ECEN 5005 Crystals, Nanocrystals and Device Applications Class 12 Group Theory For Crystals

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1 ECEN 5005 Crstals, Nanocrstals and Device Applications Class 1 Group Theor For Crstals Hierarch of Smmetr Irreducible Representations of oint Groups Transformation roperties of Functions Luminescence

2 Hierarch of Smmetr More smmetrical or less smmetrical? - The group having more smmetr elements is generall said more smmetrical. - For eample, the crstal with O h smmetr is more smmetrical than the one with D 4h smmetr, because the latter lacks several smmetr elements of the former. - This is consistent with out intuitive concepts that a cube is more smmetrical then a rectangular block elongated or compressed) along one of the four-fold rotation aes. However, it is not alwas possible to make unambiguous comparisons between two smmetr groups. - For eample, consider C, S and C 1h. - Each group contains one non-trivial smmetr element, 180 o rotation for C, 180 o improper rotation for S and reflection in a horiontal plane for C 1h. - Comparing the degree of smmetr between these groups are meaningless. Nevertheless, we can select a set of groups between which such comparisons are not onl meaningful but also helpful in recogniing the hierarchical structure of smmetr groups. - In such sets of groups, the group representing a lower smmetr is a subgroup of the one representing higher smmetr.

3 Reduction of Smmetr An eample of hierarchical chain of smmetr groups is O h D 4h C 4v C 4 C C 1 The process of transitioning from higher smmetr group to a lower one is called the reduction of smmetr. The possible reduction processes for all crstallographic point groups are shown below.

4 Irreducible Representations of oint Groups We now know all crstallographic point groups with their elements and class structures. This knowledge then fies the number of irreducible representations and their dimensionalities, according to the rules we discussed in the previous classes. Furthermore, we can also work out the complete character tables. This is done and ma be found in man books. See, for eample, Tinkham, Appendi B.) Let us take a look at the character table of D as an eample. D E C C +, A R, A 1 1-1, ), ), ) R, R ) E Column labels are the tpe and number of operations belonging to each class. articularl, C indicates the two-fold ais is perpendicular to the primar rotation ais in this particular eample). If there were additional inequivalent two-fold aes, then the would have been labeled C, C, etc. - In terms of our previous notation, C correspond to D, F and C to A, B, C. C

5 Character Tables of oint Groups Rows are labeled b the irreducible representations. Conventional notations are A and B for one-dimensional representations, E for twodimensional, and T for three-dimensional representations. The irreducible representations having a character +1 under the primar rotation, C n, is labeled A and those with a character 1 is labeled B. If there are more than one irreducible representations having the same character under the primar rotation, then subscripts 1 and are added. If the group contains inversion smmetr, subscripts g and u are used to indicate even and odd representations. The first two columns contain the coordinates, quadratic forms of coordinates, and rotations, R, R, R about the coordinate aes. These are listed in the same row as the irreducible representations according to which the transform. B saing a function, fr) or an operator) transforms according to an irreducible representation, we mean R f r) Γ i) R) f r) for all smmetr operations, R, in the group. Here, Γ R) matri representation of R in the irreducible representation, i. i) is the

6 Transformation roperties of Functions Recall that we use the conventional definition of R as ) ) 1 r R r f f R where R is the transformation of coordinates according to the smmetr operation. Before we deal with D, we digress to a simpler eample, C v, that contains onl one-dimensional representations. C v E C v σ v σ,, A R A R, B R, B where is the reflection in the plane and v σ v σ is the reflection in the plane. We can now eplicitl write out the effect of the four operations on the coordinate functions. C E v σ σv - Note that we are considering the coordinate functions and thus following the definition. ) ) 1 r R r f f R - But in this particular eample, the inverse transformation is the same as itself for all four smmetr operations.

7 Transformation roperties of Functions We then notice that the characters are in fact the representation matrices, because we onl have one-dimensional representations. B comparing the results of transformations with the character table, B1) B) A1) R Γ R), R Γ R), R Γ R) We summarie these results b saing that transforms according to B 1, transforms according to B, transforms according to A 1. Equivalentl, we can also sa belongs to B 1 and so on. Using these results, we can easil find that transforms according to A B 1 B, and so on. Also, the transformation properties of R, R, R can be verified similarl. The assignment of the functions to the corresponding irreducible representations can also be verified b using the projection operator technique. Recall that the projection operator, l * i) κκ, is defined as i) i i) i) i) i) i) κκ Γ R) κκ R and satisfies ϕ onl when is a h R κ ϕ κ basis function belonging to the κ th row of the i th irreducible representation. Let us now appl A) A) on. l A A) 1 χ R) R { + ) ) ) ) } h 4 R This confirms that indeed transforms according to A. κκ ϕ κ

