Semiconductor for beginner #5


This article is designed for beginners who want to learn about semiconductors.


I majored in physics during my university years and focused on theoretical physics.


However, during my master’s program, I unexpectedly delved into research on high-energy physics or particle physics experiments involving semiconductor detection devices.


This article compiles the knowledge I acquired about semiconductors during that time.


Given the potentially extensive content, feel free to explore based on your specific areas of interest. And most importantly, enjoy the read!

There are two types of currents

Currents include drift and diffusion currents.



So far we have gained an understanding of the number of carriers in the conduction and valence bands.


Next, let’s look at the idea of current flow in a semiconductor.


What we understand there is the movement of carriers.



The drift current mentioned at the beginning of this article is the current that flows when an electric field (electric field) is applied. When a battery is connected to a metal wire, a current flows.


This is the drift current. We will look at this in more detail later.


The English word drift means wind or something else that blows away.


In this case, the electric field plays the role of wind. More simply put, it is the electric current that Ohm’s law describes.


This is the specific example in the next section, so it is a good review for middle school.



The other diffusion current mentioned at the beginning of this section is the current that flows from an area of high carrier density to an area of low carrier density.


This is a current that has nothing to do with the electric field.



As I will explain later, please keep in mind that the former is a current involving an electric field and the latter is a current not involving an electric field.



Incidentally, an electric field is a field acting on an electric charge.


Electric charges create Coulomb forces under the influence of an electric field. Static electricity is one such example.


As a side note, I heard that people in physics are familiar with the term electric field, while people in engineering are familiar with the term electric field.


I am more familiar with the former, so I will use that one.

Drift current

Ohm’s law states that the relationship between resistance R, current I, and voltage V is.



Specifically, let us consider a circuit in which an electrical resistor is connected to a battery.


Ohm’s law : V=RI

Notice : translate from Japanese to English

抵抗 : resistance
電流 : electric current
電池 : Battery


This law is also normally applied to electric wires that we use in our daily life.


The above figure is a rough explanation, but a microscopic view will give you a deeper understanding of this law.




First, consider the case where the anode and cathode of a battery are connected by a single copper wire (excluding the resistance from the above figure).


An electric field \(E\) is created in the copper wire.


When this electric field exists, a force of \(F=qE\) acts on an electron with charge \(q\). If a certain force acts, it will gradually accelerate the electrons.


The velocity at the cathode side, which is the entrance, and at the anode side, which is the exit, appear to be different, but in fact they are the same.


The reason for this is electron scattering.


Electrons collide with something in the copper wire, changing their direction of travel, slowing them down, and causing them to bounce back.


The most common collision is with a copper atom.


When this collision occurs, the electron loses kinetic energy.


The lost energy is converted into atomic vibrations. Atomic vibration is what we call heat.


So, when an electric current is applied to a metal, heat is generated because the kinetic energy of the electrons is converted to heat in this way.


In other words, electrons move forward while colliding with atoms throughout.


Its speed can be said to be average.


Come to think of it, this also explains the heat from the hair dryer we usually use.


The nichrome wire heats up to red hot and produces heat, and the fan blows warm air.


By looking at the relationship between this scattering and Ohm’s law, we can see how the electrons behave in the resistance.


First, considering the average progression, let \(T\) be the average time between scattering and scattering of the electrons.


During this time \(T\), the electron is subjected to a force of \(F=qE\) by the electric field, so we can describe the motion of matter mechanically as \(F=ma\).


In this case, \(m\) is the mass and \(a\) is the acceleration. Using this, the acceleration can be expressed as \(a=\frac{qE}{m}\).


Therefore, the velocity after \(T\) seconds is \(\frac{qET}{m}\).



There are many different types of how electrons are scattered.


Since this is an introductory section, let us consider the problem briefly.


That is, when an electron collides with an atom, the electron is stopped (all kinetic energy is lost).


The electron is then accelerated by the electric field and stops when it collides with the atom again.


This repetition is assumed to be how the electrons move forward.


So, we consider the process from the point where it stops after colliding with an atom to the point where it collides with an atom again.


This is the state in which the electron is moving in an equi-accelerating manner.


