Chemical potential in electron-phonon interactions - Revision history

Csheldon at 09:47, 4 November 2025

2025-11-04T09:47:27Z

← Older revision		Revision as of 09:47, 4 November 2025
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	The [https://en.wikipedia.org/wiki/Chemical_potential chemical potential] <math>\mu</math> (~~we also use~~ <math>\~~eta~~</math> ~~when its meaning can be confused with~~ mobility) is the energy that can be released that due to a change in particle number. Its accurate determination is a key ingredient in the computation of [[Transport coefficients including electron-phonon scattering\|electronic transport coefficients]] such as electrical conductivity, carrier mobility, and thermopower. These quantities depend sensitively on the occupation of electronic states near the Fermi level, since only states within a few ''k<sub>B</sub>T'' of the Fermi energy contribute significantly to transport.		The [https://en.wikipedia.org/wiki/Chemical_potential chemical potential] <math>\mu</math> (make sure not to mistake the chemical potential <math>\mu</math> for the mobility <math>\mu_e</math> or <math>\mu_h</math>) is the energy that can be released that due to a change in particle number. Its accurate determination is a key ingredient in the computation of [[Transport coefficients including electron-phonon scattering\|electronic transport coefficients]] such as electrical conductivity, carrier mobility, and thermopower. These quantities depend sensitively on the occupation of electronic states near the Fermi level, since only states within a few ''k<sub>B</sub>T'' of the Fermi energy contribute significantly to transport.
	We employ the frozen-band approximation, which assumes that the electronic potential and eigenvalues computed for the undoped system remain unchanged when electrons are added or removed.		We employ the frozen-band approximation, which assumes that the electronic potential and eigenvalues computed for the undoped system remain unchanged when electrons are added or removed.

Miranda.henrique: /* Carrier density and doping */

2025-11-03T22:57:26Z

Carrier density and doping

← Older revision		Revision as of 22:57, 3 November 2025
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	The carrier concentration (number of carriers per unit volume) is then:		The carrier concentration (number of carriers per unit volume) is then:

	<math> n_e = \frac{~~N_e +~~ N_\text{carriers}}{\Omega} </math>		<math> n_e = \frac{N_\text{carriers}}{\Omega} = \frac{1}{N_\mathbf{k}\Omega} \sum_{n\mathbf{k}} \left[ f(\varepsilon_{n\mathbf{k}}, T, \mu) - f(\varepsilon_{n\mathbf{k}}, 0, \epsilon_F) \right]</math>

	where <math>\Omega</math> is the volume of the cell.		where <math>\Omega</math> is the volume of the cell.

	~~Alternatively, if~~ the target carrier concentration ''n<sub>e</sub>'' is known, the chemical potential ~~can be determined by inverting:~~		If the target carrier concentration ''n<sub>e</sub>'' is known, the chemical potential <math>\mu</math> can be determined by numerically inverting this equation.

	<math> ~~n_e = \frac{1}{\Omega N_\mathbf{k}} \sum_{n\mathbf{k}} f(\varepsilon_{n\mathbf{k}}, T,~~ \mu) </math>

	This relation links the chemical potential directly to the carrier density at a given temperature, serving as the foundation for transport coefficient calculations.		This relation links the chemical potential directly to the carrier density at a given temperature, serving as the foundation for transport coefficient calculations.

Miranda.henrique: /* Carrier density and doping */

2025-10-31T16:00:40Z

Carrier density and doping

← Older revision		Revision as of 16:00, 31 October 2025
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	In doped or non-stoichiometric systems, the total number of electrons includes additional carriers introduced by doping. The number of carriers per unit cell (or per simulation cell) is given by:		In doped or non-stoichiometric systems, the total number of electrons includes additional carriers introduced by doping. The number of carriers per unit cell (or per simulation cell) is given by:

