What Is the Bandwidth and Wave Vector Solutions for a BCC Tight Binding Model?

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SUMMARY

The bandwidth W for a Body-Centered Cubic (BCC) tight binding model, specifically for structures like iron (Fe), is calculated as W = 16t, where t is the model parameter. The energy E is expressed as E(x) = const - 8t cos(kx * a/2) * cos(ky * a/2) * cos(kz * a/2). The maximum energy Emax occurs at Emax = const + 8t, while the minimum energy Emin is given by Emin = const - 8t. The solutions for kx, ky, and kz corresponding to the band bottom and top can be derived from the cosine function values of 1 and -1.

PREREQUISITES
  • Understanding of tight binding models in solid-state physics
  • Familiarity with BCC crystal structures
  • Knowledge of wave vector notation and its implications in band theory
  • Basic proficiency in mathematical functions, particularly trigonometric functions
NEXT STEPS
  • Explore the derivation of energy bands in tight binding models
  • Study the implications of wave vector solutions in BCC structures
  • Investigate the role of the model parameter t in electronic properties
  • Learn about the relationship between crystal symmetry and band structure
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Physicists, materials scientists, and students studying solid-state physics, particularly those focusing on electronic properties of crystalline materials.

abdul-pablo
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According to tight binding moment, for BCC crystalographic structures (such as Fe), energy E depends on wave vector kx, ky, kz: E(x) = const - 8t cos(kx * a/2)*cos(ky * a/2)*cos(kz * a/2), where t>0 is the model parameter. What is the bandwidth W in terms of parameter t? Can you find kx, kz, and ky at the band bottom and at the band top?
 
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Well you need to find the minimum and maximum value of the band Energy - then W is the difference between them.. The easiest way is probably just by inspection.
 
So the maximum is Emax = const + 8t when the multiplication of cos functions gives -1 and Emin = const -8t as the multiplication of cos functions gives 1. It implies that W = 8t - (-8t) = 16t. Now I should find kx,ky and kz fow which that cos functions multiplications results in 1 and -1. I have found 4 solution for 1 on the right side of the equation (having cos on the left one) and 4 solutions for -1 on the right. Is it possible? Am I right?
 

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