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In [[solid-state physics]], '''k·p [[perturbation theory (quantum mechanics)|perturbation theory]]''' is an approximation scheme for calculating the [[band structure]] (particularly [[Effective mass (solid-state physics)|effective mass]]) and optical properties of crystalline solids.<ref name=Yu2.6/><ref name=Kittel/><ref name=Harrison> | |||
{{cite book | |||
|author=W.P. Harrison | |||
|year=1989 |origyear=1980 | |||
|title=Electronic Structure and the Properties of Solids | |||
|edition=Reprint | |||
|pages=158 ''ff'' | |||
|publisher=[[Dover Publications]] | |||
|isbn=0-486-66021-4 | |||
}}</ref> It is pronounced "k dot p", and is also called the "k·p method". This theory has been applied specifically in the framework of the '''Luttinger–Kohn''' model (after [[Joaquin Mazdak Luttinger]] and [[Walter Kohn]]), and of the '''Kane model''' (after [[Evan O. Kane]]). | |||
==Background and derivation== | |||
===Bloch's theorem and wavevectors=== | |||
{{see also|Bloch wave}} | |||
According to [[quantum mechanics]] (in the [[Hartree-Fock|single-electron approximation]]), the [[electron]]s in any material have [[wavefunction]]s which can be described by the following [[Schrödinger equation]]: | |||
:<math>\left(\frac{p^2}{2m}+V\right)\psi = E\psi</math> | |||
where '''p''' is the [[momentum operator|quantum-mechanical momentum operator]], ''V'' is the [[electrostatic potential|potential]], and ''m'' is the mass of an electron. (This equation neglects the [[spin-orbit effect]]; see below.) | |||
In a [[crystalline solid]], ''V'' is a [[periodic function]], with the same periodicity as the [[crystal lattice]]. [[Bloch wave|Bloch's theorem]] proves that the solutions to this differential equation can be written as follows: | |||
:<math>\psi_{n,\mathbf{k}}(\mathbf{x}) = e^{i\mathbf{k}\cdot\mathbf{x}} u_{n,\mathbf{k}}(\mathbf{x})</math> | |||
where '''k''' is a vector (called the ''wavevector''), ''n'' is a discrete index (called the ''[[band structure|band]] index''), and ''u''<sub>''n'','''k'''</sub> is a function with the same periodicity as the crystal lattice. | |||
For any given ''n'', the associated states are called a [[band structure|band]]. In each band, there will be a relation between the wavevector '''k''' and the energy of the state ''E''<sub>''n'','''k'''</sub>, called the [[dispersion relation|band dispersion]]. Calculating this dispersion is one of the primary applications of ''k''·''p'' perturbation theory. | |||
===Perturbation theory=== | |||
{{see also|Perturbation theory (quantum mechanics)}} | |||
The periodic function ''u''<sub>''n'','''k'''</sub> satisfies the following Schrödinger-type equation:<ref name=Yu2.6> | |||
{{cite book | |||
|author=P. Yu, M. Cardona | |||
|year=2005 | |||
|title=Fundamentals of Semiconductors: Physics and Materials Properties | |||
|url=http://books.google.com/books?id=W9pdJZoAeyEC&pg=PA244&dq=isbn=3540254706#PPA68,M1 | |||
|edition=3rd | |||
|page=Section 2.6, pp. 68 ''ff' |nopp=yes | |||
|publisher=[[Springer (publisher)|Springer]] | |||
|isbn=3-540-25470-6 | |||
}}</ref> | |||
:<math>H_{\mathbf{k}} u_{n,\mathbf{k}}=E_{n,\mathbf{k}}u_{n,\mathbf{k}}</math> | |||
where the [[Hamiltonian (quantum mechanics)|Hamiltonian]] is | |||
:<math>H_{\mathbf{k}} = \frac{p^2}{2m} + \frac{\hbar \mathbf{k}\cdot\mathbf{p}}{m} + \frac{\hbar^2 k^2}{2m} + V </math> | |||
Note that '''k''' is a vector consisting of three real numbers with units of [[inverse length]], while '''p''' is a vector of operators; to be explicit, | |||
:<math>\mathbf{k}\cdot\mathbf{p} = k_x (-i\hbar \frac{\partial}{\partial x}) + k_y (-i\hbar \frac{\partial}{\partial y}) + k_z (-i\hbar \frac{\partial}{\partial z})</math> | |||
In any case, we write this Hamiltonian as the sum of two terms: | |||
:<math>H=H_0+H_{\mathbf{k}}', \;\; H_0 = \frac{p^2}{2m}+V, \;\; H_{\mathbf{k}}' = \frac{\hbar^2 k^2}{2m} + \frac{\hbar \mathbf{k}\cdot\mathbf{p}}{m}</math> | |||
