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有效质量（Effective mass）

2008年05月31日星期六上午 10:05

有效质量，一般来讲出现在量子力学研究中为了与经典理论(如：F=ma)联系起来易于理解，而引进的与质量有同等效应的物理量。因此不同的研究体系中，所谓的有效质量的形式也是不同的。值得注意的是，有效质量只适用于量子力学领域。

유효질량이란 일반적으로 양자역학 연구에서 어떠한 현상을 고전이론의 형태로 나타냄으로써 이해하기 쉽게 하기 위하여 가정한 질량과 같은 효과를 나타내는 새로운 형태의 물리적량이다. 때문에 같지 않은 모형의 연구체계에서 유효질량에 대한 구체적인 정의는 조금씩 다르다. # 주의 할 점은 유효질량은 양자역학에서만 적용된다.

参考：固体中的有效质量

In solid state physics, a particle's effective mass is the mass it seems to carry in the semiclassical model of transport in a crystal. It can be shown that, under most conditions, electrons and holes in a crystal respond to electric and magnetic fields almost as if they were free particles in a vacuum, but with a different mass. This mass is usually stated in units of the ordinary mass of an electron m_e (9.11×10^-31 kg).

Effective mass is defined by analogy with Newton's second law $\vec{F}=m\vec{a}$ . Using quantum mechanics it can be shown that for an electron in an external electric field E:

$a = {{1} \over {\hbar^2}} \cdot {{d^2 \varepsilon} \over {d k^2}} qE$

where $a$ is acceleration, $\hbar$ is reduced Planck's constant, $\hbar = h/2\pi$ , $k$ is the wave number (often loosely called momentum since $k$ = $p/\hbar$ ), $ε(k)$ is the energy as a function of $k$ , or the dispersion relation as it is often called. From the external electric field alone, the electron would experience a force of $\vec{F}=q\vec{E}$ , where q is the charge. Hence under the model that only the external electric field acts, effective mass $m *$ becomes:

$m^{*} = \hbar^2 \cdot \left[ {{d^2 \varepsilon} \over {d k^2}} \right]^{-1}$

For a free particle, the dispersion relation is a quadratic, and so the effective mass would be constant (and equal to the real mass). In a crystal, the situation is far more complex. The dispersion relation is not even approximately quadratic, in the large scale. However, wherever a minimum occurs in the dispersion relation, the minimum can be approximated by a quadratic curve in the small region around that minimum. Hence, for electrons which have energy close to a minimum, effective mass is a useful concept.

In energy regions far away from a minimum, effective mass can be negative or even approach infinity. Effective mass, being generally dependent on direction (with respect to the crystal axes), is a tensor. However, for most calculations the various directions can be averaged out.

Effective mass should not be confused with reduced mass, which is a concept from Newtonian mechanics. Effective mass can only be understood with quantum mechanics.

(译) 在固体物理中,有效质量是指在半经典的理论模型下,粒子在晶体中运动时具有的等效质量.可以证明多数情况下,晶体中的电子与空穴对电场或磁场的响应与其在真空条件下的响应相似,仅仅是质量不同罢了.此质量通常以普通的电子质量m_e (9.11×10^-31 kg)为单位.

有效质量可以通过牛顿定律 $\vec{F}=m\vec{a}$ 来定义.用量子力学可以证明处于外电场E中的电子满足:

$a = {{1} \over {\hbar^2}} \cdot {{d^2 \varepsilon} \over {d k^2}} qE$

这里 $a$ 是指加速度, $\hbar$ 是约化普朗克常数, $\hbar = h/2\pi$ , $k$ 是指波数(由于 $k$ = $p/\hbar$ ,又常不严密地称为动量),能量 $ε(k)$ 是 $k$ 的函数,通常称之为能量色散关系.当电子仅处于电场中时,他会受到电场力 $\vec{F}=q\vec{E}$ ,其中q是电荷.因而在仅有外电场作用时,有效质量 $m *$ 为:

$m^{*} = \hbar^2 \cdot \left[ {{d^2 \varepsilon} \over {d k^2}} \right]^{-1}$

对于自由电子,色散关系是二次关系,因此有效质量为常数(并且等于真实质量).当电子处于晶体中时,境况就变得复杂多了.色散关系在很大程度上甚至连近似的二次关系都不是. 然而,只要色散关系中存在极小值,就可以在此极小值附近用二次曲线做近似.因此,对于能量处于色散关系中极小值的电子,有效质量是一个很有用的概念.

