# Electronic Band Structure, Ionization Energy

Hi, thanks for reading my questions.

I've been reading and reading and reading and reading and I'm trying to understand the difference between conductors, semiconductors, and insulators at an atomic level. When reading about electricity I often find that metals have a "sea" of "free" electrons that makes conducting currents possible. I also read about insulators and their ability to withstand high voltages before breaking down and becoming conductors (dielectric breakdown). And although I know very little about semiconductors they seem relevant here.

First question: why are some electrons more difficult to remove than others?

Second question: What does the energy level of an atom mean? does it have something to do with the orbital velocity of the electrons?

Third question: voltage is a force that acts on charged particles? does that mean that voltages are measures of electrical force?

Fourth question: how can an object have an excessively high voltage like 100,000 volts for example? Does that mean that the constituent atoms have a severe charge imbalance?

What I don't understand is what causes electrons to be ripped from their atoms and move forward in a chain reaction to create an electrical current, especially whether this is caused by repulsive forces between electrons (pushing) or by attractive forces from protons (pulling).

I apologize for asking a vague question but I can't put a finger on it.

-Andrew

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1. Valence band electrons are bound more locally to individual atoms. They have a stronger bond and are closer on average to the nuclei to which they are bound. When an electron is excited into the conduction band, it becomes delocalized (pseudo-free) and is bound to the solid as a whole and not to any individual atom. This is a weaker bond, which takes less energy to break. For this reason, it is easiest to remove conduction band electrons from a solid. The electrons in the lowest atomic levels are bound most strongly to their atoms and take the most energy to remove.

2. It gets complex quickly. Any particle can be in a number of quantized states, each with different bond strengths. The "energy" of a bound particle typically refers to the energy it would take to completely free the particle, or sometimes it is taken to mean the energy it would release if it were to transition to the ground state. The total energy of a particle in a certain state is a combination of its rotational energy and potential energy. The protons and neutrons in the nuclei can have different energy states. The electrons bound to the nuclei have different energy states. The atom as a whole has different energy states. When atoms are bound into molecules, the molecules can have different energy states. When atoms are bound into solids, the solid as whole develops an energy state structure (the band structure).

3. Voltage is the electric potential difference between two points in space. It is not a force. The electric force F on a charged particle q is F = q E where E is the electric field. The electric field is the negative gradient (the three-dimensional slope) of the electric potential. For instance, if you have a constant, uniform electric field, then the voltage V and its associated force F on a charge q is given by F = qV/d where d is the distance between the two points where the voltage is measured.

4. Yes, if the object is statically charged (as opposed to a high-voltage AC power line which)

5. For conductors, the conduction electrons are already unbound from individual atoms, and are free to be accelerated when they feel the electric force of an applied voltage. Typical electric currents are an interaction between an applied field and the pseudo-free conduction electrons and have little to do with individual internal atomic forces.

I definitely don't understand the concept of an energy band structure. I don't for example understand what the difference between an atom in it's ground state and the same atom in it's excited state is. I think that the atom has kinetic energy - like that of the orbital or rotational momentum of the electrons. I've also read of vibrational energy but I'm not sure if that is the energy we refer to when discussing band structures. Another thing is that band structures are a phenomena of solid matter rather than constituents - it seems like the energy level propagates across matter somehow.

Band structure is interesting but is probably too far beyond my current understanding to productively learn about so I will focus my questions on more familiar areas.

I think that I understand what you are saying about voltage being an electric potential difference, and I understand more clearly what is meant about the "delocalized sea" of conduction band electrons in metals.

I'd like to know a little more about what is happening to the electrons when conductors are ionized or have current flow across them.

For example a copper wire is connected to the electrically negative terminal of a 10 volt battery.

Q1: what happens to the electrons in the wire? Why isnt there current flow?

My best guess is that the electric field at the terminal of the battery applies a force to the electrons of the wire and that force is applied to both the "free" electrons and the "bound" electrons, but the binding force from the protons of individual atoms is stronger than the force created by the applied field so the bound electrons do not move. The free electrons however I think they are compelled by the applied field to move and somehow collide with each other like a row of dominos that transmits the force of the field across the conductor in all directions.

At the border between the wire and its surrounding insulator (air, or plastic) the applied force is not great enough to dislocate electrons from the atoms of the insulator because they are too tightly bound to their nuclei and require more force to move.

That's all for now,
Thanks.

To understand these questions in detail (properties of metals, difference between semiconductors and metals etc.) one has to understand band structure. This does not necessarily have to be so difficult even at a popular level.

The formation of a band structure in a solid relies basically on two things: the Pauli exclusion principle and the fact that energy levels of an atom changes in an electric field.

Firstly, the Pauli exclusion principle states that two electrons cannot occupy the same energy level in an atom (in fact, the principle is more general but this is the content here). This means that when you add more electrons to an atom they will occupy higher and higher energy levels. This is a bit like filling a bucket with water - when you add more water the level rises. This is reminiscent to the higher energy of the added electrons.

The second fact needed for band structure is the fact that the energy levels change when the atom is placed in an external electric field. This is precisely what happens when atoms are moved together to form a solid. The energy levels of one atom change in the presence of the others. The effect is that the energy levels change and form something called bands - densely packed energy levels forming an almost continuous "band" of energies.

Together these two properties create the band structure of solids. In a metal or semiconductor the effect is to create one lower band, where the electrons are more tightly bound, and a higher (conduction) band where the electrons are almost free particles. A simplified version is to say that the valence electrons (the outermost electrons, not tightly bound to the nucleus) of the metals form a conduction "sea" of freely moving electrons making the solid a conductor.

In a semiconductor the conduction band and the valence band have a gap between them. In this gap no energy levels are permitted, so no electrons can have any energy in the band gap. Also, the conduction band is only partly filled in a semiconductor at room temperature. Because of this semiconductors are not as conducting as metals at room temperature since the conduction electrons are not as many as in a metal. In metals, there is no band gap between the conduction and the valence band and as a result the conduction electrons are many more. In an isolator the conduction band is empty, and the band gap is so large that it is very unlikely that electrons receive an energy sufficient to reach the conduction band. Therefore there is no conduction of electricity.

Hope this is of help! Feel free to ask more questions if you don't understand. I recommend you to read more about band structure if you really want to understand metals and semiconductors.

Please excuse any simplifications made in the above explanations and if anything is wrong, please point this out so we don't cause confusion.

Discrete states occur when a particle/object becomes bound. When a proton becomes bound to a neutron inside the nucleus, it can only exist in certain quantum states (or a combination of them). When an electron becomes bound to a nucleus, it can only exist in certain states. When an atom becomes bound to another atom in a molecule, a series of states form, etc. Each state has a definite wave function shape and energy. The single electron states of a hydrogen atom are one of the easiest to find mathematically and one of the easiest to visualize. In general, electron wavefunction states go farther away in distance from the nucleus as you go up in energy (like how it takes more energy to stretch a spring farther). If we hit an electron in an atom with a bit a of light (a photon), the electron will absorb the photon and its energy be knocked to a higher energy state, farther away from the nucleus.

A single atom does not have bands, it has a single, discrete energy state at each level. When you bring to identical atoms close together, due to the Pauli exclusion principle (no two identical fermions can be in the same state at the same time), the levels must change a little bit so that the electrons are not in the same state. One energy level goes slightly up and one goes slightly down (actually, one combination of both is slightly up and one combination of both is slightly down). So with two bound atoms, you end up with two energy states at each level instead of one. If you have three atoms bound together, you have three very close, but different energy states that are more or less identical. You say that the original state of one atom has split into three states for three atoms. If you take billions of identical atoms and bind them together, you get an every day solid. And you get each energy state splitting into billions of very close energy states. The states are so close and so numerous, that they effectively make a continuous band of states. That is why we say solids have a band structure. The electron can take on any energy within a band.

T
For example a copper wire is connected to the electrically negative terminal of a 10 volt battery.

Q1: what happens to the electrons in the wire? Why isnt there current flow?
I assume you mean only one end of the wire it connected to the battery, but the other is not. so that the circuit is open. In that case, you are right, no current flows.

My best guess is that the electric field at the terminal of the battery applies a force to the electrons of the wire and that force is applied to both the "free" electrons and the "bound" electrons, but the binding force from the protons of individual atoms is stronger than the force created by the applied field so the bound electrons do not move...
That is true that bound electrons due not move as apart of an electrical current because their binding force is too strong. But that has nothing to do with why no current flows in an open circuit. Aside from providing a little resistivity and keeping the conduction electrons in the wire, the atoms (nuclei, bound electrons, protons, etc.) do not play much of a role when you are talking about electric circuits. The conduction electrons are effectively free. A current forms whenever the conduction electrons feel a force. Electrons feel a force when they experience an electric field. You create an electric field by creating a potential difference (voltage) between two points. It's like a ball rolling down a hill. You need a high point and a low point on a hill in order to create a slope that the ball will roll down. Attaching only one end of a wire in a circuit is like placing a ball on flat ground and calling it a high point. The ball will roll nowhere even if you call the bit of ground "really high". You have to also have a low point in order to define a slope the ball can roll down.