Spin spirals: Difference between revisions
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''i.e.'', from one unit cell to the next the up-spinor and down-spinors pick up an additional phase factor of <math>\exp(-i{\bf q}\cdot {\bf R}/2)</math> and <math>\exp(-i{\bf q}\cdot {\bf R}/2)</math>, respectively. | ''i.e.'', from one unit cell to the next the up-spinor and down-spinors pick up an additional phase factor of <math>\exp(-i{\bf q}\cdot {\bf R}/2)</math> and <math>\exp(-i{\bf q}\cdot {\bf R}/2)</math>, respectively. | ||
Here, '''R''' is a lattice vector of the crystalline lattice, and '''q''' is the so-called spin-spiral propagation vector. | |||
The latter is commonly chosen to lie within the first Brillouin zone of the reciprocal space lattice. | |||
This condition gives rise to the following behaviour of the magnetization density: | This condition gives rise to the following behaviour of the magnetization density: | ||
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This is schematically depicted in the figure at the top of this page: | This is schematically depicted in the figure at the top of this page: | ||
the components of the magnization in the ''xy''-plane rotate about the | the components of the magnization in the ''xy''-plane rotate about the spin-spiral propagation vector '''q'''. | ||
Revision as of 12:43, 6 July 2018
Generalized Bloch condition
Spin spirals may be conveniently modeled using a generalisation of the Bloch condition:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \left[{\begin{array}{c}\Psi _{{{\bf {k}}}}^{{\uparrow }}({\bf {r)}}\\\Psi _{{{\bf {k}}}}^{{\downarrow }}({\bf {r)}}\end{array}}\right]=\left({\begin{array}{cc}e^{{-i{\bf {q\cdot {\bf {R/2}}}}}}&0\\0&e^{{+i{\bf {q\cdot {\bf {R/2}}}}}}\end{array}}\right)\left[{\begin{array}{c}\Psi _{{{\bf {k}}}}^{{\uparrow }}({\bf {r-R)}}\\\Psi _{{{\bf {k}}}}^{{\downarrow }}({\bf {r-R)}}\end{array}}\right],
i.e., from one unit cell to the next the up-spinor and down-spinors pick up an additional phase factor of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \exp(-i{{\bf {q}}}\cdot {{\bf {R}}}/2) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \exp(-i{{\bf {q}}}\cdot {{\bf {R}}}/2) , respectively. Here, R is a lattice vector of the crystalline lattice, and q is the so-called spin-spiral propagation vector. The latter is commonly chosen to lie within the first Brillouin zone of the reciprocal space lattice.
This condition gives rise to the following behaviour of the magnetization density:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {{\bf {m}}}({{\bf {r}}}+{{\bf {R}}})=\left({\begin{array}{c}m_{x}({{\bf {r}}})\cos({{\bf {q}}}\cdot {{\bf {R}}})-m_{y}({{\bf {r}}})\sin({{\bf {q}}}\cdot {{\bf {R}}})\\m_{x}({{\bf {r}}})\sin({{\bf {q}}}\cdot {{\bf {R}}})+m_{y}({{\bf {r}}})\cos({{\bf {q}}}\cdot {{\bf {R}}})\\m_{z}({{\bf {r}}})\end{array}}\right)
This is schematically depicted in the figure at the top of this page: the components of the magnization in the xy-plane rotate about the spin-spiral propagation vector q.