Atoms

Josephson Junction and SQUID

		The Josephson effect occurs in both bulk and 2-dimensional super-conductors when a piece of thin barrier (the junction) is inserted into the circuit. The weak link can be a thin insulating barrier (S-I-S) about 3 nm thick, a short section of non-superconducting metal (S-N-S) ~ 10 nm, or a physical constriction that weakens the superconductivity at the point of contact (S-s-S). The circuit consists of a biasing voltage V to control the current I flowing through the apparatus (Figure 13-08j).
Figure 13-08j Josephson Junction, Circuit	Figure 13-08k Josephson Junction, Mechanism [view large image]

SQUID is a very sensitive instrument for measuring magnetic flux which is closely related to magnetic field. Figure 13-08l,c shows the range of measurable objects below and above the non-SQUID limit. It is made of two Josephson junctions on each side of a loop. The flux through the hole is detected via its relationship with the output current or voltage as shown in detail below. The superconductive materials can be either type I or type II (Figure 13-08l,a,b). Type I superconductors require very low temperature for the transition. It depends on the formation of Cooper pairs to attain the superconductive state (see list of Type I Critical Temperatures). The type II super-components are the vortices (see 2-D Superconductivity). The critical temperature for type II materials in 2-D is even lower. However, many compounds exhibit superconductive at much high Critical Temperature (see list of type II superconductors).

Figure 13-08l SQUID [view large image]

		The SQUID is a tiny equipment with a planar area of about 10-3m x 10-3m. It has to be kept inside a dewar with liquid helium (< 4.2K) or nitrogen (< 77K) to maintain the superconductive state (see Figure 13-08m,a for the basic components). It relies on the conversion of magnetic flux to voltage for performing the measurement.
Figure 13-08m SQUID Flux-locked Loop [view large image]	Figure 13-08n SQUID, Operating Principle [view large image]	See a detailed description of a commercial SQUID system (as shown in Figure 13-08m,a).

Operating Principle of SQUID (Figure 13-08n) :


• Quantization of Magnetic Flux in Superconducting Loop - Magnetic flux can have arbitrary amount through a normal conducting loop. The situation is different in superconductive loop because the Cooper pairs move as a coherent whole. The mathematical derivation is rather complicated but such phenomenon is essential for understanding the operation of SQUID. Starting from the Uncertainty Principle as shown in Figure 13-08o, when the momentum of a particle p is known precisely. Then the position cannot be determined and is in the form of an infinite wave train with wave length = h/p where p = mv + qA for the Cooper pairs in magnetic field B which is related to the vector potential A by the formula B = ( X A).

Figure 13-08o Matter Wave [view large image]	Figure 13-08p Stokes' Theorem



• SQUID Equations -
  
  
  
  Following is a qualitative description according to these formulas with the help of pictorial illustration in Figure 13-08n.
  
  
• Initial State with no external flux (i.e., the external magnetic flux = 0) -
  
  
  • Initially, ₁ = ₂ ~ /2.
  • The critical current at each junction is I_c, and I = I_b ~ 2I_c for the SQUID.
  • The constant bais current I_b is slightly greater than the critical current 2I_c, so that the SQUID is in normal conductive state. It splits evenly to I_b/2 in opposite side of the loop (Figure 13-08n,d).
  • Thus, the output voltage V is slightly above zero.
  
  
• Detection of n₀ Flux where n = (0 - 1/2), (1 - 3/2), ... -
  
  
  • When the input flux is in such range, the loop generates a screening current I_s to have it neutralized, i.e., there would be no flux going through the loop.
  • The net effect is to reduce the critical current of the SQUID to 2I_c - 2I_s (Figure 13-08n,c).
  • The SQUID is operating in normal conductive state up to n = 1/2, 3/2, ... as shown by the rightmost V-I curve in Figure 13-08n,a.
  • The output current I = I_b + 2I_s is illustrated by the same curve in Figure 13-08n,a.
  • The output voltage V increases from minimum to maximum as shown by the same diagram (step 1 in Figure 13-08n,a). The V-I relationship is approximately V ~ RI, where R is the shunt resistance (low resistance path for current measurement).
  • A simple way to visualize the variation of ₁ is to notice that I_s has to be positive between /₀ = 0 and 1/2. Then according to the formula to relate I_s and /₀, ₁ + /₀ /2, or ₁ /2 - /₀. For example, ₁ could be equal to /2 when /₀ = 0; and it would become 0 when /₀ = 1/2. Actually, it changes suddenly to negative at this point to ₁ ~ -/2.
  • The corresponding variations for ₂ (see formula) : ₂ = ₁ ~ /2 at /₀ = 0; and ₂ ~ (jump to /2) when /₀ = 1/2 (see steps 1 and 2 of the constructive interference pattern in Figure 13-08n,c).
  
  
• Detection of (n+1/2)₀ Flux where n = (0 - 1/2), (1 - 3/2), ... -
  
  
  • When the input flux is in such range, it is energetically more favorable for the loop to generate a screening current I_s (in opposite direction) to make the flux a whole number of ₀, i.e., to allow the flux through the loop.
  • The effect of such rapid reversal is to make a dis-continuous switch of the SQUID critical current to 2I_c + 2|I_s| (Figure 13-08n,c).
  • The output current suddenly lowers to I = I_b - 2|I_s| as illustrated in Figure 13-08n,a (see the sudden variation from step 1 in normal state to steps 2,3, and onto step 4 in super-conductive state, which does not produce output voltage, i.e., V ~ 0).
  • Now the SQUID is operating in superconductive state up to n = 1/2, 3/2, ... as shown by the V-I curve (step 4 in Figure 13-08n,a).
  • When /₀ = 1, ₁ = ₂ ~ /2, I_s = 0, the cycle repeats anew (see formula, starting from n = 0, then 1, ...). In other words, the difference between ₁ and ₂ runs from 0 to and then back to 0 (see steps 3 and 4 of the destructive interference pattern in Figure 13-08n,c). .
  
  
• Measurement - The SQUID depends on variation of output voltage V to detect minuscule flux down to 10^-6₀ ~ 2x10^-21 Wb. Since the magnetic field B = /S, where S ~ 10^-6 m² is the planar area of the SQUID, the minimum detectable magnetic field B ~ 10^-15 T (T = tesla = 10⁴ Gauss = Wb/m² = joule/amp-m²). Figure 13-08l,c shows the range of magnetic field for various sources. The detectable limit is

  imposed by magnetic noise in the environment and the instrument itself, e.g, from muli-channels interference (by hundreds of SQUID within a device). The external sources can be minimized by metallic shielding (Figure 13-08q), while the contribution from the instrument can be reduced by signal processing software (see "Novel Noise Reduction Methods" for a review of the problem in Magnetoencephalography operation). In addition, the signal is very weak and cyclic, the output has to be amplified and integrated, this step is performed by the Flux-locked Loop as shown in Figure 13-08m,b.

  Figure 13-08q Magnetic Shielding [view large image]

  
  
• Some Applications -
  
  

  Magneto-Encephalo-Graphy (MEG) - This device measures the magnetic field in the order of 10-13 T (as shown in Figure 13-08l,c) generated by the collective action potentials of about 105 neurons in the brain. Such feat is possible only with the SQUID detectors. It is bathed in liquid helium and operated in magnetically shielded room (see Figures 13-08q and 13-08r,a) and the skull is placed a few centimeters from the sensors. The technology complements with both fMRI and EEG, but only MEG provides timing as well as spatial information about brain activity.

  Figure 13-08r SQUID Applications [view large image]

  In addition, SQUIDs are also used to check electrical activity in the stomach (Magnetogastrography, MGG), and in cardiology for magnetic field imaging (Magnetocardiography, MCG).

  
  
  • Scanning Microscope - It is a sensitive near-field imaging system for the measurement of weak magnetic fields (about 10^-6 T) generated by current down to 10^-8 Amp at a distance of 10^-4 m. This is very useful to check out defects in micro-electronic circuit. The high temperature SQUID is made from YBa₂Cu₃O₇ cooled to below 80K while the other part of the unit and the sample are at room temperature and in air (Figure 13-08r,b). The insert in the same picture shows the discovery of ferromagnetic transition in the LaMnO₃ layer (about 3 nm thick) via electron transfer to the substrate.
  
  

  Flux Qubit - The SQUID can be fabricated with another loop to make it acting like two spin states corresponding to the two directions of the super-current (Figure 13-08r,c). As shown in the schematic diagram, 1x and 2x are the external biasing flux. The spin state is determined by the direction of 1 and hence the direction of the super-current. The different state has distinct ground state energy. The two states coupled into superposition forming a qubit when the flux is close to n0/2 (Figure 13-08s,a). Computational operations are performed by pulsing the qubit

  Figure 13-08s Flux Qubit [view large image]

  with microwave frequency radiation which has an energy comparable to the gap energy V. Figure 13-08s,b shows the state detection by another SQUID sensor.

  
  
  • Magnetometer and Gradiometer -
    • The SQUID can be used as magnetometer for exploration of mineral resources as shown in Figure 13-08r,d. In the transient electromagnetics method, in which electric and magnetic fields are induced by transient pulses of electric current and the subsequent decay response measured, the SQUID is the HTS (High Temperature Superconductor) variety with low noise interference and light weight.
    • A SQUID gradiometer has been developed to monitor the magnetic field gradient from submarine. By using three independent rotating sensors, the system will be able to measure the range, depth and bearing of a submarine, how fast it is going and if it is diving - all from one flyby (Figure 13-08r,d; and see detail in "Submarine detection").
  
  

  The full name of SQUID is Superconducting QUantum Interference Device. The interference is referred to the Cooper pairs emerging from the two Josephson junctions. Their quantum mechanical wave functions interfere with each other much like light waves from the double slits. The wave function's amplitude is proportional to the probability of the electron pairs, rather than to brightness as in the case of optical interference (Figure 13-08t). The pattern is the variation of the total current I as a function of /0 (see formula, and Figure 13-08n,c).

  Figure 13-08t Two Slits Optical Interference [view large image]

  
  

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