The distance \(l\) traveled during this time is


The average speed \(v\) can be expressed as Since the average speed \(v\) in this case can be obtained by dividing the distance \(l\) by the time \(T\),


$$=\mu _eE…①$$$

From this equation we can see that the electron velocity is proportional to the electric field \(E\), and this proportionality constant \(\mu _e\) is called the electron mobility.


$$\mu _e≡\frac{\frac{qT}{2}}{m}=\frac{q\tau}{m}$$


where the electron relaxation time \(\tau\) is defined by \(\tau≡\frac{T}{2}\).


Electron mobility is the amount of speed that is produced when an electric field is applied.


Therefore, it is an important physical quantity that determines the performance of semiconductors.


The larger the electron mobility, the faster the speed can be produced even with a small electric field.


(Just to recap, the magnitude of the current is determined by the “number of carriers” and the “speed of the carriers. In this case, we are talking about the latter.)




If the density of electrons is \(n\), the current density (current per unit cross-sectional area) \(J\) is charge x current density x speed.


Image of current I = current density J × cross-sectional area S

Notice : translate from Japanese to English

電子の速さ : Electron velocity
電子の密度 : Density of electrons
断面積 : Cross-sectional area
電流密度 : Current density
電流 : Electric current


Cross-sectional area is just what the name implies (for example, the cross-sectional area of a radish when cut is the cross-sectional area).


The current density is


and can be expressed as If the potential difference of a copper wire (or semiconductor) of length \(L\) is \(V\), the magnitude of the electric field can be expressed as \(E=\frac{V}{L}\).


Therefore, the current, together with equation ②,



Solving this for \(V\),





If we put, Ohm’s law \(V=RI\) holds.



This is what the specific resistance is. From this formula, we can understand about the resistance \(R\).


For example, how can we reduce the resistance \(R\)? Looking at the denominator of equation ③, the larger the density \(n\) of electrons or the larger the mobility \(\mu_e\), the smaller the resistance \(R\) can be.


This means that the greater the number of electrons, the smaller the resistance.


This means that the greater the number of electrons, the greater the current flow and the smaller the resistance.


Also, the greater the mobility, the greater the speed of the electrons, and thus the smaller the resistance.

(*The magnitude of the current is determined by the “number of carriers” and the “speed of carriers.)



Looking at the numerator of equation ③, the longer the length \(L\) of the copper wire or the smaller the cross-sectional area \(S\), the greater the resistance.

Diffusion current

Another current in a semiconductor is the diffusion current.


If there is a solid mass (bulk) of semiconductor and a group of conduction electrons (free electrons) exists solidly in one corner of the bulk, the free electrons have kinetic energy and can move around in various directions when the temperature is not absolute zero (for example, at room temperature).


Since there is no electric field applied to the semiconductor, the free electrons can spread within this bulk.


The current caused by the movement of electrons during this diffusion (spreading) is called the diffusion current.


Image of diffusion current

Notice : translate from Japanese to English

伝導電子(自由電子) : Conduction electrons (free electrons)
拡散 : Diffusion
しばらくすると・・・ : After a while…
半導体バルク : Bulk semiconductor

As you can see in the figure above, the direction of the diffusion current is from the area of high electron density (left side of the bulk) to the area of low electron density (right side of the bulk).



Diffusion current is the same phenomenon as diffusion of gas molecules.


Electrons have an electric charge, and electrons repel each other by Coulomb force due to their charge.


However, this is not the direct cause of diffusion. In other words, the electric current is not caused by the electric field.


This is the difference from the drift current.


The diffusion current is expressed by the equation


We will derive this equation later.


First, a brief explanation of equation ④ will be given, where \(n(x)\) is the carrier density at position \(x\). \(e\) is the electric elementary quantity.


\(D\) is a quantity called the diffusion coefficient.


The equation states that when the density slope \(\frac{dn(x)}{dx}\) is large, the current density \(J\) is large.


If the slope is small, the current will be small.


If the slope is zero, there is no current flow and zero diffusion because the density in the bulk is constant.


In other words, if the mass of electrons in the bulk is unevenly distributed, the electrons will diffuse and there will be a density slope.


If the electrons are uniformly present in the bulk, no diffusion occurs and the density slope is zero.

Derivation of the equation for diffusion current

Derivation of the equation for the diffusion current.




To simplify the story, let us consider it in 1-dimensional space.



Let the left corner of the bulk be position \(x_1\), its right neighbor \(x_2\), and so on for position \(x_i\).


Naturally, the right neighbor of \(x_i\) will be \(x_i+1\).



