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!"#$%& '#"#$()*+", -+.($ !"#$%&

Silvana Botti1,2 and Julien Vidal3

1Laboratoire des Solides Irradies and ETSF, Ecole Polytechnique, CNRS, CEA-DSM,

Palaiseau, France2LPMCN, CNRS, Universite Lyon 1, Villeurbanne, France

3Physics Department, King’s College London, London, UK

/0 12*"34*.5 62+)+7+.)(*89

#B1 1=13C5 31(1<D1E 25 ,B1 F+3,B G3.? ,B1 "4= H/<,B + 3+,1 .G 8%A ;! <= ,B1 4II13+,?.JIB131K <J 25 G+3 -+3C13 ,B+= ,B1 (4331=, /.3-E 1=13C5 (.=J4?I,<.=L M= D<1/ .G,B1 I31J1=, (3<J<J .G ,B1 E.?<=+=, 1=13C5 31J.43(1J +=E ,B1 3+<J<=C (.=(13= +2.4,1=D<3.=?1=,7 J.-+3 I./13 1?13C1J +J + (.=(31,1 +-,13=+,<D1 G.3 + J4J,+<=+2-1 +=E1=D<3.=?1=,'G3<1=E-5 C3./,B .G ,B1 /.3-E 1=13C5 J4II-5 N8OL

#B1 E<J(.D135 .G ,B1 IB.,.D.-,+<( H;PK 1GG1(, 25 Q-1R+=E31 FE?.=E S1(T4131-<= 89&U I3.D<E1E + /+5 ,. E<31(,-5 (.=D13, J.-+3 3+E<+,<.= <=,. 1-1(,3<(<,5L Q-,B.4CB,B1 ,B13?.E5=+?<( -<?<, G.3 ,B1 1GV(<1=(5 .G + ;P E1D<(1 <J +2.4, 9WL)X7 +((.3E<=C,. ,B1 *+3=., -+/7 ,B1 V3J, J.-+3 (1--J7 ?+E1 .G C.-E'(.+,1E J1-1=<4?7 /131 .=-5+3.4=E 8X1GV(<1=,L #B1 V3J,?.E13= ;P (1--J7 /<,B += 1=13C5 (.=D13J<.= 1GV(<1=(5.G +2.4, 6X7 /131 24<-, .=-5 <= ,B1 8UW)J +, S1-- Y+2.3+,.3<1J 25 *B+I<= et al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

Computational Approaches to Energy Materials7 0<3J, FE<,<.=L FE<,1E 25 Q3.= !+-JB7 Q-1R15 QL ".Z.- +=E *L ]<(B+3E QL *+,-./L

^ $)8& .B= !<-15 _ ".=J7 Y,EL ;42-<JB1E $)8& 25 .B= !<-15 _ ".=J7 Y,EL

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30 Computational Approaches to Energy Materials

,B+, +31 C1=13+,1E /B1= + IB.,.= <J +2J.321EL #B1 (B+3C1J G43,B13 E<GG4J1 ,. ,B11-1(,3.E1J +=E V=+--5 (31+,1 + (4331=, 25 31(.?2<=<=C <= ,B1 1R,13=+- (<3(4<, N&OL

