0 10 20 30 40 50 F z (pn) z (pn)

Part II: controlled single-vortex manipulation

Motivation: soft condensed matter physics Do vortices split or tilt and do vortices entangle? surface pancake vortex vortex core interlayer Josephson vortex Clem, PRB 43, 7837 (1991) Reichhardt & Hastings, PRL 92, 157002 (2004)

Motivation: Luttinger liquid physics STM: twin boundary in CuO chain plane of YBa 2 Cu 3 O 7-x y pin vortices in twin boundary plane x 0.21nm 100 Å Eric Hudson, et al Vortices are like 1D bosons in Luttinger liquid!! L τ 000 0.00nm z x L x Polkovnikov, PRB 71, 014511 (2005) vortex wandering in z-direction 1D particles moving in time y x

Motivation: vortex ratchet computing Fabricated defects in a superconductor, used to pin vortices Real material parameters for MgB 2 : Maximum speed: simulations show 315 MHz, pair-breaking is in the THz range. Minimum size: λ ~ 100nm Power dissipation: 10-17 Joules / single cell flip NAND gate: Hastings, Reichhardt, Reichhardt, PRL 90, 247004 (2003)

Our Procedure for Single Vortex Manipulation using Magnetic Force Microscopy surveillance height manipulation height

Vortex Manipulation Results A B C D E Δ f=79mhz Δ f=87mhz Δ f=96mhz Δ f=74mhz Δ f=70mhz 2μ m T = 7 7.2 K h = 288 nm 1μm T = 5.53 K h = 86.4 nm Δf=3.21Hz Δf = Hz

Vortex studies in YBCO Future directions Better MFM tips: - model as monopole 20μm 200nm Deng, APL (2004) nanotube on cantilever tip - better AFM spatial resolution, correlate with topography 100 nm Hawley, Science, 1991 STM studies: image with ξ ~ 15 Å resolution, 100 better than λ ~ 150 nm

Spatial Resolution 8μm SQUID 0.5μm Hall Probe MFM STM 0.5 Gauss 1.1 Gauss 100 Gauss 5 Tesla 10μm 550Å

Future: How to Pin Vortices in YBCO? (1) Screw dislocations (2) Chemical inclusions (3) Oxygen dopants (4) Twin boundaries STM: spiral growth patterns 100 nm Hawley, Science, 1991 STM: impurities in Bi 2 Sr 2 CaCu 2 O 8+x Ni 10Å Zn Vacancy(?) Hudson et al, Nature 411, 920 (2001). Pan et al, Nature 403, 746 (2000). Hudson et al, Physica B 329, 1365 (2003). STM: superconducting gap STM: twin boundary in CuO inhomogeneity in Bi 2 Sr 2 CaCu 2 O 8+x chain plane of YBa 2 Cu 3 O 7-x Vortex imaging in Bi 2 Sr 2 CaCu 2 O 8+x 0.21nm 100 Å 20 Δ (mev) Lang et al, Nature 415, 412 (2002). 70 100 Å Eric Hudson, unpublished. 0.00nm 100Å Hoffman, Science (2002).

ummary Manipulate single vortices with nanoscale control Measured directly the depinning i forces in Nb 1μm already applying same technique to YBCO P Δf=3.21Hz Δf = Hz Eric Straver Nick Koshnick Ophir Ausleander # vort tices Next: correlate depinning forces with topography (Moler) STM studies to explore higher fields (Hoffman) 12 10 500nm 8 17 Gauss 6 4 Gauss 4 2 vortex motion event 0 0 10 20 30 40 50 F z (pn)