2.1 Boltzmann Transport Equation
Let us regard carriers as classical particles; which dynamics are completely determined by assigning their positions 𝑟⃑ and momentums 𝑝⃑ at time t. Here, it is not necessary to consider about the size of carriers/particles. Next, let us consider a small region in the phase space about the point (𝑥, 𝑦, 𝑧, 𝑝𝑥, 𝑝𝑦, 𝑝𝑧), as shown in Figure 2-1, and the distribution function 𝑓(𝑟⃑, 𝑝⃑, 𝑡) which represents the probability to find a particle within this small region in phase space. The distribution function will change by 𝑓 during a time period 𝑡 as follows:
𝑓 = 𝑓(𝑥, 𝑦, 𝑧, 𝑝𝑥, 𝑝𝑦, 𝑝𝑧)
−𝑓(𝑥 + 𝑣𝑥 𝑡, 𝑦 + 𝑣𝑦 𝑡, 𝑧 + 𝑣𝑧 𝑡, 𝑝𝑥+𝐹𝑥 𝑡, 𝑝𝑦 + 𝐹𝑦 𝑡, 𝑝𝑧+ 𝐹𝑧 𝑡), (1)
where 𝑣⃑ is the velocity of the considered carrier and 𝐹⃑ is the force field affecting on particle.
Assuming the period 𝑡 is infinitesimally small and the distribution function is continuous in phase space, we can obtain:
𝑓
𝑡 + 𝑣⃑ ∙ ∇𝑟𝑓 + 𝐹⃑ ∙ ∇𝑝𝑓 = 0, (2)
or using the 𝑘-representation:
𝑓
𝑡 + 𝑣⃑ ∙ ∇𝑟𝑓 +𝐹⃑
ℏ∙ ∇𝑘𝑓 = 0. (3)
Points in phase space express dynamical states. These points can enter into and be emitted out of a region while satisfying this equation just like a hydrodynamic flow. Generally speaking, we can involve scattering process into this equation. With the rate of change of distribution function due to scattering (𝜕𝑓𝜕𝑡)
𝑐𝑜𝑙𝑙, the distribution function turns out to be satisfied to the equation:
𝑓
𝑡+ 𝑣⃑ ∙ ∇𝑟𝑓 +𝐹⃑
ℏ∙ ∇𝑘𝑓 = (𝜕𝑓
𝜕𝑡)
𝑐𝑜𝑙𝑙. (4)
The right-hand side is called “collision-term”, which can break the continuity and cannot be defined uniquely. Here, it is noteworthy to say that the BTE is a strange equation. Firstly,
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the left hand side should be continuous since it is written in terms of differentiation. Secondly, the right hand side can be discontinuous. In general, a discontinuous quantity cannot be equal to any continuous quantity. Accordingly, the treatment of the collision term has been a most sensitive issue in the history of modern physics. In terms of scattering theory, the contribution of scattering processes from a state 𝑘 to another state 𝑘 , and vice versa, to the collision term is written as:
𝑘 𝑘 𝑘 ( − 𝑘) − 𝑘 𝑘 𝑘( − 𝑓𝑘 ), (5) where the 𝑘 𝑘, and 𝑘 𝑘 , represent the transition rates from 𝑘 to 𝑘 and from 𝑘 to 𝑘, respectively. Here 𝑘 depicts the occupation rate of state 𝑘. According to the Fermi’s Golden Rule [16], we can write:
𝑘 𝑘 = 2𝜋
ℏ |⟨𝑘|𝑉|𝑘 ⟩|2𝛿(𝐸𝑘− 𝐸𝑘 ), (6)
where V is the considered scattering potential, and 𝐸𝑘 and 𝐸𝑘 are the initial and final state energies of the particle that scatters.
We must take into account more cases of contributions to the collision term. Most important in engineering is the process in which the number of particles is changed. For example, in electron tunneling phenomena, an electron disappears in anode where the tunneling starts, while an electron appears in cathode where the tunneling ends. The tunneling of electron decreases the number of electrons in anode and increase the number of electron in cathode. As long as we assume the energy band in semiconductor, the pair-creation and pair-annihilation of electron and hole can appear; which is called Shockley-Read-Hall (SRH) process. Since the creation of hole means the annihilation of valence electron, the total number of electrons in conduction band and valence band is unchanged. However, once we formulated the basic equations of electrons and holes separately, we must consider that the number of electrons (negative carriers; i.e., conduction electron) and the number of holes (positive carrier; a lack of valence electron) are increased at the same moment by SRH process. Moreover, we should note that the initial and end points of tunneling phenomena are regarded as spatially different, while no position change occurs in SRH process. Auger process can also increase the number of carriers.
