Fixing typos, style and grammar

src/13_semiconductors.md

@@ -47,8 +47,10 @@ _(based on chapters 17–18 of the book)_
 Up until this point, we focused on calculating and understanding the band structures.
 However, the dispersion of a band is only part of the story.
 An empty band is not gonna lead to any interesting physical properties no matter how sophisticated it is.
-Therefore, it is also important *how* bands are filled by the particles. By carefully controlling the distribution of particles in the bands, we are able to engineer material properties that we require. Without a doubt, the greatest example is the bedrock of all electronics - *semiconductors*. 
-In this lecture, we shall grasp the basics of semiconductors by learning how to treat different levels of particle filling inside the bands.
+Therefore, it is also important *how* bands are filled by the particles.
+By carefully controlling the distribution of particles in the bands, we are able to engineer material properties that we require.
+Without a doubt, the greatest example is *semiconductors* - the bedrock of all electronics. 
+In this lecture, we shall grasp the basics of semiconductors by learning how to treat bands at different filling.
 
 ## Review of band structure properties
 
@@ -62,7 +64,7 @@ Tells us how hard it is to *accelerate* the particles and is related to the curv
 The amount of states per infinitesimal interval of energy at given energy. 
 The quantity is vital in order to calculate any bulk property of the material such as conductivity, heat capacity, etc.
 
-In order to check that everything makes sense, we apply them to the free electron model:
+In order to check that everything makes sense, we apply the concepts to the free electron model:
 
 $$H = \hbar^2 k^2/2m$$
 
@@ -73,7 +75,8 @@ So in the simplest case the definitions match the usual expressions.
 
 ## Filled vs empty bands
 
-To begin our understanding of band filling, let us start with the most extreme cases - filled and empty bands.
+We distinguish three different band filling types: filled, empty and partially filled. 
+Let us start with the most extreme cases - filled and empty bands.
 We treat these two cases together, because a completely filled band is very similar to a completely empty band.
 For example, both filled and empty bands lead to zero current:
 $$
@@ -86,16 +89,15 @@ An empty band has no electrons and thus no current.
 On the other hand, a filled band has an equal number of electrons going forwards and backwards which thus cancel and lead to zero current. 
 Similar results apply to many other physical quantities such as heat capacity and magnetisation. 
 Therefore, filled and empty bands do not affect most physical properties and can be disregarded.
-Such an observation offers great simplification.
-Rather than consider thousands of bands that a material can contain, we forget about most of them and just focus on several partially filled bands around Fermi level.
+As a result, rather than to consider thousands of bands that a material contains, we neglect most of them and just focus on partially filled bands around Fermi level.
 
 ## From electrons to holes
 
 In order to understand partial filling, we begin with a simple analogy.
 Let's say we have 100 chairs: 99 are occupied and 1 is empty.
-To keep track which chair is occupied/empty, we could write down the occupation of each and every one.
-However, that is a lot to keep track of!
-Instead, we need only to write down which chair is empty, because we know the rest wil be occupied. 
+To keep track which chair is occupied/empty, we could write down the occupation of each and every chair.
+However, that would require a lot of unnecessary writing!
+Instead, we only need to write down which chair is empty, because we know the rest wil be occupied. 
 The same philosophy is applied to band filling. 
 Instead of describing a lot of electrons that are present in an almost filled band, we focus on those that are absent.
 The absence of an electron is called a **hole**: a state of a completely filled band with one particle missing.
@@ -104,7 +106,7 @@ The absence of an electron is called a **hole**: a state of a completely filled
 
 In this schematic we can either say that 8×2 electron states are occupied (the system has 8×2 electrons counting spin), or 10×2 hole states are occupied.
 A useful analogy to remember: glass half-full or glass half-empty.
-Electrons and holes is nothing more than a way to explain the occupation of a band in two equivalent ways.
+Electron and hole pictures correspond to two different, but equivalent ways of describing the occupation of a band.
 In practise, we choose to deal with electrons(holes) whenever a band is almost empty(full).
 
 ## Properties of holes
@@ -112,11 +114,11 @@ In practise, we choose to deal with electrons(holes) whenever a band is almost e
 Let us compare the properties of electrons and holes.
 The probability for an electron state to be occupied in equilibrium is given by $f(E)$:
 
-$$f(E) = \frac{1}{e^{(E-E_F)/kT} + 1}$$
+$$f(E) = \frac{1}{e^{(E-E_F)/kT} + 1}.$$
 
-On the other hand, since a hole is a missing electron, the probability for a hole state to be occupied is:
+On the other hand, since a hole is a missing electron, the probability for a hole state to be occupied is
 
-$$f_h(E) = 1 - f(E) = 1 - \frac{1}{e^{(E-E_F)/kT} + 1} = \frac{1}{e^{(-E+E_F)/kT} + 1}$$
+$$f_h(E) = 1 - f(E) = 1 - \frac{1}{e^{(E-E_F)/kT} + 1} = \frac{1}{e^{(-E+E_F)/kT} + 1},$$
 
 therefore for holes both energy $E_h= -E$ and $E_{F,h} = -E_F$.
 
@@ -126,15 +128,15 @@ Therefore $p_h = -\hbar k$, where $k$ is the wave vector of the electron.
 Similarly, the total **charge** should be the same regardless of whether we count electrons or holes, so holes have a positive charge $+e$ (electrons have $-e$).
 
 On the other hand, the velocity of a hole is **the same**:
-$$\frac{dE_h}{dp_h} = \frac{-dE}{-d\hbar k} = \frac{dE}{dp}$$
+$$\frac{dE_h}{dp_h} = \frac{-dE}{-d\hbar k} = \frac{dE}{dp}.$$
 
 Finally, we derive the hole effective mass from the equations of motion:
 
-$$m_h \frac{d v}{d t} = +e (E + v\times B)$$
+$$m_h \frac{d v}{d t} = +e (E + v\times B).$$
 
 Comparing with
 
-$$m_e \frac{d v}{d t} = -e (E + v\times B)$$
+$$m_e \frac{d v}{d t} = -e (E + v\times B),$$
 
 we get $m_h = -m_e$.
 
@@ -154,14 +156,13 @@ Therefore we can approximate the dispersion relation of both bands as parabolic.
 
 Or in other words
 
-$$E_e = E_c + \frac{\hbar^2k^2}{2m_e},$$
+$$E_e = E_c + \frac{\hbar^2k^2}{2m_e}$$
 $$E_h = E_{v,h} + \frac{\hbar^2k^2}{2m_h} = -E_{v} + \frac{\hbar^2k^2}{2m_h},$$
 
 with the corresponding density of states
 
 $$ g(E) = (2m_e)^{3/2}\sqrt{E-E_c}/2\pi^2\hbar^3$$
-
-$$ g(E_h) = (2m_h)^{3/2}\sqrt{E_h+E_v}/2\pi^2\hbar^3.$$
+$$ g_h(E_h) = (2m_h)^{3/2}\sqrt{E_h+E_v}/2\pi^2\hbar^3.$$
 
 Here $E_c$ is the energy of an electron at the bottom of the conduction band and $E_v$ is the energy of an electron at the top of the valence band.