#B1 ;P ,1(B=.-.C5 .G ,B1 V3J, C1=13+,<.= 31-<1E +-?.J, 1R(-4J<D1-5 .= .=1?+,13<+-7 J<-<(.=L M=E11E7 J<-<(.= /+J ,B1 .=-5 J1?<(.=E4(,.3 /B.J1 C3./,B /+J?+J,131E ,. ,B1 1R,1=, 31T4<31E G.3 ,B1 I31(<J1 (.=,3.- .G E.I<=C +=E <?I43<,5(.=(1=,3+,<.= NAOL "<=(1 ,B1 8UW)J7 ,B1 <=(31+J<=C I13G.3?+=(1 .G (35J,+--<=1 J<-<'(.= J.-+3 (1--J B+J 21=1V,1E G3.? ,B1 ,1(B=.-.C<(+- I3.C31JJ1J <= J1?<(.=E4(,.3<=E4J,35L `1JI<,1 31+(B<=C 31(.3E 1GV(<1=(<1J .G $WX NWO +=E E1JI<,1 <,J E.?<=+,<.=.= ,B1 J.-+3 (1-- ?+3Z1,7 ,B1 ;P ,1(B=.-.C5 2+J1E .= (35J,+--<=1 J<-<(.= J,34CC-1J,. 231+Z ,B1 J5?2.-<( $8a! 2+33<13L #B1 31E4(,<.= .G (.J,J <J ?+<=-5 -<?<,1E 25,B1 31T4<31?1=, .G C3./<=C + ,B<(Z -+513 .G B<CB-5 I431 J<-<(.=L M= G+(,7 E41 ,. ,B1<=E<31(, C+I .G J<-<(.= +=E ,B1 (.=J1T41=, I..3 +2J.3I,<.= .G -<CB, +, ,B1 .=J1,7+ (35J,+--<=1 -+513 .G 8))µ? <J =1(1JJ+35 <= .3E13 ,. C1=13+,1 1=.4CB 1-1(,3.=[B.-1 I+<3JL b.31.D137 + B<CB I43<,5 <J 31T4<31E ,. <=(31+J1 ,B1 E<GG4J<.= -1=C,B .G1-1(,3.=J +=E B.-1J <= ,B1 T4+J<'=14,3+- 31C<.=7 <= .3E13 ,. +JJ431 ,B+, ,B1 (B+3C1J31+(B ,B1 1-1(,3<(+- (.=,+(,J 21G.31 31(.?2<=<=CL 0.3 51+3J ,B<J <JJ41 B+J J13<.4J-5<?I+<31E ,B1 D<+2<-<,5 .G ;P J.-+3 I./137 ?+Z<=C <, B+3E ,. (.?I1,1 /<,B ,3+E<,<.=+-1=13C5 31J.43(1J N6OL

Q J1(.=E'C1=13+,<.= ,1(B=.-.C5 2+J1E .= ,B<='V-? +2J.3213J 1?13C1E +-31+E5&) 51+3J +C.7 /<,B ,B1 ?+<= C.+- .G E1(31+J<=C ,B1 I3.E4(,<.= (.J, .G J.-+3 I+=1-JN%OL Q, ,B1 B1+3, .G ,B1 E1D<(17 ,B131 <J J,<-- + I[= \4=(,<.= ,B+, <J 31JI.=J<2-1 G.3,B1 J1I+3+,<.= .G (B+3C1 (+33<13JL M= (.=,3+J, /<,B (35J,+--<=1 +2J.32<=C -+513J .GB4=E31EJ .G ?<(3.?1,13J7 ,B<='V-? +2J.3213J +31 \4J, $[&µ? ,B<(ZL #B1 ,B<(Z=1JJ.G ,B1 +2J.3213 -+513 (+= 21 31E4(1E .=-5 <G 1R,31?1-5 1GV(<1=, -<CB,'B+3D1J,<=C?+,13<+-J +31 1?I-.51E: ,B+, <J /B5 ?+,13<+-J J4(B +J *E#1 N9O +=E *4 HM=7 c+KH"17"K$ NUO7 /<,B 31?+3Z+2-5 -+3C1 +2J.3I,<.= (.1GV(<1=,J7 +31 <E1+- +2J.3213J <= ,B<='V-? ;PJL #B1 4J1 .G ,B<= V-?J <?I-<1J J<C=<V(+=, J+D<=C .= ,B1 T4+=,<,5 .G ?+,13<+-31T4<31EL S1J<E1J7 <, 23<=CJ + J<?I-<V(+,<.= <= ,B1 C3./,B I3.(1JJ1J +=E +--./J G.3,B1 <=,3.E4(,<.= .G =1/ J42J,3+,1JL #B+=ZJ ,. ,B+,7 ,B1 $8a! 2+33<13 G.3 C3<E I+3<,521(.?1J +((1JJ<2-1 1D1= <G ,B<='V-? J.-+3 (1--J J,<-- E<JI-+5 -./13 (.=D13J<.=1GV(<1=(<1J ,B+= ,B1<3 (35J,+--<=1 J<-<(.= (.4=,13I+3,JL