在远离极小值点的区域,有效质量可能为负值甚至趋于无穷大.有效质量普遍依赖于方向,是一个张量.然而,大多数计算中可以用根据不同方向计算出的平均值.

注意不要将有效质量与约化质量相混淆,约化质量是牛顿力学中的概念,有效质量仅仅适用用于量子力学中.

目录

1 Effective mass for some common semiconductors (for density of states calculations)

2 Experimental determination

3 Significance

4 External link

Effective mass for some common semiconductors (for density of states calculations)

Material Electron effective mass Hole effective mass

Group IV

Si 0.36 m_e 0.81 m_e

Ge 0.55 m_e 0.37 m_e

III-V

GaAs 0.067 m_e 0.45 m_e

InSb 0.013 m_e 0.6 m_e

II-VI

ZnO 0.19 m_e 1.21 m_e

ZnSe 0.17m_e 1.44 m_e

Sources:
S.Z. Sze, Physics of Semiconductor Devices, ISBN 0-47-105661-8.
W.A. Harrison, Electronic Structure and the Properties of Solids, ISBN 0-48-666021-4.

Experimental determination

Traditionally effective masses were measured using cyclotron resonance, a method in which microwave absorption of a semiconductor immersed in a magnetic field goes through a sharp peak when the microwave frequency equals the cyclotron frequency $\omega_c = \frac{eB}{m^* c}$ . In recent years effective masses have more commonly been determined through measurement of band structures using techniques such as angle-resolved photoemission (ARPES). Effective masses can also be estimated using the coefficient γ of the linear term in the low-temperature electronic specific heat at constant volume $c v$ . The specific heat depends on the effective mass through the density of states at the Fermi level and as such is a measure of degeneracy as well as band curvature. Very large estimates of carrier mass from specific heat measurements have given rise to the concept of heavy fermion materials. Since carrier mobility depends on the ratio of carrier collision lifetime $τ$ to effective mass, masses can in principle be determined from transport measurements, but this method is not practical since carrier collision probabilities are typically not known a priori.

(译) 测量有效质量的传统方法是回旋共振.当微波的频率与处于磁场中的半导体所具有的回旋频率 $\omega_c = \frac{eB}{m^* c}$ 一致时,微波将被共振吸收.近年来有效质量常通过测量能带结果的方法来确定,比如角分辨光发射谱(ARPES).有效质量也可以通过低温电子恒压比热容 $c v$ 的线性部分的系数γ来估计.比热容依赖于有效质量是因为它与费米能级处的态密度相关,而态密度又决定与能带的简并度与弯曲情况.一大批通过比热容测量估计的载流子有效质量引入了重费米子材料的概念. 既然载流子迁移率决定于载流子碰撞寿命 $τ$ 与有效质量的比值,原则上有效质量可以通过传输来测量,但是这种方法并不实际,因为通常是不会事先知道碰撞概率的.

Significance

As the table shows, III-V compounds based on GaAs and InSb have far smaller effective masses than tetrahedral group IV materials like Si and Ge. In the simplest Drude picture of electronic transport, the maximum obtainable charge carrier velocity is inversely proportional to the effective mass: $\vec{v} = \begin{Vmatrix}\mu\end{Vmatrix} \cdot \vec{E}$ where $\begin{Vmatrix}\mu\end{Vmatrix} = \frac{e \tau}{\begin{Vmatrix}m^*\end{Vmatrix}}$ with $e$ being the electronic charge. The ultimate speed of integrated circuits depends on the carrier velocity, so the low effective mass is the fundamental reason that GaAs and its derivatives are used instead of Si in high-bandwidth applications like cellular telephony.

External link

NSM archive
Template:Cite book This book contains an exhaustive but accessible discussion of the topic with extensive comparison between calculations and experiment.wp:Effective mass

取自"http://chemwiki.net/index.php/%E6%9C%89%E6%95%88%E8%B4%A8%E9%87%8F"

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