This expression is the basis for [[perturbation theory (quantum mechanics)|perturbation theory]]. The "unperturbed Hamiltonian" is ''H''<sub>0</sub>, which in fact equals the exact Hamiltonian at '''k'''=0 (i.e., at the [[Gamma point]]). The "perturbation" is the term <math>H_{\mathbf{k}}'</math>. The analysis that results is called "k·p perturbation theory", due to the term proportional to ''k''·''p''. The result of this analysis is an expression for ''E''<sub>''n'','''k'''</sub> and ''u''<sub>''n'','''k'''</sub> in terms of the energies and wavefunctions at '''k'''=0. | |||
Note that the "perturbation" term <math>H_{\mathbf{k}}'</math> gets progressively smaller as '''k''' approaches zero. Therefore, k·p perturbation theory is most accurate for small values of '''k'''. However, if enough terms are included in the [[perturbation theory (quantum mechanics)|perturbative expansion]], then the theory can in fact be reasonably accurate for any value of '''k''' in the entire [[Brillouin zone]]. | |||
===Expression for a nondegenerate band=== | |||
For a nondegenerate band (i.e., a band which has a different energy at '''k'''=0 from any other band), with an [[extremum]] at '''k'''=0, and with no [[spin-orbit coupling]], the result of ''k''·''p'' perturbation theory is (to [[perturbation theory|lowest nontrivial order]]):<ref name=Yu2.6/> | |||
:<math>u_{n,\mathbf{k}} = u_{n,0}+\frac{\hbar}{m}\sum_{n' \neq n}\frac{\langle u_{n,0} | \mathbf{k}\cdot\mathbf{p} | u_{n',0} \rangle}{E_{n,0}-E_{n',0}} u_{n',0}</math> | |||
:<math>E_{n,\mathbf{k}} = E_{n,0}+\frac{\hbar^2 k^2}{2m} + \frac{\hbar^2}{m^2} \sum_{n'\neq n} \frac{|\langle u_{n,0} | \mathbf{k}\cdot\mathbf{p} | u_{n',0} \rangle |^2}{E_{n,0}-E_{n',0}}</math> | |||
The parameters that are required to do these calculations, namely ''E''<sub>''n'',0</sub> and <math>\langle u_{n,0} | \mathbf{p} | u_{n',0} \rangle</math>, are typically inferred from experimental data. (The latter are called "optical matrix elements".) | |||
In practice, the sum over ''n''' often includes only the nearest one or two bands, since these tend to be the most important (due to the denominator). However, for improved accuracy, especially at larger '''k''', more bands must be included, as well as more terms in the perturbative expansion than the ones written above. | |||
====Effective mass==== | |||
{{main|Effective mass (solid-state physics)}} | |||
Using the expression above for the energy dispersion relation, a simplified expression for the effective mass in the conduction band of a semiconductor can be found.<ref name=Harrison/> To approximate the dispersion relation in the case of the conduction band, take the energy ''E<sub>n0</sub>'' as the minimum conduction band energy ''E<sub>c0</sub>'' and include in the summation only terms with energies near the valence band maximum, where the energy difference in the denominator is smallest. (These terms are the largest contributions to the summation.) This denominator is then approximated as the band gap ''E<sub>g</sub>'', leading to an energy expression: | |||
:<math>E_c(\boldsymbol k ) \approx E_{c0} +\frac{(\hbar k)^2}{2m} +\frac{\hbar ^2}{{E_g}m^2}\sum_n {|\langle u_{c,0}|\mathbf{k}\cdot\mathbf{p}| u_{n,0} \rangle |^2} </math> | |||
The effective mass in direction ℓ is then: | |||
:<math> \frac{1} {m}_{\ell} = {{1} \over {\hbar^2}} \sum_{ m} \cdot {{\partial^{\ 2} E_{c} (\boldsymbol{k})} \over {\partial k_{\ell} \partial k_ m}} \approx \frac{1}{m}+\frac{2}{E_gm^2}\sum_{m,\ n} {\langle u_{c,0}|p_{\ell}| u_{n,0} \rangle }{\langle u_{n,0}|p_{m}| u_{c,0} \rangle } </math> | |||
Ignoring the details of the matrix elements, the key consequences are that the effective mass varies with the smallest bandgap and goes to zero as the gap goes to zero.<ref name=Harrison/> A useful approximation for the matrix elements in [[direct gap]] semiconductors is:<ref name=DirectGap>A ''direct gap'' semiconductor is one where the valence band maximum and conduction band minimum occur at the same position in '''k'''-space, usually the so-called Γ-point where '''k''' = 0.</ref> | |||
:<math>\frac{2}{E_gm^2}\sum_{m,\ n} {|\langle u_{c,0}|p_{\ell}| u_{n,0} \rangle |}{|\langle u_{c,0}|p_{m}| u_{n,0} \rangle |} \approx 20\mathrm{eV} \frac{1}{mE_{g}} \ , </math> | |||