M= (.?I+3<J.= ,. ,B1 V3J, J.-+3 (1--J I3.E4(1E <= ,B1 8UW)J7 ,B1 E1J<C= .G ,B<='V-? E1D<(1J 4=E13/1=, J.?1 1JJ1=,<+- (B+=C1JL 0<C431 $L8 E<JI-+5J ,B1 J(B1?1 .G+ ,B<='V-? J.-+3 (1--d+, -1+J, G.43 E<GG131=, ?+,13<+-J /<,B JI1(<V( 1-1(,3<(+- +=E.I,<(+- I3.I13,<1J +31 J,+(Z1EL #B1 J<=C-1 ?+,13<+- I3.I13,<1J ,B+, E1,13?<=1 ,B1;P 1GV(<1=(57 +=E ,B+, .=1 /.4-E -<Z1 ,. (.=,3.-7 +31 2+=EC+IJ7 E.I<=C -1D1-J7 +=E+2J.3I,<.= (.1GV(<1=,JL M= +EE<,<.=7 2+=E .GGJ1,J +, ,B1 <=,13G+(1J7 /B<(B E1I1=E.= ,B1 J,34(,431 .G ,B1 <=,13G+(17 .= E.I<=C -1D1-J7 +=E .= 2+=E'1EC1 +-<C=?1=,J7B+D1 + J,3.=C <=e41=(1 .= ,B1 E1D<(1 I13G.3?+=(1L #B1J1 1-1(,3.=<( I3.I13,<1J +31=./+E+5J +((1JJ<2-1 +=E I31E<(,+2-1 D<+ ab initio (+-(4-+,<.=JL

M= ,B<J (B+I,13 /1 I31J1=, ,B1 J,+,1'.G',B1'+3, ab initio +II3.+(B1J +2-1 ,. E1,13'?<=1 1-1(,3.=<( J,+,1J +=E .I,<(+- I3.I13,<1J .G (.?I-1R (.?I.4=EJ 4J1E <= ,B<='V-? J.-+3 (1--JL "4(B +II3.+(B1J +31 2+J1E .= ,B1 GW +II3.R<?+,<.= /<,B<=

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Energy Generation: Solar Energy 31

Doping

Substrate

Buffer layer

Back-contact layer

Absorber layer

Contact

Figure 2.1 Schematic description of a thin-film solar cell. TCO, transparent conducting oxide.

?+=5'2.E5 I13,432+,<.= ,B1.35 HbS;#KL #B<J <J + 3+I<E-5 1RI+=E<=C V1-E .G <=D1J',<C+,<.= +=E <, <J =., I.JJ<2-1 ,. <=(-4E1 B131 +-- 31(1=, E1D1-.I?1=,JL !1 E1(<E1E,B131G.31 ,. G.(4J .= ,/. +II-<(+,<.=Jd*4'2+J1E +2J.3213J +=E *4'2+J1E ,3+=J'I+31=, (.=E4(,<=C .R<E1Jd+=E ,. 4J1 ,B1J1 1R+?I-1J ,. E<J(4JJ +(B<1D1?1=,J +=E-<?<,+,<.=J .G ,B1 ?.J, +((43+,1 ,B1.3<1JL