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𝑡 𝑓
𝑓 + , + , 𝑡 + 𝑡 𝑓
𝑓 , , 𝑡 P
r r+dr r
P p+dp
Figure 2-1 Carrier Transport in Phase Space
2.2 Relaxation-Time Approximation
The Boltzmann equation is very difficult to solve, because it has both the continuous terms (related to the differential of the distribution function) and the discontinuous term (the collision term). In order to solve it, we have to make some simplification. Within the first order approximation, the most well-known is the relaxation-time approximation (RTA). In this approximation, we can replace the collision term with (𝑓0− 𝑓)/τ , where 𝑓0 is the distribution function at equilibrium condition and τ is the so-called relaxation time that specifies the time needed for the system to relax to its equilibrium state. Through a plenty of collisions, the distribution function shall be relaxed to the equilibrium state, i.e., equilibrated.
In RTA, all the collisions are averaged and the essence of the collisions is involved to the relaxation time. We can no more discuss about each collision. In other words, the RTA gives us the view point of independent particle (with no collision). Here, we can distinguish three different types of relaxation times: 1) The collision time, which represents the average time to the next collision. This means that all primitive processes are averaged by the collision time, which is the same as the mean-free time in mean-field theory. The concept of this relaxation is valid with or without the conservation laws of energy and momentum. 2) The momentum relaxation time, which is the time needed for relaxing the momentum of a considered particle (individual from others within mean-field theory). Let us consider a particle with momentum higher than the average. The momentum of this particle is relaxed progressively with time.
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This relaxation time is usually longer than the collision time, because it needs a plenty of collisions to relax the momentum of the considered particle. 3) The energy relaxation time, which is the time needed for relaxing the energy of considered particle. Let us consider a particle with energy higher than the average. The energy of this particle is relaxed progressively with time. This relaxation time is usually longer than the collision time, because it needs a plenty of collisions to relax the energy of considered particle. To consider all possibilities of the relaxations mentioned above from the view point of independent particle, we need to sum up all the inverse relaxation times to get the total relaxation time: ⁄ =𝜏𝑡
∑ 𝜏𝑖 ⁄ k, where the subscript k represents the different relaxation view point, that is, k=1) collision, k=2) momentum relaxation and k=3) energy relaxation.
To see more about the meaning of relaxation time, we assume a condition without external field and within a homogeneous semiconductor material. The BTE is reduced to:
∂𝑓
∂𝑡 = −𝑓 − 𝑓0
𝜏k , ℎ𝑒𝑟𝑒 ∇𝑟𝑓 = 0 𝑎 𝐹⃑ = 0. (7)
Solving this differential equation with assuming the initial state of distribution function to be 𝑓𝑖, we can obtain:
𝑓 = 𝑓0+ (𝑓𝑖− 𝑓0)𝑒−𝑡 𝜏⁄ k. (8)
Here note that this result is valid for any k, i.e. the classification of relaxation. And we assume the relaxation time is independent of the velocity, position, and time.
8 approximation discussed above and the effective mass approximation (𝑝⃑ = 𝑚∗𝑣⃑), the BTE is reduced to: eliminated at steady state.
Thereby, we can obtain:
Substituting this into the current equation, we can obtain:
𝐽(𝑥) = 𝑒 ∫ 𝑣𝑥𝑓0 𝑣𝑥
Since the equilibrium distribution function 𝑓0 is an even function, it is symmetrical in 𝑣⃑.
The integral of the first term in the current equation must thereby be zero, that is, 𝑣𝑥𝑓0 is an odd function and then the integration must be zero over an even integration interval. Since the relaxation time is assumed to be independent of 𝑣⃑, we have the following expression:
𝐽(𝑥) = −𝑒 ∫ 𝜏𝑣𝑥2𝜕𝑓(𝑣𝑥, 𝑥)
The first term in the right hand side is rewritten as: