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Compare and Contrast electrical resistivity measurements in linear 4 probe and Montgomery
Rabindra Bhattarai Department of Physics Iowa State University
What is Electrical Resistivity Electrical resistivity Fundamental property of a material that quantifies how strongly the material opposes the flow of electric current. A low resistivity indicates a material that readily allows the flow of electric current https://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity
Alternative expression E = ρJ E= Electric Field , J = Current Density
Motivation for finding electrical resistivity To know Insulator
material ---- Metal,
Resistivity gives us idea about other physical quantities of interest – Mobility of electron and hole in semiconductor ( Grain size , mid-gap states)
(Condensed Matter Physics) Electrical Transport measurement
Phase transitions in conducting materials are observable as a sharp feature or change in slope of the temperature dependence of the resistivity. Presence of anisotropies in electrical transport can reflect the presence of broken rotational symmetries. Measurement of these anisotropies can contribute to understanding the nature and origin of associated phase transitions, particularly those that are driven by interactions at the Fermi-level
Problem !!!!!! Contact Resistance Resistance due to interface of electrical leads and connections (Parasitic resistance) Due to oxidation of metal on the surface of the metal used Ways to avoid contact resistance Metal and Metal --- soldering, welding
Semiconductor ------Metal in contact with highly doped polycrystalline leading to quantum tunneling
Resistivity measurements April 16, 2014 590B Makariy A. Tanatar
American Physicist Frank Wenner (1873-1954) (American Bureau of Standards) invented Wenner array to measure resistivity of soil
Geometry of sample : Pandora’s Box • Finite shape and size Two length scales -------Spacing between probes and Dimension of sample Two Special case Semi infinite 3D sample Infinite 2D sheets It’s all about how current distribute in sample We have opened Pandora’s box.
Surface Review and Letters 10(6):963 · December
Special Case:Semi –infinite 3D and infinite 2D sheets I Miccoli et al 2015 J. Phys.: Condens. Matter 27 223201
Contd……… I Miccoli et al 2015 J. Phys.: Condens. Matter 27 223201
• For thickness t << S (probe spacing)
Correction Factor for finite thickness Resistivity 𝜌 = F1.F2.F3 (V/I) Correction factor It’s all about how current distribute in sample a) Thickness of sample (F1) b) Position of probe from edge of sample (F2) c) Lateral dimension of the sample (F3)
I Miccoli et al 2015 J. Phys.: Condens. Matter 27 223201
Anisotropy in crystal structure Electrical resistivity : Scalar or Tensor Alternative expression E = ρJ E= Electric Field , J = Current Density ρ𝑥𝑥 𝐸𝑥 𝐸𝑦 = ρ𝑦𝑥 ρ𝑧𝑥 𝐸𝑧
Symmetric tensor by Reciprocity theorem Six independent components Further reduces down to one two or three or more depending on crystal symmetry Cubic ---1 , Trigonal and Tetragonal -2 ……. Orthorhombic -3 Monoclinic ----6 Why not only three ? Diagonalize symmetric matrix and find 3 eigenvectors( principal axis)
Yes. For specific orientation only 3 independent components is sufficient . But its impossible to know the required orientation apriori from crystal structure alone.
Montgomery Method: Exploring unknown with known In 1970 Montgomery proposed a GRAPHICAL method for specifying the resistivity of anisotropic materials cut in the form of a parallelepiped with the three orthogonal edges 𝑙1′ 𝑙2′ 𝑙3′ sides Consists of basically two steps. Step 1: Wasscher mapping : Mapping isotropic parallelepiped on an anisotropic parallelepiped Step 2: Relating the bulk resistivity of a rectangular isotropic prism to voltage- current measurements ( Taking account of geometry(size and shape) of sample and probe
Montgomery Method Step 1: Wasscher mapping Let 𝑙1 𝑙2 𝑙3 and 𝑙1′ 𝑙2′ 𝑙3′ are sides of isotropic and anisotropic material 𝜌1 , 𝜌2 𝑎𝑛𝑑 𝜌3 𝑟𝑒𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 𝑎𝑛𝑖𝑠𝑜𝑡𝑟𝑜𝑝𝑖𝑐 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 and 𝜌 = 𝑟𝑒𝑠𝑖𝑠𝑡𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 𝑖𝑠𝑜𝑡𝑟𝑜𝑝𝑖𝑐 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙
Mapping 𝜌3 = 𝜌1 𝜌2 𝜌3 and 𝑙𝑖 = 𝑙𝑖′ 𝜌𝑖/ 𝜌
Mapping preserves preserve voltage and current relationship, i.e. they do not affect the resistance R I Miccoli et al 2015 J. Phys.: Condens. Matter 27 223201
Anisotropic solid can be mapped to an equivalent isotropic solid. Equivalent means that when identical currents flow into corresponding contact areas on the two solids the potential differences between corresponding points are identical.