0.3 ,B1 I+J, 51+3J7 ,B131 B+J 211= + J42J,+=,<+- 1GG.3, ,. J,4E5 ,B1 I3.I13,<1J .G*4'2+J1E (.?I.4=EJ G.3 ;PJ7 =., .=-5 ,. 4=E13J,+=E ,B1? G3.? + G4=E+?1=,+-I.<=, .G D<1/7 24, +-J. ,. .I,<?<f1 ,B1<3 I3.I13,<1J +=E E1J<C= J.-+3 (1--J /<,B21,,13 I13G.3?+=(1L g./1D137 ,B131 <J J,<-- + (-1+3 -+(Z .G (.=J<J,1=, ,B1.31,<(+- +=E(.?I4,+,<.=+- J,4E<1J ,B+, (+= 1RI-+<= +=E E1J(3<21 1-1(,3.=<( 1R(<,+,<.=J <= ,B<J(-+JJ .G ?+,13<+-JL QJ + (.=J1T41=(17 /B1=1D13 J<-<(.= <J =., 4J1E +J += +2J.32137+ ?+\.3 .2J,+(-1 G.3 + G+J, E1D1-.I?1=, .G B<CB-5 1GV(<1=, E1D<(1J <J ,B1 -+(Z .GZ=./-1EC1 .G 2+J<( ?+,13<+- I3.I13,<1J7 <= I+3,<(4-+37 31-+,1E ,. E1G1(, IB5J<(JL M=G+(,7 J<-<(.= +=E .,B13 MMM'P J1?<(.=E4(,.3J B+D1 +-J. + 23.+E 4J1 .4,J<E1 ,B1 /.3-E;PJ H1LCL7 <= 1-1(,3.=<(J7 -+J13J7 +=E YF`JK7 +=E ,B131G.31 + ?4(B -+3C13 1GG.3, B+J211= E1D.,1E ,. ,B1 4=E13J,+=E<=C .G ,B1<3 2+J<( J1?<(.=E4(,.3 I3.I13,<1J /<,B21=1V(<+- (.=J1T41=(1J G.3 ;P ,1(B=.-.C<1JL

*4'2+J1E (B+-(.I53<,1 J1?<(.=E4(,.3J .G ,B1 G.3? *4Hc+7M=KH"7"1K$ H*Mc"KB+D1 +-31+E5 C+<=1E + 31I4,+,<.= +J ,B1 -1+E<=C (+=E<E+,1J G.3 J1(.=E'C1=13+,<.=;PJ7 /<,B ,B1 B<CB1J, 1GV(<1=(5 +?.=C ,B<='V-? J.-+3 (1--J H$)L8XK NU7 8)OL #B<J<J I.JJ<2-1 ,B+=ZJ ,. ,B1 I1(4-<+3 .I,<(+- +=E J,34(,43+- I3.I13,<1J .G *Mc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

Q=.,B13 (-+JJ .G (B+-(.C1=<E1J JB+31J /<,B (B+-(.I53<,1 *Mc" ,B1<3 1R(1--1=,1-1(,3.=<( I3.I13,<1JL M, <J ,B1 G+?<-5 .G T4+,13=+35 *4$h="=H"7"1KA H*h#"K /<,B

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Z1J,13<,1 .3 J,+==<,1 J,34(,431L *h#" B+D1 += 1R,3+ +ED+=,+C1d,B15 +31 .=-5 (.?'I.J1E .G 1-1?1=,J ,B+, +31 =+,43+--5 +24=E+=, +=E =.=,.R<(L "<?<-+3-5 ,. *Mc"7,B1 +--.5J *4$h="="x"18−x B+D1 .I,<?+- C+IJ G.3 ;P (.=D13J<.=7 +((.3E<=C ,. ,B1"B.(Z-15[i41<JJ13 -<?<,7 +=E B<CB +2J.3I,<.= (.1GV(<1=,JL #B1<3 4J1 +J +2J.3213J<= ,B<='V-? J.-+3 (1--J <J C1,,<=C 1J,+2-<JB1E /<,B + C3./<=C 1=13C5 (.=D13J<.=1GV(<1=(5 H+-?.J, 8)X N8W7 86O G.3 -+2 (1--JKL g./1D137 ,B1 4=E13J,+=E<=C .G ,B1I3.I13,<1J .G ,B1 E<GG131=, IB+J1J <J J,<-- 3+,B13 J4I13V(<+-L

