Effective mass

This article presents the concept of solid state physics. For the historical physical concept, see Electromagnetic Mass .

In solid-state physics, the effective mass is the apparent mass of a particle in a crystal in the context of a semiclassical description. Similar to the reduced mass , the effective mass allows the use of a simplified equation of motion .

In many situations, electrons and holes in a crystal behave as if they were free particles in a vacuum , only with a changed mass. This effective mass is usually given in units of the electron mass ( m _e = 9.11 × 10 ⁻³¹ kg ). Experimental methods for determining the effective mass make use of cyclotron resonance , among other things . The basic idea is that the energy-momentum relationship (i.e. the dispersion relation ) of a particle or quasiparticle in the vicinity of a local minimum turns out to be

${\displaystyle E=E_{0}+{\frac {1}{2m^{*}}}(p-p_{0})^{2}+{\mathcal {O}}\left((p-p_{0})^{3}\right)}$

with p for the momentum and for the higher terms. The quadratic term looks like the kinetic energy of a particle of mass m *. ${\displaystyle {\mathcal {O}}}$

Definition and characteristics

Effective mass in the crystal lattice

The effective mass is defined in analogy to Newton's second law ( , acceleration equals force per mass ). A quantum mechanical description of the crystal electron in an external electric field E is provided by the equation of motion ${\displaystyle a={\tfrac {1}{m}}\cdot F}$

${\displaystyle a={\frac {1}{\hbar ^{2}}}{\frac {d^{2}\varepsilon }{dk^{2}}}\cdot qE}$,

wherein a the acceleration, the Planck's constant , k the wave number of the ascribed to the electron Bloch wave (often somewhat lax as pulse referred to as the quasi-momentum of the particle), the energy as a function of k (the dispersion relation ), and q the charge of the electron are. A free electron in a vacuum, however, would accelerate ${\displaystyle \hbar }$ ${\displaystyle p=\hbar k}$ ${\displaystyle \varepsilon (k)}$

${\displaystyle a={\frac {1}{m_{\mathrm {e} }}}\cdot qE}$

Experienced. Thus the effective mass of the electron in the crystal is m *

${\displaystyle m^{*}=\hbar ^{2}\left[{\frac {d^{2}\varepsilon }{dk^{2}}}\right]^{-1}}$.

For a free particle the dispersion relation is quadratic , and thus the effective mass would then be constant (and equal to the actual electron mass). In a crystal the situation is more complex: The dispersion relation is generally not quadratic, which leads to a velocity-dependent effective mass, s. a. in the band structure . The concept of the effective mass is therefore most useful in the area of ​​minima or maxima of the dispersion relation, where it can be approximated by quadratic functions. The effective mass is therefore proportional to the inverse curvature of the strip edge . The interesting physics of the semiconductor takes place in a minimum of the conduction band (curvature positive = effective mass of the electrons positive) and in a maximum of the valence band (curvature negative = effective mass of the electrons negative). A hole is assigned the negative effective electron mass in the valence band, which is thus positive again.

With electron energies far away from such extremes, the effective mass can also be negative or even infinite in the conduction band (see Gunn effect ). One can this strange at first glance property in the wave pattern by the Bragg reflection explain the one-dimensional lattice: With the Bragg condition

${\displaystyle \ 2d\sin \theta =n\lambda }$

for the reflection at the ions “planes”, and follows ${\displaystyle \theta =90^{\circ }}$ ${\displaystyle \lambda =2\pi /k}$

${\displaystyle k={\frac {n\pi }{d}}}$.

The condition is hardly fulfilled for small amounts of , the electrons move according to their free mass m _e . For larger amounts of k, there is an increasing amount of reflection until no acceleration by an electric field is actually possible. Is now . In the case of even larger k values, an acceleration by an external field due to the effect of the internal forces (interaction with phonons in the particle image ) may lead to an acceleration opposite to the expected direction, the effective mass is consequently negative. ${\displaystyle k}$ ${\displaystyle m^{*}=\infty }$

Effective mass without a crystal field

By modifying the energy-momentum relation of the atoms in a Bose-Einstein condensate , it was possible in 2017 to give them a negative effective mass (according to the above formula) in a certain momentum range. The authors clearly write of “effective mass”, speculations about the generation of “negative mass” as such (such as in Spiegel Online ) currently appear to be unfounded. ${\displaystyle \varepsilon (k)}$

Effective mass as a tensor

The effective mass is generally direction-dependent (with respect to the crystal axes) and is therefore a tensile quantity. The following applies to the tensor of the effective mass:

${\displaystyle \left({\frac {1}{m^{*}}}\right)_{ij}={\frac {1}{\hbar ^{2}}}\cdot {\frac {\partial ^{2}\varepsilon }{\partial k_{i}\partial k_{j}}}}$

This means in particular that the acceleration of the electrons in an electric field does not have to be parallel to the field vector . In particular (analogous to the inertia tensor) there will be a principal axis system due to the symmetry of m ^* , in which (1 / m ^* ) _ij assumes a diagonal form , with the associated eigenvalues on the diagonal. If the electric field then lies along one of these main axes (which can be achieved by rotating the crystal in a constant field), only the associated eigenvalue is included. Since not all eigenvalues ​​have to be the same, there are i. A. Principal axes with large and small eigenvalues ​​of the effective mass. With a constant electric field, small eigenvalues ​​lead to a higher acceleration of the charge carriers. The effective masses increase with increasing temperature. ${\displaystyle m^{*}{\vec {a}}=q{\vec {E}}}$ ${\displaystyle {\vec {E}}}$ ${\displaystyle {\vec {E}}}$