	<math> N_\text{carriers} = \frac{1}{N_\mathbf{k}} \sum_{n\mathbf{k}} \left[ f(\varepsilon_{n\mathbf{k}}, T, \mu) - f(\varepsilon_{n\mathbf{k}}, T, \~~mu_0~~) \right] </math>		<math> N_\text{carriers} = \frac{1}{N_\mathbf{k}} \sum_{n\mathbf{k}} \left[ f(\varepsilon_{n\mathbf{k}}, T, \mu) - f(\varepsilon_{n\mathbf{k}}, 0, \epsilon_F) \right] </math>

	where ~~''μ~~<~~sub~~>0</~~sub~~>'' is ~~the chemical potential~~ of the neutral (undoped) system.		where <math>\epsilon_F</math> is fermi energy of the neutral (undoped) system.
	Positive ''N<sub>carriers</sub>'' corresponds to n-type doping (extra electrons), while negative ''N<sub>carriers</sub>'' corresponds to p-type doping (holes).		Positive ''N<sub>carriers</sub>'' corresponds to n-type doping (extra electrons), while negative ''N<sub>carriers</sub>'' corresponds to p-type doping (holes).

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	<math> n_e = \frac{N_e + N_\text{carriers}}{\Omega} </math>		<math> n_e = \frac{N_e + N_\text{carriers}}{\Omega} </math>

	where ~~''Ω''~~ is the ~~unit~~ cell ~~or supercell volume~~.		where <math>\Omega</math> is the volume of the cell.

	Alternatively, if the target carrier concentration ''n<sub>e</sub>'' is known, the chemical potential can be determined by inverting:		Alternatively, if the target carrier concentration ''n<sub>e</sub>'' is known, the chemical potential can be determined by inverting:

Csheldon: /* Why the chemical potential matters */

2025-10-22T10:28:55Z

Why the chemical potential matters

← Older revision		Revision as of 10:28, 22 October 2025
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	== Why the chemical potential matters ==		== Why the chemical potential matters ==
	At zero temperature in an undoped system, the chemical potential coincides with the Fermi energy. However, in real materials and realistic conditions:		At zero temperature in an undoped system, the chemical potential coincides with the Fermi energy. However, in real materials and realistic conditions:
	* The system may be doped, i.e. it contains additional electrons or holes.		* The system may be doped, i.e., it contains additional electrons or holes.
	* The system may be at finite temperature, which modifies the balance between electron and hole occupations.		* The system may be at finite temperature, which modifies the balance between electron and hole occupations.
	* The transport coefficients are dominated by states within a narrow energy window around the chemical potential, so even small inaccuracies can lead to large errors.		* The transport coefficients are dominated by states within a narrow energy window around the chemical potential, so even small inaccuracies can lead to large errors.

Miranda.henrique at 15:07, 21 October 2025

2025-10-21T15:07:06Z

← Older revision		Revision as of 15:07, 21 October 2025
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	The [https://en.wikipedia.org/wiki/Chemical_potential chemical potential] <math>\mu</math> (we also use <math>\eta</math> when its meaning can be confused with mobility) is the energy that can be released that due to a change in particle number. Its accurate determination is a key ingredient in the computation of [[Transport coefficients including electron-phonon scattering\|electronic transport coefficients]] such as electrical conductivity, carrier mobility, and thermopower. These quantities depend sensitively on the occupation of electronic states near the Fermi level, since only states within a few ''k<sub>B</sub>T'' of the Fermi energy contribute significantly to transport.		The [https://en.wikipedia.org/wiki/Chemical_potential chemical potential] <math>\mu</math> (we also use <math>\eta</math> when its meaning can be confused with mobility) is the energy that can be released that due to a change in particle number. Its accurate determination is a key ingredient in the computation of [[Transport coefficients including electron-phonon scattering\|electronic transport coefficients]] such as electrical conductivity, carrier mobility, and thermopower. These quantities depend sensitively on the occupation of electronic states near the Fermi level, since only states within a few ''k<sub>B</sub>T'' of the Fermi energy contribute significantly to transport.
			We employ the frozen-band approximation, which assumes that the electronic potential and eigenvalues computed for the undoped system remain unchanged when electrons are added or removed.

	Because the occupation of states is governed by the Fermi–Dirac distribution, an accurate evaluation of the chemical potential as a function of temperature and doping is essential. Even small inaccuracies in <math>\mu</math> can lead to large errors in transport properties.		Because the occupation of states is governed by the Fermi–Dirac distribution, an accurate evaluation of the chemical potential as a function of temperature and doping is essential. Even small inaccuracies in <math>\mu</math> can lead to large errors in transport properties.

