Electrons and holes: how can holes have a mass?

[fisico30]

Hello Forum,

in semiconductors, there are two types of charge carriers: electrons and holes.

Holes are fictitious positive charges...How can they have a mass?

thanks,
fisico30

 

[DrDu]

The electrons that are missing have a negative effective mass. So the holes have a positive effective mass.

 

[Aeroslicer821]

Lets say you have a classroom of 100 chairs and 99 people, so that there is one empty chair. You can track the movement of the 99 people to see how the chairs are being filled and from there derive their equations of motion. Rather than go the many body approach to tracking 99 different people at the same time, you can do the same thing by just tracking the single empty chair. From those new equations of motion you will be able to derive an effective mass for the empty chair since you are now treating the empty chair as an entity.

 

[enotstrebor]

Historically, Dirac himself initially voiced his own difficulties with the analogy of electron holes, but over time as it became "popular" so to did his acceptance (not implying cause and effect just historical accuracy).

But as a concept holes in electronics is usefull and things can be given mass as related to its behavior by the mathematical theory.

So if you don't feel comfortable with the physicality of the hole concept you are in good company.

I always looked at it as Dirac originally did as a conceptual tool. Though today, such a view is admittedly in the minority.

You will find a fundamental lack of natural physicality in quantum mechanics, It is however and excellent mathematical tool with vast experimental validation so QM is clearly correct about how nature behaves.

 

[enotstrebor]

Hello Forum,

in semiconductors, there are two types of charge carriers: electrons and holes.

Holes are fictitious positive charges...How can they have a mass?

thanks,
fisico30

The key word here is "effective" so the mass assigned to the hole is no more real than the mass assigned to the electron.

When an electron is moving inside a solid material, the force between other atoms will affect its movement and it will not be described by Newton's law. So we introduce the concept of effective mass to describe the movement of electron (and holes) in Newton's law terms. Generally, in the absence of an electric or magnetic field, the concept of effective mass does not apply.

Donor impurities create states near the conduction band while acceptors create states near the valence band. In either case the energy required to conduct electricity is lowered by the change in the band gap.

The effective mass is related to the new dopant valence band gap (associated with holes) and the dopant conduction band gap (associated with electrons).

Effective mass is defined by analogy with Newton's second law F = m a. Since we can measure the applied force and the resultant acceleration, we can compute "effective" mass.

So just as one assigns and effective mass to the electron related to the dopant conduction band gap so one can assign an effective mass of the "hole" related to the dopant valance band gap. So in general the "effective mass" of the electron and hole are not equal.

[I was hoping some "authority" would present this type of answer, as I admittedly am not an expert but I believe this is correct (and maybe more informative than the previous answers including my own).]

 

[jambaugh]

Remember mass essentially tells us how much energy is required to move the object at a certain speed (via KE= 1/2 m V^2). To move a hole you must move the electrons in the opposite direction to fill the hole's previous location.

There are also interaction effects which change the effective masses of both electrons and holes moving in a material because they change the surrounding field which must be carried around with the moving charges relative to what they would be in a vacuum.

 

[ZapperZ]

Also note that the so-called "electrons" that you are measuring are NOT the same electrons that you measure in vacuum! These electrons are quasiparticles, the same way the holes are quasiparticles, via Landau's Fermi Liquid theory. One only needs to look at the effective mass of these electrons to know that they are not the same as the bare electrons. In fact, electrons in the ruthenates can have 200 times the mass of a bare electron, while those in graphene can act as if they have NO mass!

So if one can "renormalize" these electrons into such quasiparticles, one can certainly do the same for these holes.

Zz.

 

[fisico30]

Hi ZapperZ,

thanks for the answer. But could you explain what a quasi-particle is, compared to a normal particle? In what sense is it quasi?

thanks
fisico30

 

[jambaugh]

Hi ZapperZ,

thanks for the answer. But could you explain what a quasi-particle is, compared to a normal particle? In what sense is it quasi?

thanks
fisico30

If I may, a quasi-particle is when the aggregate behavior of many actual particles behaves like a single particle. The best example is the electron hole in the semi-conductor. Phonons in a crystal are also quasi-particles.

Note that quasi-particles are not just composite particles, i.e. an alpha particle = helium nucleus is a particle not a quasi-particle.

 

[ZapperZ]

Hi ZapperZ,

thanks for the answer. But could you explain what a quasi-particle is, compared to a normal particle? In what sense is it quasi?

thanks
fisico30

I'm sure I've done this already in some other threads, but here it is one more time. It will be a very naive explanation, but if you don't know enough about many-body physics, it will give you a good idea.

Let's say you have a system where there are many electrons. In the classical gas case, they only interact when they bump into each other, and thus, you have the typical Drude-type model. But what if they start to interact with each other, say, via their coulombic interactions? Now, try to write down this interaction. It is difficult because there are a gazillion of then, and you have to consider the interaction of particle 1 with particle 2, 3, 4, ... N. Then you have to do the same for particle 2 interacting with particle 1, 3, 4, 5, ... N.