which applies within about 15% or better to most group-IV, III-V and II-VI semiconductors.<ref name=Table2.22>See [http://books.google.com/books?id=W9pdJZoAeyEC&pg=PA244&dq=isbn:3540254706#PPA71,M1 Table 2.22] in Yu & Cardona, ''op. cit.''</ref> | |||
In contrast to this simple approximation, in the case of valence band energy the ''spin-orbit'' interaction must be introduced (see below) and many more bands must be individually considered. The calculation is provided in Yu and Cardona.<ref name=valence>See Yu & Cardona, ''op. cit.'' pp. 75-82</ref> In the valence band the mobile carriers are ''[[Electron hole|holes]]''. One finds there are two types of hole, named ''heavy'' and ''light'', with anisotropic masses. | |||
===''k''·''p'' model with spin-orbit interaction=== | |||
Including the [[spin-orbit interaction]], the Schrödinger equation for ''u'' is:<ref name=Kittel> | |||
{{cite book | |||
|author=C. Kittel | |||
|year=1987 | |||
|title=Quantum Theory of Solids | |||
|edition=Second Revised Printing | |||
|pages=186–190 | |||
|isbn=0-471-62412-8 | |||
|publisher=[[John Wiley & Sons|Wiley]] | |||
|location=New York | |||
}}</ref> | |||
:<math>H_{\mathbf{k}} u_{n,\mathbf{k}}=E_{n,\mathbf{k}}u_{n,\mathbf{k}}</math> | |||
where | |||
:<math>H_{\mathbf{k}} = \frac{p^2}{2m} + \frac{\hbar \mathbf{k}\cdot\mathbf{p}}{m} + \frac{\hbar^2 k^2}{2m} + V + \frac{1}{4 m^2 c^2} (\vec \sigma \times \nabla V)\cdot (\hbar\mathbf{k}+\mathbf{p})</math> | |||
where <math>\vec \sigma=(\sigma_x,\sigma_y,\sigma_z)</math> is a vector consisting of the three [[Pauli matrix|Pauli matrices]]. This Hamiltonian can be subjected to the same sort of perturbation-theory analysis as above. | |||
===Calculation in degenerate case=== | |||
For degenerate or nearly degenerate bands, in particular the [[valence band]]s in certain materials such as [[gallium arsenide]], the equations can be analyzed by the methods of [[perturbation theory (quantum mechanics)|degenerate perturbation theory]].<ref name=Yu2.6/><ref name=Kittel/> Models of this type include the "[[Luttinger-Kohn model]]" (a.k.a. "Kohn-Luttinger model"),<ref> | |||
{{cite journal | |||
|author=J. M. Luttinger, W. Kohn | |||
|year=1955 | |||
|title=Motion of Electrons and Holes in Perturbed Periodic Fields | |||
|journal=[[Physical Review]] | |||
|volume=97 |issue= |pages=869 | |||
|doi=10.1103/PhysRev.97.869 | |||
|bibcode = 1955PhRv...97..869L }}</ref> and the "[[Kane model]]".<ref> | |||
{{cite journal | |||
|author=Evan O. Kane | |||
|year=1957 | |||
|title=Band Structure of Indium Antimonide | |||
|journal=[[Journal of Physics and Chemistry of Solids]] | |||
|volume=1 |pages=249 | |||
|doi=10.1016/0022-3697(57)90013-6 | |||
|bibcode = 1957JPCS....1..249K }}</ref> | |||
==Notes and references== | |||
{{reflist}} | |||
==See also== | |||
{{Col-begin}} | |||
{{Col-1-of-2}} | |||
'''Electronic band structure''' | |||
* [[Electronic band structure]] | |||
* [[Nearly free electron model]] | |||
* [[Kronig-Penney model]] | |||
'''Band properties''' | |||
* [[Band gap]] | |||
* [[Effective mass (solid-state physics)|Effective mass]] | |||
* [[Density of states]] | |||
* [[Fermi surface]] | |||
{{Col-2-of-2}} | |||
'''Wavefunctions''' | |||
* [[Wannier functions]] | |||
* [[Bloch waves]] | |||
'''Fundamental theory''' | |||
* [[Kohn-Sham equations]] | |||
* [[Local-density approximation]] | |||
{{col-end}} | |||
{{DEFAULTSORT:K P Perturbation Theory}} | |||
[[Category:Electronic structure methods]] | |||
Revision as of 14:27, 24 December 2013
Template:Lowercase Template:Electronic structure methods In solid-state physics, k·p perturbation theory is an approximation scheme for calculating the band structure (particularly effective mass) and optical properties of crystalline solids.[1][2][3] It is pronounced "k dot p", and is also called the "k·p method". This theory has been applied specifically in the framework of the Luttinger–Kohn model (after Joaquin Mazdak Luttinger and Walter Kohn), and of the Kane model (after Evan O. Kane).
