Drude Model of Electrical Conduction in Metals

Start with classical ideas to obtain some useful results for conduction electrons in metals

• In the absence of an applied electric field the electrons move in random directions colliding with random impurities and/or lattice imperfections in the crystal arising from thermal motion of ions about their equilibrium positions.

 
• The frequency of electron-lattice imperfection collisions can be described by a mean free path l  -- the average distance an electron travels between collisions.

 
• When an electric field is applied the electron drift (on average) in the direction opposite to that of the field with drift speed 

 
• The drift speed is much less than the effective instantaneous speed  of the random motion

 
• In copper  cm/s while  cm/s where


 
• The drift speed can be calculated in terms of the applied electric field E and of v and l

 
• When an electric field is applied to an electron in the metal it experiences a force eE resulting in acceleration

 

 



 Then the electron collides with a lattice imperfection and changes its direction randomly
 

• Just before the next collision the electron accelerates on average by

 

 


is the mean time between collisions

• The drift speed is



 If n is the number of conduction electrons per unit volume and j is the current density



 Combining with the definition of resistivity  gives



 Since  and l do not depend on the applied electric field this says that metals obey Ohm's law
 

• Conductivity



 Defining Mobility, the magnitude given by the ratio of the drift speed to applied electric field



 Since 



 If we have conduction by positive carriers as well as negative carriers



 where , ,  are the mobilities, the charges, and the number of the carriers per unit volume of the +ve and -ve carriers.

• The sign of q is -ve or +ve for -ve and +ve carriers

 
• also depends on the sign of q so  is always +ve