#B1 T41J, G.3 <?I3.D<=C ,B1 1GV(<1=(5 .G 1=13C5 (.=D13J<.= E.1J =., (.=(13=.=-5 ,B1 +2J.32<=C -+513L Q=.,B13 J.43(1 .G -.JJ .G 1GV(<1=(5 <J ,B1 G3.=, (.=,+(,.G ,B1 E1D<(17 ?+E1 .G ,3+=JI+31=, (.=E4(,<=C .R<E1J H#*jJKL #B1J1 +31 <=J4-+,<=C.R<E1J ,B+,7 G.3 + (13,+<= 3+=C1 .G E.I<=C7 21(.?1 (.=E4(,<D1 /B<-1 31,+<=<=C,3+=JI+31=(5 <= ,B1 D<J<2-1 JI1(,34?L #B1 ?.J, (.??.= 1R+?I-1J +31 1-1(,3.=H='KE.I1E "=j$7 M=$j&7 +=E h=jL g.-1 HI'KE.I<=C .G /<E1'C+I J1?<(.=E4(,.3J/+J7 G.3 + -.=C ,<?17 D135 B+3E ,. .2,+<= N8%789OL M, <J ,B131G.31 =., J43I3<J<=C ,B+,,B1 E<J(.D135 .G I'E.I<=C <= *4Q8j$ ,B<= V-?J /<,B + (+33<13 ?.2<-<,5 .G +2.4,8) (?$aHPJK +,,3+(,1E C31+, <=,131J, N8U7 $)OL #*jJ +31 Z15 1-1?1=,J G.3 E<GG131=,+II-<(+,<.=J7 <=(-4E<=C e+, J(311=J7 /B<(B <?I-<1J ,B1 =11E G.3 + E<D13J<V(+,<.= .G,B1 .GG13 <= ,13?J .G (B1?<(+- 1-1?1=,J 4J1EL !1 /<-- +=+-5f1 B131 ,B1 1-1(,3.=<(+=E .I,<(+- I3.I13,<1J .G *4'2+J1E ,3+=JI+31=, (.=E4(,<D1 ?+,13<+-J 21-.=C<=C,. ,B1 E1-+G.JJ<,1 G+?<-57 J4(B +J *4Q-j$7 *4c+j$ N$8O7 +=E *4M=j$ N$$OL #B1-+,,13 (.?I.4=E <J I+3,<(4-+3-5 <=,131J,<=C +J <, 1RB<2<,J 2<I.-+3 H=' +=E I',5I1K(.=E4(,<D<,5 25 E.I<=C /<,B +II3.I3<+,1 <?I43<,<1J +=E ,4=<=C ,B1 V-?'E1I.J<,<.=(.=E<,<.=J N$$OL #B<J .I1=J ,B1/+5 ,. ,B1 E1D1-.I?1=, .G ,3+=JI+31=, I[= \4=(,<.=J7+=E ,B131G.31 G4--5 ,3+=JI+31=, .I,.1-1(,3.=<( E1D<(1JL

/ 4*$9)36$*"8*:.#9 ;#)2+<9 =+$ !.#8)$+"*8 !>8*)()*+"9

Q 31-<+2-1 E1J(3<I,<.= G3.? V3J, I3<=(<I-1J .G 1-1(,3.=<( 1R(<,+,<.=J <= (.?I-1R?+,13<+-J J,+3,J G3.? <E1=,<G5<=C ,B1 21J, ,B1.31,<(+- +II3.+(B1J7 /B<(B (+= J<?I-<G5<E1+--5 2.,B ,B1 G.3? .G ,B1?+=5'2.E5g+?<-,.=<+= +=E ,B1 +=+-5J<J .G ,B1 IB5J<(+-T4+=,<,<1J <=D.-D1E7 /B<-1 (.=J13D<=C ,B1 E1J<31E +((43+(5 .G 31J4-,JL

