Energy Levels of Free Electrons In Solids
(One Dimensional Case)
- Consider A Free Electron Gas In A One Dimensional Case.
- By Quantum Mechanics We Can Find Out The Energy Levels And
Wave Function For Electron Gas.
- For Simplicity, First We Consider One Dimensional Case And We
Can Extend The Same Theory For Three Dimensional Case.
- Suppose An Electron Of Mass M Confined To A Length L By
Infinite Barrier And 𝜀𝑛 Is The Energy Of
Nth Orbit.
- If The Wave Function For Such Electron Is 𝜓𝑛 , Then
Schrodinger Wave Equation Can Be Given As
𝐸𝜓 = 𝐻𝜓
For Free Electron We Neglect The Potential Energy Term 𝑉 Then
Mathematically The
Solution Of Above Equation Can Be Given As
𝜓𝑛
= 𝐴
sin 𝑘𝑥
+ 𝐵
sin 𝑘𝑥......(4)
where A and B are
constants.
- Using Boundary Conditions In Eq.(4) As
At x = 0, 𝜓𝑛
= 0 then 0 = 0 + 𝐵 sin 𝑘𝑥 ⇒
𝐵
= 0
- At x = L, 𝜓𝑛 = 0 then 0 = 𝐴
sin 𝑘𝐿
⇒
𝑘𝐿
= 𝑛𝜋
⇒
𝑘
= 𝑛𝜋/L
Thus The Wave Function
From Eq.(3) Energy Of nth Level
- Equation (5) And (6) Represent The Wave Function And Energy
Of Free Electron In A Solid, As Shown In Figure Wave Function
Has A Sine Wave Shape.
- If There Is N Number Of Electrons In The Solid Than According
To Pauli Principle, No Two Electrons Have Same Set Of All
Quantum
Numbers.
- Thus The Orbits Can Be Filled In Such A Manner That Each
Electron Has Different Quantum Number.
- However, One Orbit May Have Same Energy. The Number Of
Orbital With Same Energy Is Called Degeneracy.
- If We Start The Filling Of Electrons From Bottom N=1, And
Continue The Filling The Higher Levels Until All N Electrons Are
Accommodated Then The Topmost Filled Level Is Called Fermi
Level And Denoted By 𝑛𝐹.
#Energylevels
#Freeelectrons
#Solidstatephysics
#Quantummechanics
#Electronicstructure
#Bandtheory
#Conductionband
#Valenceband
#Onedimensional
#Crystallattice
#Electronstates
#Energybands
#Quantumstates
#Electrondispersion
#Fermienergy
#Densityofstates
#Electronmotion
#Electronconductivity
#Semiconductorphysics
#Materialsscience