Now, set up the electron density \(n(x)\) (the number of electrons) to be biased as you go to the left corner.


Then the relationship between electron density \(n(x)\) and position \(x_i\) can be graphically imaged as follows.


Electron density vs. position in the bulk

Notice : translate from Japanese to English

電子密度 : electron density
位置 : Position
バルクの左端 : Left end of bulk
距離 : Distance
バルク内のイメージ : Image in the bulk
電子 : Electrons
左端にたくさん電子が詰まっている!! : A lot of electrons are packed in the left end!


Since it is one-dimensional, there are only two possible positions for the electron to move next: right or left.


So, probabilistically, right or left is expressed as \(\frac{1}{2}\).


The electron density \(n(x_i)\) (number of electrons) at position \(x_i\) can be expressed as \(l(=|x_i-x_{i+1}|)\), where the average collision time is \(t\) and the distance travelled during this time is \(l(=|x_i-x_{i+1}|)\) (\(i\) is a natural number), half of the electron density \(\frac{1}{2}n(x_i)\) will move to the right and the other half \(\frac{1}{2}n(x_i)\) to the left.



Now consider electrons moving from \(x_i\) to \(x_{i+1}\) and from \(x_{i+1}\) to \(x_i\).


The distance between \(x_i\) and \(x_{i+1}\) is \(l\).


Consider a cross section between these distances \(l\) (shaded area in the figure below).


Consider an electron passing through this cross section.

Electron transfer occurring at \(x_i\) and \(x_{i+1}\)


Notice : translate from Japanese to English

この断面を通過する電子密度は : The electron density passing through this cross-section is


The above figure looks three dimensional, but it is only one dimensional (it is a matter of pictorial impression).


From a numerical point of view, the electron density (number of electrons) that can pass through the shaded line is



which is $$\frac{1}{2}n(x_i)$$. i+1})[/latex])[/latex], the difference was subtracted from \(\frac{1}{2}n(x_i)\).



This is rewritten as current density \(J\).


Current density\(J\)= charge\(-e\)× electron densitylatex)[/latex]× velocity\(\frac{l}{t}\)


So, a slight variation of this formula is,



which is the same as the following.


Using \(x_{i+1}=x_i+l\), consider the inside of the brackets. In this case, take the limit so that the distance \(l→0\) (first-order derivative).


Recall the definition of derivatives.


$$ f'(x)=\frac{df(x)}{dx}=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$



In other words, I want you to think of it as a mathematical process when you consider the distance \(l\) that we have been considering so far with the minute distance \(l→0\) (when it is as close to zero as possible).



Then it can be approximated as follows.


$$ \frac{n(x_i+l)-n(x_i)}{l}≈\frac{dn(x)}{dx}$$


Using this, the diffusion current is,





If we further define the diffusion constant \(D≡\frac{l^2}{2t}\)

can be written as $$J=eD\frac{dn(x)}{dx}$$.


We can now derive equation ④ (the diffusion current equation).


What is important is that the larger the slope of the electron density \(\frac{dn(x)}{dx}\), the larger the diffusion current, and if the slope is zero, no diffusion current flows.



It means that the diffusion current flows when the electron density (number of electrons) is unevenly distributed in the hole.

Equation for current continuity

The equation for current continuity is: think of a section of a semiconductor.


This is the section where electrons move in and out.


Think of the current continuity equation as counting those electrons.


For example, consider the middle of an intersection.


Cars pass through that space as they enter and exit.


You can think of it as counting the number of cars passing through that space.



The current continuity equation is an important equation to express in mathematical terms when dealing with drift and diffusion currents. So let’s understand the concept properly.



Consider the case of electrons.


Consider a current flowing through an elongated semiconductor with cross section \(S\).


Consider the region between the position \(x\) and \(x+dx\).


Assuming that the current flows from left to right as shown below, in that section, electrons may be produced (and holes produced at the same time) if light irradiation takes place.


Or the electrons may be annihilated by emission of light (or the electrons and holes may recombine).


Whatever the case may be, think of it as a space where electrons flow in one direction only, but are also produced and annihilated.



We will discuss the calculation of diffusion current in pn junctions later, so for the time being, let’s consider only the case of recombination.