QJ <=,3.E4(1E <= *B+I,13 87 E1=J<,5 G4=(,<.=+- ,B1.35 H`0#K N$&7 $AO /+J 4=,<-31(1=,-5 ,B1 .=-5 ab initio ,..- (+I+2-1 ,. (+-(4-+,1 ,B1 1-1(,3.=<( J,+,1J .G ?+,13<+-J/<,B + -+3C1 =4?213 .G 1-1(,3.=J <= ,B1 4=<, (1--L g./1D137 J,+=E+3E `0# <J E1J<C=1E,. +((1JJ C3.4=E'J,+,1 I3.I13,<1J +=E =., 1R(<,1E J,+,1JL #B<J 31J,3<(,<.= <J =., +T41J,<.= .G +D+<-+2-1 +II3.R<?+,<.=JL M= G+(,7 1D1= <G .=1 (.4-E (+-(4-+,1 ,B1 1R+(,k.B=["B+? Hk"K 1<C1=D+-41J7 ,B1<3 E<GG131=(1J /.4-E =., =1(1JJ+3<-5 21 (-.J1,. ?1+J431E 1R(<,+,<.= 1=13C<1JL >1<,B137 25 E1V=<,<.=7 ,B15 J,+=E G.3 1-1(,3.=+EE<,<.= .3 31?.D+- 1=13C<1J N$WOL g1=(17 ,B1 G+(, ,B+, ,B1 k" C+I <J <= C1=13+-31I.3,1E ,. 21 ,.. J?+-- /<,B 31JI1(, ,. ?1+J431E C+IJ E.1J =., ,1-- 4J a priori

+=5,B<=C +2.4, ,B1 T4+-<,5 .G + (B.J1= +II3.R<?+,<.= G.3 ,B1 1R(B+=C1[(.331-+,<.=I.,1=,<+- +II1+3<=C <= ,B1 k" 1T4+,<.=JL

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

#B1 (+J1 .G ?+,13<+-J /<,B -.(+-<f1E d .3 f J,+,1J <J I+3,<(4-+3-5 E1-<(+,1L M= G+(,7,B1?.J, (.??.= +II3.R<?+,<.=J G.3 ,B1 1R(B+=C1[(.331-+,<.= G4=(,<.=+- H1LCL7 ,B1-.(+- E1=J<,5 +II3.R<?+,<.=7 Y`Q N$AO +=E C1=13+-<f1E C3+E<1=, +II3.R<?+,<.=J7ccQJ N$6OK7 /B<(B +31 Z=./= ,. 21 31-<+2-1 G.3 ,B1 C3.4=E'J,+,1 I3.I13,<1J .G sp

J1?<(.=E4(,.3J7 ?+5 21(.?1 <=+E1T4+,1 G.3 ,B1 J5J,1? /<,B d .3 f 1-1(,3.=JL M=,B1 (+J1 .G ,B1 *4'2+J1E (.?I.4=EJ ,B+, /1 +31 <=,131J,1E <=7 ,B<J I3.2-1? (+= 21,3+(1E ,. + I..3 E1J(3<I,<.= .G ,B1 B523<E<f+,<.= .G ,B1 d J,+,1J .G ,B1 ,3+=J<,<.=?1,+-/<,B ,B1 p J,+,1J .G "7 "17 +=E j (-.J1 ,. ,B1 013?< -1D1-7 -1+E<=C ,. 31?+3Z+2-5 -+3C1133.3J <= ,B1 1D+-4+,<.= .G 2.=E<=C -1=C,BJL QJ + (.=J1T41=(17 +-J. C3.4=E'J,+,1I3.I13,<1J7 J4(B +J ,B1 1T4<-<23<4? +,.?<( J,34(,431 .3 IB.=.= G31T41=(<1J7 (+= 214=J+,<JG+(,.3<-5 E1J(3<21E 25 J,+=E+3E E1=J<,5 G4=(,<.=+-JL