What you end up with is what we call a many-body interaction, which is unsolvable! This is a one, many-body problem. Now, Landau stated that, in cases where these interactions are "weak", then we can get away with changing the one many-body problem into many one-body problem which is simpler. What this means is that we take all the many-body interactions and naively lump it into a "renormalized" particle, called a quasiparticle. This particle will have some properties that are different than the bare particle. It has a self-energy term (both real and imaginary) that can correspond to its lifetime, scattering rate, effective mass, etc.. etc.

So the "electron" that we measure in solids are really these quasiparticle. In some system, the quasiparticle is almost the same as the bare electron because the interactions are so weak, they approach that of the bare electron. But in many other systems, the so-called strongly-correlated electron system, this is no longer the case. In fact, in some systems, the interactions are so strong, even the Landau's Fermi Liquid theory no longer works well. It is why in condensed matter physics, the study of such system (and this includes many things such as High-Tc superconductors) is a very active area.

Zz.

 

[fisico30]

Thanks ZapperZ,

great explanation. So too many interactions to keep up with. Each particle has its own many interactions. There are many particles. So when u say the many body problem should involve an equation for each electron that takes into account all interactions.
For example, say we have 3 particles: A, B, C.

we would write one equation of motion for A. This equation taks into account the interaction A,B and A,C.

Another equation for particle B: here we see the interaction terms B,A and B,C.
Same for particle C: interaction terms A,C and C,B...
The 3 equations form a system of equation, right?

Now, we want to turn this single 3-body problem into 3 one-body problem...What would that mean?
thanks
fisico30

 

[ZapperZ]

Now, we want to turn this single 3-body problem into 3 one-body problem...What would that mean?
thanks
fisico30

It means that knowing what one particle does is sufficient to characterize the entire system. That single particle defines the state (ground state and low-lying excited state) of the system, which is what you want in many cases. For a Fermi liquid, it means that you know fully the single-particle spectral function.

Zz.

 

[jambaugh]

Let me add a bit of kindling to the camp fire...

At another level all particles are in a sense "quasi". Even in vacuum e.g. an electron cannot be separated from its surrounding electric field (sea of virtual and actual photons).

I recall the classical calculation of the energy of point charge's E field. You get a divergent result and you must invoke a small radius cut-off. You can thereby define a classical electron radius whereby treating the electron as a spherical charge distribution all of its mass is accounted for by the field.

Take this as a model for an electron and you then can recalculate the mass (= energy of its field) when it is no longer in a vacuum but rather within a dielectric medium (e.g. semiconductor).

In the quantum version of this one gets the analogous QFT "self interaction" calculation for effective mass and must invoke renormalized regularization (the quantum analog of picking a finite radius for the electron) to remove the divergence. With the semiconductor case a modified effective QFT with modified effective masses results. But there is in this analysis no real distinction between "particle" and "quasi-particle" and one leaves with a sense of the inseparability of system and environment.

 

[ZapperZ]

jambaugh has brought up an interesting point that has also been brought up by many condensed matter physicists, namely Anderson and Laughlin. There are strong arguments that ALL elementary particles are essentially "emergent", meaning that they came out of vacuum many-body interactions. The Higgs mechanism is one such example that endows some particles with a portion of their masses.

What this means is that the things that we learn in solid state physics, and which has been designated as "applied physics", is really quite possibly fundamental in nature. This is because the physics and mathematics that we gain such a condensed matter system could possibly be "universal" in the way we have to view our elementary particles.

Zz.

 

[jambaugh]

...There are strong arguments that ALL elementary particles are essentially "emergent", meaning that they came out of vacuum many-body interactions. The Higgs mechanism is one such example that endows some particles with a portion of their masses.

Yes and that is in part why some turn to e.g. strings and M-theory. However this brings up a "chicken & egg" issue. I think the answer may lie, not in emergence from a lower level of fundament, but rather in a relativity principle in how we decompose systems and resolve them into components. Note also that the Higgs mechanism has yet to be confirmed. "It's only a model"

 

[ZapperZ]

Yes and that is in part why some turn to e.g. strings and M-theory. However this brings up a "chicken & egg" issue. I think the answer may lie, not in emergence from a lower level of fundament, but rather in a relativity principle in how we decompose systems and resolve them into components. Note also that the Higgs mechanism has yet to be confirmed. "It's only a model"

Actually, an analogous Higgs mechanism has been confirmed in condensed matter systems. We just haven't confirmed it for the electroweak symmetry breaking.

I'm not sure I understand what you mean by decomposing of system. In emergent properties, once you try to break it apart, the effect goes away (example: superconductivity).

In any case, I don't think this is the proper thread to carry this discussion since it is a bit off-topic.

Zz.

 

[Naty1]

Fascinating discussion above...

enot posts:

You will find a fundamental lack of natural physicality in quantum mechanics, It is however and excellent mathematical tool with vast experimental validation so QM is clearly correct about how nature behaves.

I can't agree with the first sentence: what seems "natural physicality" is not so 'natural' as it might first appear, not as we might expect from everyday experience. The physicality is what we measure it to be.
The above posts remind me of where the energy in a gravitational field resides...as jambaugh and zapper just discussed 'decomposition' in this analysis.