Career Flow

Notice : translate from Japanese to English

再結合の割合 : Percentage of recombination
断面積 : Cross-sectional area
電流(面)密度 : Current (areal) density


Consider the sum of the number of electrons flowing into a space of thickness \(dx\), the number of electrons flowing out of this space, and the number of electrons disappearing by recombination


(free electrons in the conduction band emit light and their energy drops to the valence band) with a ratio \(R_n\).




R is taken from the English word “recombination.


Expressed in terms of the areal density of current \(J_n\) (electrons flowing per cross section S),


the current flowing in can be expressed as \(J_n(x)S\) and the current flowing out as \(J_n(x+dx)S\).


Dividing this by the electron charge \(-e\) gives the number of electrons.


Consider the time variation of the electron density \(n(x,t)\) in the part of width \(dx\)


(when differentiating two or more variables, write in the form \(\frac{\partial n(x,t)}{\partial t}\). (This is the time derivative, but the same applies to the spatial derivative.)


$$\frac{\partial n}{\partial t} Sdx=(\frac{J_n(x)S}{-e}-\frac{J_n(x+dx)S}{-e})-R_nSdx$$

can be written as.


Note that we use two variables for electron density, position \(x\) and time \(t\).


This time variation is taken into account in the current continuity equation.



Here, using a first-order Taylor expansion, we can approximate as follows.


$$J_n(x+dx)≈J_n(x)+\frac{\partial J_n}{\partial x}dx$$.


*This equation shows that we are also dealing with two variables \(x,t\) here for the current density.


Using this and dividing by both sides \(Sdx\), the time variation of the electron density at the width \(dx\) just shown is as follows.


$$\frac{\partial n}{\partial t} =\frac{1}{e}\frac{\partial J_n}{\partial x}-R_n$$



The same equation holds for the holes.


This is the equation for the current continuity that includes the recombination term.



Drift and diffusion currents correspond to this current.



We will focus on the case of diffusion current, which will be useful later at the pn junction.


The recombination term can also be expressed using the carrier lifetime \(\tau\).


If the number of carriers is \(\delta n\) more than the equilibrium state


(no free electrons in the conduction band, no holes in the valence band, and stable)

(this number can recombine),


then the reciprocal ratio of the carrier lifetime \(\frac{1}{\tau}[/latex and disappear (the reciprocal of the lifetime can also be interpreted as the rate at which the carrier decays.


In this case, recombination corresponds to decay.


For example, suppose we have a toy that explodes after 100 hours of use, and we used it for 80 hours.


The rate at which this toy explodes is \(\)\frac{80}{100}\). Multiply this by 100 to express it as a percentage).


From this, the fraction of electrons that disappear in recombination is \(R_n=\frac{\Delta n}{\tau}\).


The expression for the current continuity in this case can be expressed as follows by substituting equation ④ for the diffusion current.


$$\frac{\partial n}{\partial t} =D\frac{\partial^2 n}{\partial x^2}-\frac{\Delta n}{\tau}…⑤$$$



This equation will be used later, so keep it in mind.

Hall effect

The hole in the Hall effect does not refer to a hole, but comes from an American man named Hall.


In other words, this person discovered it, hence the name Hall effect.


In conclusion, the Hall effect is a phenomenon in which an electric field is created by a pooled electric charge.


We will look at this now.



So far, we have understood the drift current due to the electric field and the diffusion current due to the diffusion of carrier density.


And by understanding the current continuity equation, we have understood how the number of electrons in a semiconductor changes under time variation.


From equation ②, what is needed for drift current is carrier density and mobility.



From equation ④, the diffusion coefficient and carrier density are necessary for the diffusion current.



When a semiconductor is created, we need to know these quantities in order to examine its quality.



Of these physical quantities, the Hall measurement tells us information about carrier density (electron density and hole density) and mobility.


 We will follow this step by step using diagrams.


The first stage when an electric current is applied to a semiconductor


Notice : translate from Japanese to English

y軸方向磁場 : y-direction magnetic field
ローレンツ力 : Lorentz force
電圧 : Voltage


The box in the figure above is a semiconductor.


Since we are now considering electrons,


which are carriers, we will discuss n-type semiconductors (p-type semiconductors can be explained in the same way).


In the circuit shown above, the current flows clockwise.


The electrons move in the opposite direction of the current, so in the above figure, the moment they enter the semiconductor, they move in the -x direction.



Now, let us apply a magnetic field in the y direction.


Then the Lorentz force acts on the electrons and they move in the z direction.


This is all that can be seen from the above figure.


Next, consider the case where a large number of electrons move in the z direction and accumulate on the upper surface of the semiconductor in the z direction.


This is shown in the following figure.

Second stage when current is passed through a semiconductor

Notice : translate from Japanese to English

y軸方向磁場 : y-direction magnetic field

z方向電場 : z-direction electric field
電流 : Electric current
-x方向に電子が進行 : Electrons traveling in -x direction
z方向ロレンツ力 : Lorentz force in z direction
-z方向電場による力 : Force due to electric field in -z direction
電圧 : Voltage


Free electrons gather on the top surface in the z direction, and at the same time, the bottom surface in the z direction becomes positively charged due to the lack of free electrons.



One side is negatively charged and the other side is positively charged, creating a potential difference.


Where the potential difference occurs, there is an electric field. Since the electric field goes from positive to negative, an electric field is generated in the z direction.



At this time, electrons entering the semiconductor are subject to two forces: one is the Lorentz force due to the magnetic field, and the other is the force due to the electric field.


Interestingly, these forces are balanced. If we write the equation


$$∴ E=vB$$

and so on. 



By the way, the current density per unit cross-sectional area \(J=-e×n×v\) can be expressed by the formula charge × current density × velocity. Using this, the electric field is



can be expressed



This phenomenon in which an electric field is generated by an accumulated electric charge is called the Hall effect, which also occurs in the case of p-type semiconductors.



By looking at the ☆ equation, we can distinguish between unknown and measurable quantities.


First, the electric elementary quantity \(e\) is known.


The magnetic field \(B\) and current density \(J\) are measurable quantities.


The electric field \(E\) can also be obtained from the relation shown later.



Therefore, everything except current density \(n\) can be obtained from measurements.


From these quantities, the current density \(n\) can also be obtained.


The same is true for holes. So we now know how to find the carrier density.


The top bottom surface of the semiconductor is connected to a voltmeter and the potential difference \(V\) is measured


Notice : translate from Japanese to English

y軸方向磁場 : y-direction magnetic field

z方向電場 : z-direction electric field
断面積 : cross-sectional area
電圧計 : Voltmeters
電圧 : Voltage


Now, we need to understand how to find the electric field generated to determine the carrier density.


This is from the relationship between voltage (potential difference) and electric field,


if the thickness of the semiconductor is \(W\),


can be written as follows.


The potential difference \(V\) can be measured using a voltmeter as shown above.


Using this, the ☆ equation can be rewritten as.


which is the same as the following. \(S\) is the cross-sectional area of the semiconductor.


Furthermore, since all the quantities on the right-hand side are measured values, the carrier density can be obtained with this.


Since current density is the current flowing per unit cross-sectional area, multiplying current density by area gives the current flowing in the semiconductor.




By the way, in the section on drift current, we learned that current density can be expressed as in equation ②.

$$J=-env=-en\mu_eE_L (∵②)$$


In this case, \(E_L\) is the electric field facing in the same direction as the electrons are heading.


Note that this means that it is a transverse electric field and is different from the electric field \(E\) generated by the Hall effect.


(The transverse electric field \(E_L=\frac{V_B}{L}\), so it is a measurable quantity from the lateral length of the semiconductor \(L\) and the external voltage \(V_B\))


Substituting this for \(I=JS\),



which is the same as $$I=-en\frac{I}{enE_LS}$$. Since the right-hand side is all measurable quantities,


we can find the electron mobility.


(Review : Electron mobility is how fast the electrons go when an electric field is applied. So it is an important physical quantity that determines the performance of semiconductors. The larger the electron mobility, the faster the speed can be produced even with a small electric field.)



Naturally, the same goes for hole mobility.


Therefore, we now know how to find the mobility.


Incidentally, I once did a student experiment on the Hall effect when I was a student.


But at that time, I didn’t understand the importance of this experiment that much.


We now have a better knowledge of currents in semiconductors.


In the fourth lecture, we will use the knowledge we have acquired so far to learn about pn junctions, which are the most important in semiconductor devices.


As the name suggests, we will learn about the phenomena and mechanisms that occur when p-type and n-type semiconductors are combined.


高校数学でわかる半導体の原理―電子の動きを知って理解しよう (ブルーバックス) (日本語) 新書 – 2007/3/21

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