Fermi-Dirac vs Boltzmann Distribution in Semiconductors
The number of electrons that carry current in a semiconductor depends on two things. One is how many energy "states" (places an electron can sit) there are. The other is the probability that each state holds an electron. This probability is given by the Fermi-Dirac distribution (Fermi distribution), and the energy where it is exactly one half is called the Fermi level. When the states are almost empty, you can count with the simpler Boltzmann form, exp(−E/kT). This explains why the number of electrons changes by orders of magnitude with temperature and band gap.
What you'll learn
- kT, the scale of thermal energy (about 26 meV at room temperature), and what the Boltzmann factor means
- What the Fermi level is, and what the Fermi-Dirac distribution looks like
- When the Fermi-Dirac distribution can be approximated by the Boltzmann factor
- Why the number of electrons in silicon changes so much with temperature and band gap
- Why impurities move the Fermi level
- How leakage current depends on temperature, and the "60 mV/decade" limit of transistors
Prerequisites: a rough idea of band gap, n-type and p-type, and holes, as explained in "What Is a Semiconductor?" Each equation is explained in words before it is shown.
In everyday life
The chips in phones and computers get hot as they work. When a chip heats up, more current leaks from transistors that are supposed to be off. This leakage wastes power and makes even more heat. So chip designers always estimate leakage current at high temperature.
Talk about low-power transistors often mentions the number "60 mV/decade." A decade here means a factor of 10. In an ordinary transistor near its off state, a tenfold rise in current takes a gate voltage change of at least about 60 mV at room temperature.
Both stories have the same root. The question is how heat decides how many electrons in a semiconductor can carry current. This article walks through that "electron counting" step by step.
How it works 1: kT, the scale of thermal energy, is about 26 meV at room temperature
Because of heat, the atoms and electrons in matter constantly trade energy. A rough measure of that energy is kT: the absolute temperature T times the Boltzmann constant k. Absolute temperature is the °C value plus 273.15, in units of K (kelvin).
Energy here is given in eV (electron volts). 1 eV is the energy one electron gains when pushed by a voltage of 1 V. At room temperature (300 K, about 27 °C), kT is about 0.026 eV, or about 26 meV (millielectron volts).
Compare this with the band gap of silicon. It is the energy step an electron must climb to move from the lower band, which is full of electrons, to the empty upper band. In silicon it is about 1.12 eV at 300 K. That is about 43 times kT, so very few electrons get over the step by heat alone.
How it works 2: the Boltzmann factor, about 1/2.7 for each kT
Of the electrons that get energy from heat, only some get a lot of it. Roughly, each extra kT of energy cuts the share that reaches that height to about 1/2.7. The number 2.7… is called e.
Written as a formula, this share is the Boltzmann factor, exp(−E/kT). E is the height of the energy target. exp(−3) means "e to the power −3," that is, 1/2.7 multiplied three times. Because the same fraction is multiplied again and again, a small change in height changes the result by orders of magnitude. When particles are spread out in energy according to this factor, it is called the Boltzmann distribution.
For example, a height of 10 times kT gives about 1 in 20,000, and 20 times kT gives about 1 in 500 million. The same factor is why chemical reactions and wear-out speed up so sharply with temperature. "Arrhenius Equation in Chips" explains that side of the story.
How it works 3: the Fermi-Dirac distribution and the Fermi level
Electrons, however, follow a rule that other particles do not. In a crystal, electrons can only take certain energies. Each place an electron can sit at a given energy is called a state, and each state holds at most one electron. This rule is called the Pauli exclusion principle. To keep things simple, we count electrons with different spin directions as being in different states.
At absolute zero (0 K), electrons fill the states from the bottom up, and everything above a certain height is empty. As temperature rises, heat moves electrons near that boundary up into slightly higher states, and the boundary blurs. The blur is a few times kT wide.
The probability that a state at each height is filled is described by the Fermi-Dirac distribution (Fermi distribution). The height where this probability is exactly one half is called the Fermi level. States well below the Fermi level are almost full, and states well above it are almost empty. This single marker is well defined only in thermal equilibrium. That means the temperature is uniform and no outside voltage or light is applied.
As a formula, the probability that a state at height E is filled = 1 ÷ (1 + exp((E − EF) / kT)). EF is the height of the Fermi level. When E = EF, exp(0) = 1, so the denominator is 2 and the probability is 1/2.
How it works 4: when the Boltzmann factor is a good approximation
For states far above the Fermi level, the exp part of the denominator becomes very large. The "1 +" in the denominator then hardly matters, and the probability is almost the same as exp(−(E − EF) / kT). This is exactly the Boltzmann factor from How it works 2.
