Density of states

The density or (Engl. Density of states , abbreviated DOS ) is a physical quantity that indicates how many states per energy interval and per frequency interval exist in a physical system. ${\displaystyle D(E)}$ ${\displaystyle D(\omega )}$ ${\displaystyle \mathrm {d} E}$ ${\displaystyle \mathrm {d} \omega }$

In general, the density of states for many-particle systems is considered in the context of a model of independent particles. Then the variables or relate to the energy of the 1-particle states. Often the density of states is then also considered as a function of the momentum or wave vector of the 1-particle states and indicates their number per volume interval of momentum space ( ) or reciprocal space ( ). The density of states can refer to different types of particles, e.g. B. on photons , phonons , electrons , magnons , quasiparticles , and is given per unit of spatial volume. For free particles without spin , the density of states can be calculated from the fact that each quantum mechanical state takes up the volume in phase space . The density of states (per volume) is then constant ${\displaystyle \omega }$ ${\displaystyle E=\hbar \omega }$ ${\displaystyle {\vec {p}}=\hbar {\vec {k}}}$ ${\displaystyle {\vec {k}}}$ ${\displaystyle \mathrm {d} ^{3}{\vec {p}}}$ ${\displaystyle \mathrm {d} ^{3}{\vec {k}}}$ ${\displaystyle (2\pi \hbar )^{3}}$ ${\displaystyle D({\vec {k}})}$

${\displaystyle D({\vec {k}})={\frac {1}{(2\pi )^{3}}}\ .}$

In the case of interactions between the particles, be it with one another or with predetermined potentials, the density of states can deviate significantly from this (see e.g. ribbon model ).

definition

In general, the volume -related density of states for a countable number of energy levels is defined by: ${\displaystyle V}$ ${\displaystyle N}$

${\displaystyle D(E)={\frac {1}{V}}\cdot \sum _{i=1}^{N}\delta (E-E({\vec {k}}_{i}))}$

with the delta distribution . ${\displaystyle \delta }$

From this, by expanding with (the smallest permitted change of for a particle in a box of dimension and length ) and transitioning to a Riemann integral (Limes ), the volume-related density of states for continuous energy levels is obtained: ${\displaystyle (\Delta k)^{d}=\left({\frac {2\pi }{L}}\right)^{d}}$ ${\displaystyle k}$ ${\displaystyle d}$ ${\displaystyle L}$ ${\displaystyle L\to \infty }$

${\displaystyle D(E):=\int _{\mathbb {R} ^{d}}{\frac {\mathrm {d} ^{d}k}{(2\pi )^{d}}}\cdot \delta (E-E({\vec {k}}))\qquad (*)}$

With

• ${\displaystyle d}$ the spatial dimension of the system under consideration
• the amount of the wave vector . ${\displaystyle k}$

Equivalently, the density of states can also be understood as the derivative of the microcanonical sum of states according to the energy: ${\displaystyle Z_{m}(E)=N(E)}$

${\displaystyle D(E)={\frac {1}{V}}\cdot {\frac {\mathrm {d} N(E)}{\mathrm {d} E}}}$

The number of states with energy ( degree of degeneration ) is given by: ${\displaystyle E'}$

${\displaystyle g(E')=\lim _{\Delta E\to 0}\int _{E'}^{E'+\Delta E}D(E)\mathrm {d} E=\lim _{\Delta E\to 0}D(E')\Delta E}$, whereby the last equal sign only applies if the mean value theorem of integral calculus is applicable to the integral.

Intuition

One clearly counts the micro-states for a given energy : If one considers a system with micro-states , the density of states is described by ${\displaystyle E}$ ${\displaystyle N}$ ${\displaystyle i}$

${\displaystyle D(E)=\sum _{i=1}^{N}\delta (E-E_{i})}$

because the integral over the density of states gives the total number of microstates: ${\displaystyle N}$

${\displaystyle \int _{\mathbb {R} }\sum _{i=1}^{N}\delta (E-E_{i})\cdot \mathrm {d} E=\sum _{i=1}^{N}=N}$

and the following integral also gives the number of micro-states in energy : ${\displaystyle n(E)}$ ${\displaystyle E}$

${\displaystyle \lim _{\Delta E\to 0}\int _{E_{i}-\Delta E}^{E_{i}+\Delta E}\sum _{i=1}^{N}\delta (E-E_{i})\cdot \mathrm {d} E=n(E)}$

In the above formula , the property of delta distribution is important, at least for the visualization , which, however, only applies to finitely many and simple zeros of . ${\displaystyle (*)}$ ${\displaystyle \delta (g(x))=\sum _{i=1}^{n}{\frac {\delta (x-x_{i})}{|g^{\prime }(x_{i})|}}}$ ${\displaystyle x_{i}}$ ${\displaystyle g(x)}$

n-dimensional electron gas

The following explanations relate primarily to applications in solid state physics .