Background and derivation
Bloch's theorem and wavevectors
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According to quantum mechanics (in the single-electron approximation), the electrons in any material have wavefunctions which can be described by the following Schrödinger equation:
where p is the quantum-mechanical momentum operator, V is the potential, and m is the mass of an electron. (This equation neglects the spin-orbit effect; see below.)
In a crystalline solid, V is a periodic function, with the same periodicity as the crystal lattice. Bloch's theorem proves that the solutions to this differential equation can be written as follows:
where k is a vector (called the wavevector), n is a discrete index (called the band index), and un,k is a function with the same periodicity as the crystal lattice.
For any given n, the associated states are called a band. In each band, there will be a relation between the wavevector k and the energy of the state En,k, called the band dispersion. Calculating this dispersion is one of the primary applications of k·p perturbation theory.
Perturbation theory
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The periodic function un,k satisfies the following Schrödinger-type equation:[1]
where the Hamiltonian is
Note that k is a vector consisting of three real numbers with units of inverse length, while p is a vector of operators; to be explicit,
In any case, we write this Hamiltonian as the sum of two terms:
This expression is the basis for perturbation theory. The "unperturbed Hamiltonian" is H0, which in fact equals the exact Hamiltonian at k=0 (i.e., at the Gamma point). The "perturbation" is the term . The analysis that results is called "k·p perturbation theory", due to the term proportional to k·p. The result of this analysis is an expression for En,k and un,k in terms of the energies and wavefunctions at k=0.
Note that the "perturbation" term gets progressively smaller as k approaches zero. Therefore, k·p perturbation theory is most accurate for small values of k. However, if enough terms are included in the perturbative expansion, then the theory can in fact be reasonably accurate for any value of k in the entire Brillouin zone.
Expression for a nondegenerate band
For a nondegenerate band (i.e., a band which has a different energy at k=0 from any other band), with an extremum at k=0, and with no spin-orbit coupling, the result of k·p perturbation theory is (to lowest nontrivial order):[1]
The parameters that are required to do these calculations, namely En,0 and , are typically inferred from experimental data. (The latter are called "optical matrix elements".)
In practice, the sum over n' often includes only the nearest one or two bands, since these tend to be the most important (due to the denominator). However, for improved accuracy, especially at larger k, more bands must be included, as well as more terms in the perturbative expansion than the ones written above.
Effective mass
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Using the expression above for the energy dispersion relation, a simplified expression for the effective mass in the conduction band of a semiconductor can be found.[3] To approximate the dispersion relation in the case of the conduction band, take the energy En0 as the minimum conduction band energy Ec0 and include in the summation only terms with energies near the valence band maximum, where the energy difference in the denominator is smallest. (These terms are the largest contributions to the summation.) This denominator is then approximated as the band gap Eg, leading to an energy expression:
The effective mass in direction ℓ is then:
Ignoring the details of the matrix elements, the key consequences are that the effective mass varies with the smallest bandgap and goes to zero as the gap goes to zero.[3] A useful approximation for the matrix elements in direct gap semiconductors is:[4]
which applies within about 15% or better to most group-IV, III-V and II-VI semiconductors.[5]
In contrast to this simple approximation, in the case of valence band energy the spin-orbit interaction must be introduced (see below) and many more bands must be individually considered. The calculation is provided in Yu and Cardona.[6] In the valence band the mobile carriers are holes. One finds there are two types of hole, named heavy and light, with anisotropic masses.
k·p model with spin-orbit interaction
Including the spin-orbit interaction, the Schrödinger equation for u is:[2]
where
where is a vector consisting of the three Pauli matrices. This Hamiltonian can be subjected to the same sort of perturbation-theory analysis as above.
Calculation in degenerate case
For degenerate or nearly degenerate bands, in particular the valence bands in certain materials such as gallium arsenide, the equations can be analyzed by the methods of degenerate perturbation theory.[1][2] Models of this type include the "Luttinger-Kohn model" (a.k.a. "Kohn-Luttinger model"),[7] and the "Kane model".[8]
Notes and references
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See also
Template:Col-begin Template:Col-1-of-2 Electronic band structure
Band properties
Template:Col-2-of-2 Wavefunctions
Fundamental theory
- ↑ 1.0 1.1 1.2 1.3
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 2.0 2.1 2.2
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 3.0 3.1 3.2
20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ A direct gap semiconductor is one where the valence band maximum and conduction band minimum occur at the same position in k-space, usually the so-called Γ-point where k = 0.
- ↑ See Table 2.22 in Yu & Cardona, op. cit.
- ↑ See Yu & Cardona, op. cit. pp. 75-82
- ↑
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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang