QM- Drift & Maximum velocity of an electron

[Lisa...]

I was solving the following problem the following way:

In a solid the atoms are regularly arranged in space. The potential seen by an electron is thus periodic and the energy levels are arranged in bands in which the energy of a state k is given by:

e_k= E_at + 2t cos(ka)

The relevant band structure of the metal Silver (Ag) can be roughly modeled with a band of that form with t= 4 eV and a= 0.409 nm. Assume Ag has one conduction electron per atom.

a) Calculate the maximum velocity of an electron.

e_k= E_at + 2t cos(ka)
with:
a= 0.409* 10 ^-9 m
t= 4 eV= 4* 1.6*10^-19= 6.4*10^-19 J

Therefore:

e_k= E_at + 1.28*10^-18 cos(0.409* 10 ^-9k)

v_g= 1/h_{with line} * (d e_k/dk)

d e_k/dk= -1.28*10^-18 *0.409* 10 ^-9 sin(0.409* 10 ^-9k)

v_g=-4.96*10⁶ sin(0.409* 10 ^-9k)

The velocity is maximal when the sine is maximal, therefore when the sine=1. In that case the velocity is -4.96*10⁶ m/s.


b) Calculate the drift velocity (average velocity) of electrons in a silver wire of 1mm² cross-sectional area through which a current of 1A is flowing, knowing v_drift= I/nAq and n= 5.85*10²⁸ electrons per m³.

v_drift= I/nAq
v_drift= 1/5.85*10²⁸* 1*10^-6* 1.6*10^-19 = -1.069*10^-4 m/s


Now my question is: why are the velocities calculated in a and b so different from each other?!

 

[Physics Monkey]

Hi Lisa, here is a hint: the band calculation assumes a perfect crystal. Is this a realistic for assumption for calculating average electronic velocities?

 

[Lisa...]

Ah ok.. I already thought I needed to think in that direction to get an answer... I'll give it a shot ;-)

As you said the band calculation assumes a perfect crystal. The silver in the wire surely is not that perfect arranged (it probably has a lot of impurities and atoms that aren't arranged in a nice pattern). Therefore electrons in the silver wire will bump into impurities and lose their velocity over and over again at each collision. The average velocity will be much lower because of that, than in the case where there are no impurities and the electrons can move without colliding often. Is this correct or is there more to it?

 

[Physics Monkey]

Yes, Lisa, looks like you pretty much have it. The band structure is still pretty good for a lot of things, but those electron are constantly being scattered by the impurities in the crystal. This is often taken into account in semiclassical transport theory by including a damping term in the equation of motion which is -\frac{\hbar \vec{k}}{\tau}, where \tau is the relaxation time. This allows you to recover the result of part b from part a. In fact, conduction in a perfect crystal would display a number of interesting features not observed in ordinary metals. In particular, a DC input would result in an AC output, a phenomenon called Bloch oscillation. These oscillations are almost impossible to observe in metals because the electronic mean free path is so much smaller than the size of the electronic orbit in a Bloch oscillation. In other words, the electrons can't make even a fraction of a single oscillation! However, these oscillations have been observed in a very nice series of experiments where atomic systems were confined to an optical lattice.

 

[Lisa...]

Wow thanks a lot for your background info! You sure know a lot about it ;-) I've only started studying QM 4 weeks ago... it's hard to switch to a certain way of thinking but it's really fascinating though..

 

[mahmudhassan]

hi,
the answer lies in question.
'drift' velocity is net velocity in one direction.
'velocity' is just random velocity.
in the absence of any electric field there is no drift velocity.
when there is a field there will be a drift velocity along field, i.e. in addition to the random very high velocity of electrons (of the order of 10^6 ), there will be a net velocity along field (of order 10^-6).
in the absence of field, although electrons will be moving, the average number of electron moving from left to right (or from any direction to opposite direction) will be equal to number of electron moving from right to left. so there is no net flow. i.e. no drift, no current.
in a current carrying wire it is the field that travels at speed c, not the electrons.