Density of states over the energy depending on the dimension (3D = dotted, 2D = red, 1D = green, 0D = blue). The jumps in the density of states for the dimensions D = 0 to D = 2 are due to the fact that in these cases the densities of states are drawn around different energy states. Around these energy states the density of states then has the form calculated and shown in the table.

In a -dimensional electron gas , charge carriers can move freely in the dimensions . The corresponding portion of the energy is continuous and can be specified using the parabolic approximation: ${\displaystyle n}$ ${\displaystyle 1,\dotsc ,n}$

${\displaystyle E={\frac {\hbar ^{2}k^{2}}{2m^{*}}}}$

It is

• ${\displaystyle m^{*}}$the effective mass of the charge carrier in the solid, more precisely the effective mass of state ${\displaystyle m^{*}}$
• ${\displaystyle \hbar }$the Planck quantum of action ( divided by) . ${\displaystyle 2\pi }$

In contrast, the energy component of the other dimensions is discretized in the values . The density of states ( related to the volume ) can be described in general: ${\displaystyle E_{l}}$ ${\displaystyle V}$

${\displaystyle D(E)=2\cdot {\frac {\mathrm {d} N(E)}{\mathrm {d} E}}{\frac {1}{V}}.}$

Therein corresponds

• the prefactor 2 the two possible spin states (but it is often taken into account, this was not done here) ${\displaystyle N(E)}$
• ${\displaystyle V=L_{x}\cdot L_{y}\cdot L_{z}}$ the volume of the solid
• ${\displaystyle N(E)}$the number of all states with energy less than or equal (see: micro-canonical sum of states ): ${\displaystyle E}$ ${\displaystyle Z_{m}}$

${\displaystyle N(E)={\begin{cases}{\frac {V_{k}}{\Omega _{k}}}&{\text{wenn}}\quad n=3\\\sum _{l}\Theta (E-E_{l}){\frac {V_{k}}{\Omega _{k}}}&{\text{wenn}}\quad n=1,2\\\sum _{l}\Theta (E-E_{l})&{\text{wenn}}\quad n=0\end{cases}}}$

• • ${\displaystyle V_{k}}$describes in the -dimensional -space the total volume of all states that are accessible with the remaining energy ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle E-E_{l}}$
  • ${\displaystyle \Omega _{k}}$ is the volume of such a state.
  • ${\displaystyle \Theta }$is the Heaviside function .

Values ​​for different-dimensional electron gases

	Total volume of all states ${\displaystyle V_{k}}$	Volume of a state ${\displaystyle {\Omega _{k}}}$	Density of states (related to the volume) ${\displaystyle D(E)}$
	in k space at the remaining energy ${\displaystyle E-E_{l}}$
3D - bulk	${\displaystyle {\frac {4}{3}}\pi k^{3}}$	${\displaystyle {\frac {(2\pi )^{3}}{L_{x}L_{y}L_{z}}}}$	${\displaystyle {\frac {(2m^{*})^{\frac {3}{2}}}{2\pi ^{2}\hbar ^{3}}}{\sqrt {E}}}$
2D - quantum well / quantum film	${\displaystyle \pi k^{2}}$	${\displaystyle {\frac {(2\pi )^{2}}{L_{x}L_{y}}}}$	${\displaystyle {\frac {m^{*}}{\pi \hbar ^{2}L_{z}}}\sum _{l}\Theta (E-E_{l})}$
1D - quantum wire	${\displaystyle 2k}$	${\displaystyle {\frac {2\pi }{L_{x}}}}$	${\displaystyle {\frac {\sqrt {2m^{*}}}{\pi \hbar L_{y}L_{z}}}\sum _{l}{\frac {1}{\sqrt {E-E_{l}}}}}$
0D - quantum dot			${\displaystyle {\frac {2}{L_{x}L_{y}L_{z}}}\sum _{l}\delta (E-E_{l})}$

In the semiconductor

Density of states (colored) in an undoped semiconductor with a direct band transition . In addition, the Fermi distribution at room temperature is plotted to the left, the Fermi level E _F and the conduction band energy E _C as the energy levels .

In semiconductor materials , because of the periodically occurring atomic nuclei, a similar approach is made for the conduction and valence bands (see band model ). Semiconductors are characterized by the fact that their dispersion curves or band structure have a maximum (valence band) and a minimum (conduction band), which do not overlap but are separated by the band gap. If the extremes are offset in space ( momentum space ), this is referred to as an indirect semiconductor , and if the momentum difference is the same as a direct semiconductor. The functional behavior around such extreme values ​​can be approximated parabolically (quadratically). However, the curvature of this shape used for approximation does not have to agree with the curvature of the dispersion curve of the free electrons discussed above. Instead, the charge carriers, i.e. electrons and holes , are assigned effective masses at these extremes in the two bands so that the functional description is now identical to that of the real free electrons. ${\displaystyle k}$ ${\displaystyle m^{*}}$

Density of states (colored) in an n-doped semiconductor with a direct band transition . Energy level of the dopant atoms E _D .

