Lithuanian Journal of Physics article / Lietuvos fizikos žurnalo straipsnis

[PDF] http://dx.doi.org/10.3952/lithjphys.49311

Open access article / Atviros prieigos straipsnis

Lith. J. Phys. 49, 277–284 (2009)

APPLICATION OF CLIFFORD ALGEBRA TO ANALYSIS OF SPIN PROPERTIES OF SEMICONDUCTORS
A. Dargys
Semiconductor Physics Institute, A. Goštauto 11, LT-01108 Vilnius, Lithuania
E-mail: dargys@pfi.lt

Received 14 May 2009; revised 11 September 2009; accepted 15 September 2009

The quantum mechanics is formulated in terms of state vectors in the Hilbert space and operators that act on these vectors. However, using the Clifford algebra an alternative formulation is possible, where multivectors in n-dimensional linear Euclidean space represent both the state of the system and quantum operators. In this report a short overview is presented on how the Clifford algebra can be used to investigate electron and hole spin properties in semiconductors. In particular, calculation of free electron and hole spin precession in cubic semiconductors with the help of the Clifford algebra is discussed.

Keywords: Clifford algebra, geometric algebra, cubic semiconductors, spintronics

PACS: 75.10.Jm, 85.75.-d, 72.25.Dc, 71.70.Ej

CLIFFORDO ALGEBROS TAIKYMAS SUKINIŲ SAVYBIŲ ANALIZĖJE PUSLAIDININKIAMS
A. Dargys
Puslaidininkių fizikos institutas, Vilnius, Lietuva

Kvantinė mechanika suformuluota matematinės Hilberto erdvės sąvokomis, kur svarbų vaidmenį turi būsenų vektoriai bei operatoriai, kurie vienus Hilberto erdvės vektorius perveda į kitus. Tačiau yra žinoma alternatyvi kvantinės mechanikos formuluotė [1, 2], kurioje dalelės būsena bei fizikinius dydžius aprašantys operatoriai yra nusakomi Cliffordo algebros sąvokomis, būtent, per daugiamatės Euklido erdvės vektorius ir multivektorius. Pastarieji yra sukonstruoti iš n-matės tiesinės Euklido erdvės bazinių vektorių visų galimų sandaugų. Būdinga tokių bazinių vektorių savybė ta, kad jie tarpusavyje antikomutuoja, todėl Cliffordo algebros multivektoriais galima aprašyti spinorius bei jų dinamiką išoriniuose laukuose. Multivektoriaus projekcijos, kurios nusako matuojamus dydžius, yra realūs (o ne kompleksiniai, kaip Hilberto erdvės atveju) skaičiai. Cliffordo algebroje dalelės būsena ir operatoriai yra lygiavertės sąvokos ir nusakomos panašiai. Straipsnyje trumpai apžvelgta, kaip galima pritaikyti Cliffordo algebros formalizmą sukinių savybių analizėje. Aptarta kubinės simetrijos puslaidininkio laidumo juostos elektrono ir valentinės juostos skylės sukinio precesija bei kaip ją būtų galima apskaičiuoti naudojant Cliffordo algebrą.

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