The rule of thumb is a state at least 3 times kT above the Fermi level. There, the two differ by 5% or less. At room temperature, that means states about 0.08 eV or more above the Fermi level. In almost empty states, two electrons hardly ever compete for the same state. So you can ignore the "one per state" rule and count electrons like ordinary particles. The same goes for empty states (holes) in the lower band. For states at least 3kT below the Fermi level, the probability of being empty takes the Boltzmann form.
Most ordinary silicon devices work in the range where this approximation holds. That is why most semiconductor equations in textbooks are written in Boltzmann form. In metals, and in silicon with a very large amount of added impurities, the Fermi level sits near or inside a band. There the approximation fails, and the Fermi-Dirac distribution must be used as it is. Such a semiconductor is called a degenerate semiconductor.
How it works 5: how many electrons pure silicon has
In pure silicon, the number of electrons that have moved up to the upper band equals the number of empty states (holes) left in the lower band. This balance puts the Fermi level almost exactly in the middle of the band gap. So the lowest states in the upper band are about 0.56 eV above the Fermi level, about 22 times kT.
Counting with the Boltzmann factor, the probability that the lowest state in the upper band is filled is about 4×10⁻¹⁰. Multiply this by the number of states near the bottom edge of the upper band. You get about 1×10¹⁰ electrons per cm³ at 300 K. This number is called the intrinsic carrier concentration, written ni. Silicon has about 5×10²² atoms per cm³, so this is about 1 electron per 5 trillion atoms.
The value of ni varies slightly between sources. Some textbooks use 1.45×10¹⁰ or 1.5×10¹⁰. Recent precise estimates are about 1.0×10¹⁰ or 9.65×10⁹. The differences come from measurement methods and the values used in calculations, and all have the same order of magnitude.
This mechanism shows that ni is very sensitive to temperature. Near room temperature, it roughly doubles for every 10 °C rise, and grows about 67 times from 25 °C to 85 °C. It is also sensitive to the band gap. At 300 K, germanium, with a gap of about 0.66 eV, has about 2×10¹³. Gallium arsenide (GaAs), at about 1.42 eV, has about 2×10⁶. The reason is that half the gap, divided by kT, goes into the exponent of the Boltzmann factor.
How it works 6: impurities move the Fermi level
Adding impurity atoms such as phosphorus, called donors, puts many electrons into the upper band. The probability that upper-band states are filled goes up. So the Fermi level, the "half-probability height," moves toward the upper band. In the same way, adding acceptors such as boron creates more empty states, and the Fermi level moves toward the lower band.
How far it moves follows from the Boltzmann factor. To raise the number of upper-band electrons tenfold, the Fermi level must rise by about 60 meV (about 2.3 times kT). For example, add 10¹⁶ phosphorus atoms per cm³ (1 per 5 million atoms). The electrons then number 1 million times ni. That is six factors of 10, so the Fermi level rises about 0.36 eV from the middle.
Even so, the edge of the upper band is still about 0.2 eV (about 8 times kT) away, so the Boltzmann approximation still works. In this case, holes drop to about 10⁴ per cm³. This is because, in thermal equilibrium, the number of electrons times the number of holes equals ni squared. Above about 10¹⁸ impurity atoms per cm³, the distance to the band edge drops below 3kT, and the approximation starts to break down. Also, ni grows as temperature rises, so with the same amount of impurities the Fermi level moves back toward the middle.
Where it matters 1: leakage current rises with temperature
Where p-type and n-type regions meet, there is a PN junction. Even with voltage applied in the direction that blocks current, a tiny current leaks through it. Part of this leakage is carried by electrons and holes created by heat. So leakage current has parts proportional to ni and parts proportional to ni squared, and it rises steeply with temperature.
The first transistors were made of germanium, but silicon later took over. This is one of the reasons. With its wider gap, silicon has an ni about 1/2000 that of germanium, so it leaks less as temperature rises. Materials with even wider gaps, such as silicon carbide (SiC) and gallium nitride (GaN), are used in power semiconductors that run at high temperature and high voltage.
Where it matters 2: the 60 mV/decade limit
In the transistors used in chips (MOSFETs), electrons enter from a terminal called the source and leave through a terminal called the drain. When the transistor is off, there is an energy barrier for electrons between the two. The voltage on an electrode called the gate raises and lowers this barrier. "What Is a Transistor?" explains the structure in detail.
Only the source electrons with more energy than the barrier can get over it. The top of the barrier is well above the Fermi level, so their number follows the Boltzmann factor. Each time the barrier drops by kT (about 26 meV), the current grows about 2.7 times. Drop it by about 60 meV, 2.3 times kT, and the current grows tenfold.
In an ideal transistor, raising the gate voltage by 1 V lowers the barrier for electrons by exactly 1 eV. So a tenfold rise in current takes about 60 mV of gate voltage, and no less will do. This gate voltage needed for a tenfold change in current is called the subthreshold swing, given in mV/decade.
The name comes from the threshold voltage, the gate voltage at the boundary between on and off. The subthreshold swing describes how current changes below it, in the "subthreshold" region.
In real transistors, only part of the gate voltage acts on the barrier height, so the swing is larger than 60 mV. Also, this lower limit is proportional to kT, so at 85 °C it widens to about 71 mV. The limit rests on two assumptions. First, current is carried by electrons that get over the barrier thanks to heat. Second, the gate voltage can move the barrier by no more than the voltage itself. As long as both hold, changing the material or size cannot lower the limit.
This limit causes trouble through a trade-off between leakage current and supply voltage. To cut the current to 1/100,000 of its value at the threshold voltage, the gate voltage must drop by at least 0.3 V (60 mV × 5). That makes it hard to lower the threshold voltage, and hard to lower the supply voltage as well.
This also lets you estimate the effect of temperature. Take a transistor turned off at 0.3 V below its threshold voltage, and heat it from 25 °C to 85 °C. Counting only the Boltzmann factor, leakage current grows about 7 times. Real devices add other effects, such as the threshold voltage falling as temperature rises. So this value is only a rough guide.
Common misconceptions
Myth: There are lots of electrons at the Fermi level
In fact, the Fermi level is a marker for "the height where a state has a 50% chance of being filled." It does not mean electrons are there. In pure silicon and ordinary n-type and p-type silicon, the Fermi level lies inside the band gap, a range of heights with no states. In metals, there are many electrons around the Fermi level. People who learned it as "the top of the filled electrons" tend to fall for this myth.
Myth: The Boltzmann and Fermi-Dirac distributions describe different things
In fact, in thermal equilibrium, electrons always fill states according to the Fermi-Dirac distribution, and the Boltzmann factor is an approximation of it. It works only for states at least 3 times kT away from the Fermi level: almost empty above it, or almost full below it. Most textbook equations are written in Boltzmann form, so they look like a separate law. In regions with many impurities, and in metals, you have to go back to the Fermi-Dirac distribution.
Myth: 60 mV/decade can be beaten by shrinking transistors or changing materials
In fact, it is a limit set by temperature, through the thermal spread of electron energies in the source. An ordinary transistor carries current with electrons that get over a barrier thanks to heat, and its gate voltage moves that barrier directly. As long as it works this way, no shape or material can make it steeper than 60 mV/decade at room temperature. This limit is also known as the "Boltzmann tyranny." To get past it, researchers are studying transistors that use tunneling through the barrier, or "negative capacitance," which amplifies the gate voltage inside the device.
Key numbers
| Number | What it is |
|---|---|
| About 26 meV | kT at room temperature (300 K), the scale of thermal energy |
| About 1.12 eV | Band gap of silicon (300 K), about 43 times kT |
| 3kT (about 0.08 eV at room temperature) | For states at least this far above the Fermi level, the Boltzmann factor is within 5% |
| About 1×10¹⁰ per cm³ | Intrinsic carrier concentration of silicon (300 K); about 0.97–1.5×10¹⁰ depending on the source |
| About 2 times | Change in intrinsic carrier concentration for a 10 °C rise near room temperature |
| About 60 meV | Fermi level shift needed for ten times as many electrons (room temperature) |
| About 60 mV/decade | Lower limit of the subthreshold swing at room temperature (about 71 mV at 85 °C) |
The number 60 appears twice, and that is no coincidence. Both come from the same calculation: to change a number tenfold through the Boltzmann factor, the energy must shift by about 2.3 times kT.
Learn more
- Terms in depth: Fermi level (the equation, and how it lines up across a junction) / intrinsic carrier concentration / subthreshold swing (its link to short-channel effects)
- See it in figures: band diagram of an n-type semiconductor / PN junction and depletion layer
- Compare in tables: intrinsic vs extrinsic semiconductor / n-type vs p-type / static vs dynamic power
- Try the numbers: intrinsic carrier density calculator (change temperature and material) / PN junction depletion calculator
- The same Boltzmann factor in another role: "Arrhenius Equation in Chips"