Csheldon: Csheldon moved page Chemical potential to Chemical potential in electron-phonon interactions

2025-10-21T10:30:24Z

Csheldon moved page Chemical potential to Chemical potential in electron-phonon interactions

← Older revision	Revision as of 10:30, 21 October 2025
(No difference)

Csheldon at 10:18, 21 October 2025

2025-10-21T10:18:25Z

← Older revision		Revision as of 10:18, 21 October 2025
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	This equation implicitly defines <math>\mu(T)</math> for a given total number of electrons <math>N_e</math>.		This equation implicitly defines <math>\mu(T)</math> for a given total number of electrons <math>N_e</math>.

	== Carrier ~~Density~~ and ~~Doping~~ ==		== Carrier density and doping ==

	In doped or non-stoichiometric systems, the total number of electrons includes additional carriers introduced by doping. The number of carriers per unit cell (or per simulation cell) is given by:		In doped or non-stoichiometric systems, the total number of electrons includes additional carriers introduced by doping. The number of carriers per unit cell (or per simulation cell) is given by:

Csheldon: /* Related tags and articles */

2025-10-20T14:15:18Z

Csheldon: /* Related tags and articles */

2025-10-20T14:03:02Z

Csheldon: /* Related tags and articles */

2025-10-20T14:02:30Z

← Older revision		Revision as of 14:15, 20 October 2025
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	* [[Transport coefficients including electron-phonon scattering\|Transport calculations]]		* [[Transport coefficients including electron-phonon scattering\|Transport calculations]]
	* [[Electron-phonon accumulators]]		* [[Electron-phonon accumulators]]
			* [[Electronic transport coefficients]]
	* {{TAG\|ELPH_RUN}}		* {{TAG\|ELPH_RUN}}
	* {{TAG\|ELPH_SELFEN_CARRIER_DEN}}		* {{TAG\|ELPH_SELFEN_CARRIER_DEN}}

← Older revision		Revision as of 14:03, 20 October 2025
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	* {{TAG\|ELPH_SELFEN_MU}}		* {{TAG\|ELPH_SELFEN_MU}}
	* {{TAG\|ELPH_SELFEN_MU_RANGE}}		* {{TAG\|ELPH_SELFEN_MU_RANGE}}
	* ~~{{TAG\|NELECT}}, {{TAG\|ISMEAR}}, {{TAG\|SIGMA}},~~ {{TAG\|EFERMI}}		* {{TAG\|EFERMI}}

	[[Category:Theory]][[Category:Electron-phonon_interactions]]		[[Category:Theory]][[Category:Electron-phonon_interactions]]

← Older revision		Revision as of 14:02, 20 October 2025
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	* [[Electron-phonon accumulators]]		* [[Electron-phonon accumulators]]
	* {{TAG\|ELPH_RUN}}		* {{TAG\|ELPH_RUN}}
	* {{TAG\|ELPH_SELFEN_CARRIER_DEN}}, {{TAG\|ELPH_SELFEN_CARRIER_DEN_RANGE}}		* {{TAG\|ELPH_SELFEN_CARRIER_DEN}}
			* {{TAG\|ELPH_SELFEN_CARRIER_DEN_RANGE}}
	* {{TAG\|ELPH_SELFEN_CARRIER_PER_CELL}}		* {{TAG\|ELPH_SELFEN_CARRIER_PER_CELL}}
	* {{TAG\|ELPH_SELFEN_TEMPS}}, {{TAG\|ELPH_SELFEN_TEMPS_RANGE}}		* {{TAG\|ELPH_SELFEN_TEMPS}}
	* {{TAG\|ELPH_SELFEN_MU}}, {{TAG\|ELPH_SELFEN_MU_RANGE}}		* {{TAG\|ELPH_SELFEN_TEMPS_RANGE}}
			* {{TAG\|ELPH_SELFEN_MU}}
			* {{TAG\|ELPH_SELFEN_MU_RANGE}}
	* {{TAG\|NELECT}}, {{TAG\|ISMEAR}}, {{TAG\|SIGMA}}, {{TAG\|EFERMI}}		* {{TAG\|NELECT}}, {{TAG\|ISMEAR}}, {{TAG\|SIGMA}}, {{TAG\|EFERMI}}

	[[Category:Theory]][[Category:Electron-phonon_interactions]]		[[Category:Theory]][[Category:Electron-phonon_interactions]]