Let the energy of the lower edge of the conduction band be that of the upper edge of the valence band , the difference being equal to the band gap energy . The density of states in the conduction band is ( is the density of states of the electron in the conduction band, i.e. its mean effective mass): ${\displaystyle E_{\mathrm {C} }}$ ${\displaystyle E_{\mathrm {V} }}$ ${\displaystyle E_{\mathrm {G} }=E_{\mathrm {C} }-E_{\mathrm {V} }}$ ${\displaystyle m_{\mathrm {e,d} }^{*}}$

${\displaystyle D_{\mathrm {C} }(E)={\frac {(2m_{\mathrm {e,d} }^{*})^{\frac {3}{2}}}{2\pi ^{2}\hbar ^{3}}}{\sqrt {E-E_{\mathrm {C} }}}}$

The density of states in the valence band is ( is the density of states of the hole in the valence band): ${\displaystyle m_{\mathrm {p,d} }^{*}}$

${\displaystyle D_{\mathrm {V} }(E)={\frac {(2m_{\mathrm {p,d} }^{*})^{\frac {3}{2}}}{2\pi ^{2}\hbar ^{3}}}{\sqrt {E_{\mathrm {V} }-E}}}$

In the case of doped semiconductors, there are also states in the band gap in addition to these possible states. With -doping these are close to the conduction band and with -doping close to the valence band. By supplying energy, the activation energy can be overcome and more occupied states are formed in conduction or valence bands. In addition, doping changes the position of the Fermi level: it is raised with -doping, or decreases with -doping towards the valence band. In the case of doping, far more states are occupied in the conduction band than with an undoped material because of the thermal energy . The additional free charge carriers can thus increase the transport of electricity . ${\displaystyle n}$ ${\displaystyle p}$ ${\displaystyle n}$ ${\displaystyle p}$ ${\displaystyle n}$

The thermal population of the states is determined by the Fermi distribution. The probability density that a state is occupied by the energy is written ${\displaystyle [E,E+\mathrm {d} E]}$

${\displaystyle W_{\mathrm {e} }(E)={\frac {1}{\exp {\left({\frac {E-\mu }{k_{\mathrm {B} }T}}\right)}+1}}}$

The probability density that a state is not occupied by the energy or, expressed equivalently, is occupied by a hole, is written ${\displaystyle [E,E+\mathrm {d} E]}$

${\displaystyle W_{\mathrm {h} }(E)=1-W_{\mathrm {e} }(E)={\frac {1}{\exp {\left(-{\frac {E-\mu }{k_{\mathrm {B} }T}}\right)}+1}}}$

This allows the charge carrier densities, i.e. electron density in the conduction band and hole density in the valence band, to be given: ${\displaystyle n}$ ${\displaystyle p}$

${\displaystyle n=\int _{E_{\mathrm {C} }}^{\infty }W_{\mathrm {e} }(E)\,D_{\mathrm {C} }(E)\,\mathrm {d} E}$

such as

${\displaystyle p=\int _{-\infty }^{E_{\mathrm {V} }}W_{\mathrm {h} }(E)\,D_{\mathrm {V} }(E)\,\mathrm {d} E}$

Actually, the integration limits shouldn't be extended to infinity, but only to the end of the respective band. However, the Fermi distribution there is already approximately zero - the chemical potential lies in the band gap area - so that the error is negligible. For the calculation of these integrals see Fermi-Dirac integral .

literature

• Wolfgang Demtröder: Experimentalphysik Vol. 3 - Atoms, Molecules and Solids . 3. Edition. Springer, Berlin 2005, ISBN 3-540-21473-9 .

Web links

• Semiconductor Physics: Density of States. In: Britney Spears' Guide to Semiconductor Physics. Carl Hepburn, accessed April 7, 2009 . 
• CRWie: Carrier concentration in Si (or in any Semiconductor) versus the Fermi Energy Level and the Density of States. Accessed April 7, 2009 (English, Java applet on density of states in semiconductors). 
• M. Alam: Online lecture: ECE 606 Lecture 8: Density of States (English).
• Density of states of the free electron gas - Chapter 2.2.3. In: Introduction to Materials Science II, Uni Kiel. Retrieved on August 12, 2010 (very detailed and quite understandable script). 
• Derivation of the density of states of free particles in 3D ( Memento from October 19, 2013 in the Internet Archive )