8 Transformation roperties of Functions Now we return to the group D. In order to discuss specific transformations, we first need to define coordinate sstem, which is shown in the right. It is important to clearl define the sense of C rotations because their inverse transformations are not the same as themselves. We define the operations D and F as clockwise and counter-clockwise rotations b 10 o, respectivel. A B C We can now work out a matri representation, keeping in mind the definition,. ) ) 1 r R r f f R F D C B A E

9 Transformation roperties of Functions First, we observe that alwas transforms into a multiple of itself, indicating that itself forms a basis of a one-dimensional irreducible representation. Furthermore, we notice that the character under operation A which is one of the C operations) is 1. Thus, we identif that A is the irreducible representation for which forms a basis function. With and, the situation is more complicated since cross terms appear. However, we take note of the fact that and mi with each other onl and not with. Thus, we can see that and together form basis functions for a twodimensional irreducible representation, which must be E. The matrices can be obtained b eamining the transformation equations above and the are equal to the matrices given before. In this eample, we used simple basis functions,, and, to obtain representation matrices. But this does not mean that the choice of basis functions is limited. In general, there are infinitel man functions that transform the same wa. This means that with relativel small number of matrices we can describe the transformation properties of a large number of functions, to be more specific, ALL functions that are eigenfunctions for the given problem.

10 Luminescence Luminescence generation of light b a material when it is relativel cool, in contrast to incandescence or black bod radiation). - Means cold light. - Luminescence is due to electrons undergoing transitions between quantum states, while incandescence is due to atomic vibrations. - Broadl used to include ultra-violet and infrared emission as well as visible light generation. - A more general term including both fluorescence and phosphorescence - Can be ecited a man different was: cathodoluminescence electron beam), photoluminescence high energ photons), electroluminescence high electric field), thermoluminescence thermal energ), etc. Fluorescence luminescence that onl lasts less than 10-8 sec. - In organic materials, fluorescence means luminescence due to an electronic transition between states with the same multiplicit. hosphorescence long, persistent luminescence that sometimes lasts for several seconds. - In organic materials, phosphorescence is due to transitions between different multiplicit. - In inorganic materials, distinctions are not clear.

11 Luminescent rocesses in Crstals There are man different processes that can induce luminescence. a) eciton recombination recombination of an electron and hole pair bound together b Coulomb force b) conduction band CB)-to-acceptor recombination recombination of a free electron in CB with a hole bound to an acceptor level c) donor-to-valence band VB) recombination - recombination of an electron bound to a donor level with a free hole in the VB. d) donor-acceptor pair recombination recombination between an electron at a donor level with a hole at an acceptor level e) localied ionic transition transition within an ionic luminescent center such as transition metal ions and rare earth ions. conduction band a b c d e valence band

12 Applications of Luminescent Crstals hotoluminescent Crstals - fluorescent lamp: uses a low pressure Hg discharge to generate 54nm UV radiation, which in turn ecites photoluminescent crstals deposited inside the fluorescent tube wall. E. Ca 5 O 4 ) F or Ca 5 O 4 ) Cl doped with Sb and Mn - d-d transition of Mn + Y O :Eu f-f transition of Eu + Ce,Tb)MgAl 11 O 19 f-f transition of Tb + BaMgAl 10 O 17 :Eu d-f trasnsition of Eu + - plasma displa: similar operation to the fluorescent tube but uses Xe discharge emitting 147nm vacuum UV radiation. E. Man fluorescent lamp phosphors Y,Gd)BO 4 :Eu f-f transition of Eu + Zn SiO 4 :Mn - d-d transition of Mn + - solid-state lighting device: a GaN-based blue or UV laser ecites photoluminescent crstals deposited on the laser surface to generate white light. E. Y Al 5 O 1 :Ce d-f transition of Ce + SrS:Eu d-f transition of Eu + SrGa S 4 :Eu d-f transition of Eu +

13 Applications of Luminescent Crstals hotoluminescent crstals - photo-pumped laser: uses intense lamp to create population inversion in the laser crstal leading to laser action. E. Al O :Cr rub) d-d transition of Cr + Y Al 5 O 1 :Nd YAG:Nd) f-f transition of Nd + - Er-doped fiber amplifier: diode pump beam 980nm or 1.49µm) ecites Er ions doped in optical fiber to obtain laser action at 1.54µm. E. SiO :Er f-f transition of Er + Cathodoluminescent crstals - cathode ra tube: electron beams are used to ecite luminescent crstals deposited on the screen E. Y O :Eu f-f transition of Eu + ZnS:Cu,Al donor-acceptor pair transition ZnS:Ag,Cl donor-acceptor pair transition Electroluminescent crstals - Electroluminescent displa: high E-field ecites luminescence E. ZnS:Mn d-d transition of Mn + SrS:Ce d-f transition of Ce + - Light emitting diode and laser diode: low E-field injects electrons and holes into the active laer where the recombine.