Montgomery Method Step 2:Relating bulk resistivity with voltage and current Relating the bulk resistivity of a rectangular isotropic prism to voltage- current measurements Taking account of geometry(size and shape) of sample and probe. 𝜌=𝐻
𝐸 𝑅 where E is the effective thickness
H is the correction factor for finite size (lateral dimension) • Effective thickness equal to real thickness for
𝑙3 𝑙1 𝑙2 1/2
Montgomery Method Step 2:Relating bulk resistivity with voltage and current • H and E are independent parameters For
𝑙3 𝑙1 𝑙2 1/2
E/ 𝑙1 𝑙2 1/2 is fairly independent of 𝑙2 /𝑙1
Not end yet but its getting there….Procedure…… Step 1: Van der Pauw resistivity measurements R1 = V1/I1
V1 voltage between the two contacts on an edge 𝑙1 ’ I1 the current between the opposite two contacts 𝑙2 ’
Current and Voltage connections are rotated by 90 degree to get R2. R2 /R1 is independent of thickness From 𝑙𝑖 = 𝑙𝑖′ 𝜌𝑖/ 𝜌 Using 𝜌 = 𝐻
we get (𝜌2/ 𝜌1) ½ = (𝑙2/𝑙1 )(𝑙1 ’/𝑙2 ’) ………..i
𝐸 𝑅 and 𝑙𝑖 = 𝑙𝑖′ 𝜌𝑖/ 𝜌
𝑙3 𝑙1 𝑙2 1/2
(𝜌2 𝜌1) ½ =H𝑙3′ R1 ……………………….ii Solving (i) and (ii) we get 𝜌1
We use Graph to find value of H and (𝑙2/𝑙1 ) ----------GRAPHICAL METHOD Second sample face normal to first one is used to find 𝜌3
If sample thickness is not small and all three resistivity are different We assume any value of 𝜌3 and make multiple measurement on two perpendicular face until we get self consistent solution.
Why to take all this trouble ? Why not Linear Four Probe We can measure resistivity in various direction with single sample , same sets of electrodes in quick succession.
No Free lunch Montgomery method also has many limitations. • Too many assumptions ( e.g. Validity of Wasscher mapping )
• For the Montgomery method to be valid the sample must be square or rectangular in the plane and of constant thickness.
• Highly sensitive to current path (Sample homogeneity) • Measured voltage have non linear relationship with sample thickness.
Comparative study Linear Four Probe
Isotropic Material 𝜌
Anisotropic Material 𝜌1 , 𝜌2 𝑎𝑛𝑑 𝜌3
Less susceptible to sample thickness and homogeneity compared to Montgomery
Highly sensitive to sample thickness. Doesn’t account for probe position . Based on validity assumptions
Availability of Sample
Thin films or very bulky sample Correction Factor are well defined for finite thickness
Good for parallelepiped but can have various thickness
Time and Simultaneity
Simultaneous measurements of 𝜌1 , 𝜌2 𝑎𝑛𝑑 𝜌3 𝑖𝑠 𝑛𝑜𝑡 possible
****Simultaneous measurement of 𝜌1 , 𝜌2 𝑎𝑛𝑑 𝜌3 is possible *** Ubiquitous signatures of nematic quantum criticality in optimally doped Fe-based superconductors. Kuo HH1, Chu JH2, Palmstrom JC2, Kivelson SA3, Fisher IR