When calculating the density of states , the effective mass is included. In order to be able to maintain the shape of the isotropic case, one defines a density of states

${\displaystyle m_{d}=N^{2/3}\left(m_{1}^{*}\,m_{2}^{*}\,m_{3}^{*}\right)^{1/3}}$,

where the degeneracy factor N indicates the number of equivalent minima (N usually 6 or 8) and are the eigenvalues ​​of the effective mass tensor. ${\displaystyle m_{i}^{*}}$

The conductivity or mobility is proportional to the reciprocal effective mass. In anisotropic systems, an average mobility can be specified using the conductivity mass: ${\displaystyle m_{c}}$

${\displaystyle {\frac {1}{m_{c}}}={\frac {1}{3}}\left({\frac {1}{m_{1}^{*}}}+{\frac {1}{m_{2}^{*}}}+{\frac {1}{m_{3}^{*}}}\right)}$

Effective mass for silicon

Conduction band

For electrons in the conduction band at a temperature close to absolute zero, the following applies : ${\displaystyle T=1{,}4\,\mathrm {K} }$

Formula symbol	Effective mass
${\displaystyle m_{1}^{*}}$	${\displaystyle 0{,}19\,m_{\mathrm {e} }}$
${\displaystyle m_{2}^{*}}$	${\displaystyle 0{,}19\,m_{\mathrm {e} }}$
${\displaystyle m_{3}^{*}}$	${\displaystyle 0{,}91\,m_{\mathrm {e} }}$

The two equal masses are called the transverse mass and the longitudinal mass . The density of states ( ) bei is , bei is it . The conductivity mass at is . ${\displaystyle m_{1}^{*}=m_{2}^{*}}$ ${\displaystyle m_{\mathrm {t} }}$ ${\displaystyle m_{3}^{*}}$ ${\displaystyle m_{\mathrm {l} }}$
${\displaystyle N=6}$ ${\displaystyle T=4{,}2\,\mathrm {K} }$ ${\displaystyle m_{d}=1{,}06\,m_{\mathrm {e} }}$ ${\displaystyle T=300\,\mathrm {K} }$ ${\displaystyle m_{d}=1{,}09\,m_{\mathrm {e} }}$ ${\displaystyle T=1{,}4\,\mathrm {K} }$ ${\displaystyle m_{c}=0{,}26\,m_{\mathrm {e} }}$

Valence band

In the valence band there are two sub-bands due to spin-orbit interaction ( ) at the band edge. One is the heavy holes (“heavy holes” with and ), the other is the light holes (“light holes” with and ). Both have different effective masses, at is and . In addition, there is another sub-band (“split off band” with ), which is energetically lowered compared to the valence band edge. At is . The density of states of the valence band at is and at is . ${\displaystyle l=1,s=1/2}$ ${\displaystyle j=3/2}$ ${\displaystyle m_{j}=3/2}$ ${\displaystyle j=3/2}$ ${\displaystyle m_{j}=1/2}$ ${\displaystyle T=4{,}2\,\mathrm {K} }$ ${\displaystyle m_{\mathrm {hh} }=0{,}54\,m_{\mathrm {e} }}$ ${\displaystyle m_{\mathrm {lh} }=0{,}15\,m_{\mathrm {e} }}$ ${\displaystyle j=1/2}$ ${\displaystyle T=4{,}2\,\mathrm {K} }$ ${\displaystyle m_{\mathrm {so} }=0{,}25\,m_{\mathrm {e} }}$ ${\displaystyle T=4{,}2\,\mathrm {K} }$ ${\displaystyle m_{d}=0{,}59\,m_{\mathrm {e} }}$ ${\displaystyle T=300\,\mathrm {K} }$ ${\displaystyle m_{d}=0{,}81\,m_{\mathrm {e} }}$

Web links

• Applications and values ​​for some semiconductors (University of Colorado, English)

Individual evidence

1. ↑ Khamehchi, MA; Hossain, Khalid; Mossman, ME; Zhang, Yongping; Busch, Th .; Forbes, Michael McNeil; Engels, P .: Negative-Mass Hydrodynamics in a Spin-Orbit-Coupled Bose-Einstein Condensate . In: Physical Review Letters . tape 118 , no. 15 , 2017, p. 155301 , doi : 10.1103 / PhysRevLett.118.155301 ( online [PDF; accessed April 19, 2017]).
2. ↑ koe: Washington: Researchers create negative mass. In: Spiegel Online . April 18, 2017, accessed April 13, 2020 . 
3. ^ Martin Green: Intrinsic concentration, effective densities of states, and effective mass in silicon . In: Journal of Applied Physics . 67, No. 6, 1990, pp. 2944-2954. doi : 10.1063 / 1.345414 .
4. ^ Landolt-Börnstein : Condensed Matter (III); Semiconductors (41); Group IV Elements, IV-IV and III-V Compounds (A1); Electronic, Transport, Optical and Other Properties (β); Silicon: conduction band, effective masses; Silicon: valence band, effective masses