QM- Drift & Maximum velocity of an electron

In summary, the conversation discussed the potential and energy levels of electrons in a solid, specifically in the metal silver. The maximum velocity and drift velocity of electrons were calculated using the band structure model, with the drift velocity being significantly lower due to impurities and collisions in the crystal. It was also mentioned that in a perfect crystal, interesting phenomena such as Bloch oscillations can occur. The difference between drift velocity and regular velocity was also explained.
  • #1
Lisa...
189
0
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:

ek= Eat + 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.

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

Therefore:

ek= Eat + 1.28*10-18 cos(0.409* 10 -9k)

vg= 1/hwith line * (d ek/dk)

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

vg=-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 vdrift= I/nAq and n= 5.85*1028 electrons per m3.

vdrift= I/nAq
vdrift= 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?!
 
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  • #2
Hi Lisa, here is a hint: the band calculation assumes a perfect crystal. Is this a realistic for assumption for calculating average electronic velocities?
 
  • #3
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?
 
  • #4
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 [tex] -\frac{\hbar \vec{k}}{\tau} [/tex], where [tex] \tau [/tex] 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.
 
  • #5
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..
 
  • #6
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.
 

What is QM- Drift?

QM- Drift, also known as quantum mechanical drift, is a phenomenon in which electrons in a material move in response to an electric field. This movement is described by the laws of quantum mechanics.

What is the maximum velocity of an electron?

The maximum velocity of an electron is the speed of light, which is approximately 3 x 10^8 meters per second. This is the upper limit for the velocity of any particle in the universe.

How is QM- Drift related to electronics?

QM- Drift is a fundamental concept in electronics, as it describes the movement of electrons in a material. This movement is essential for the functioning of electronic devices such as transistors and diodes.

What factors affect the QM- Drift of electrons?

The QM- Drift of electrons is affected by various factors, such as the strength of the electric field, the properties of the material (such as its conductivity and bandgap), and the temperature of the material.

Why is understanding QM- Drift important?

Understanding QM- Drift is crucial for the development of advanced electronic devices and technology. It also plays a significant role in fields such as solid-state physics and materials science, as well as in the study of fundamental quantum mechanics.

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