Q=.,B13 J,3.=C 31+J.= ,. C. 215.=E `0# +=E <,J J,+=E+3E +II3.R<?+,<.=J31?+<=J ,B1 =11E ,. (+-(4-+,1 1R(<,1E J,+,1J G.3 +II-<(+,<.=J <= ,B1 E.?+<= .G;PJ H<L1L7 ,. 1D+-4+,1 T4+J<'I+3,<(-1 1=13C<1J7 +2J.3I,<.= JI1(,3+7 1R(<,.=<( 2<=E<=C1=13C<1J7 +=E J. .=KL Q I.JJ<2-1 J.-4,<.= (.=J<J,J <= J,4E5<=C I+3,<(-1 I3.I+C+,<.=J+=E e4(,4+,<.=J <= ,B1 J5J,1?L #B<J C<D1J (.331-+,<.= G4=(,<.=J ,B+, (+= ,B1= 2131-+,1E ,. 31JI.=J1 G4=(,<.=J 5<1-E<=C7 G.3 1R+?I-17 ,B1 -<=1+3 31JI.=J1 G.3 .I,<(+-+2J.3I,<.=L #B1J1 (.331-+,<.= G4=(,<.=J +31 .=1' .3 ,/.'2.E5 c311=lJ G4=(,<.=JN$%OL #B1 .=1'2.E5 c311=lJ G4=(,<.= H,B+, (+= 1JJ1=,<+--5 21 4=E13J,..E +J + ,<?1'E1I1=E1=, I+3,<(-1 +=E B.-1 E1=J<,5 ?+,3<RK B+J IB+J1 e4(,4+,<.=J H<L1L7 I.-1J <=G31T41=(5 JI+(1K C<D1= 25 1-1(,3.= +EE<,<.= +=E 31?.D+- 1=13C<1J HT4+J<'I+3,<(-11=13C<1JK7 ?1+J431E <= IB.,.1?<JJ<.= .3 <=D13J1 IB.,.1?<JJ<.= 1RI13<?1=,JL #B1I+3,<(-1'B.-1 I+3, .G ,B1 ,/.'I+3,<(-1 c311=lJ G4=(,<.= B+J <=J,1+E I.-1J +, ,B1 1=13'C<1J .G =14,3+- 1R(<,+,<.=JL bS;# 5<1-EJ + G3+?1/.3Z /B131 .=1 (+= V=E J4<,+2-1+II3.R<?+,<.=J G.3 ,B.J1 c311=lJ G4=(,<.=JL M= I+3,<(4-+37 ,B1 GW +II3.R<?+,<.=7<=,3.E4(1E <= 8U6W 25 Y+3J g1E<= N$9O7 B+J 211= 1R,31?1-5 J4((1JJG4- <= E1J(3<2'<=C T4+J<'I+3,<(-1 1=13C<1J G.3 ?1,+-J7 J1?<(.=E4(,.3J7 +=E <=J4-+,.3J7 <= ,B1 24-Z+J /1-- +J +, J43G+(1J +=E <= (.=V=1E J,34(,431JL ]1J,3<(,1E J1-G'(.=J<J,1=, I3.'(1E431J /<,B<= ,B1 GW +II3.R<?+,<.= N$U[&8O B+D1 211= 31(1=,-5 I3.I.J1E +=E4J1E /<,B J4((1JJ G.3 ,3+=J<,<.= ?1,+- (.?I.4=EJ7 /B131 I13,432+,<D1 GW G+<-JLmJ<=C bS;#7 ,B1 I3<(1 ,. I+5 G.3 + IB5J<(+--5 <=,4<,<D1 +=E <= C1=13+- T4<,1 31-<+2-1E1J(3<I,<.= <J ,B1 B<CB (.?I4,+,<.=+- (.J, 21(+4J1 =./ c311=lJ G4=(,<.=J +II1+3<=J,1+E .G ,B1 1-1(,3.=<( E1=J<,5L M= D<1/ .G ,B+,7 ?.31 1GV(<1=, H+=E -1JJ 31-<+2-1K+II3.+(B1J7 .G,1= E1I1=E1=, .= J.?1 1?I<3<(+- I+3+?1,13J7 J4(B +J `0#+ U N&$O7`0#'(.331(,1E ?1,B.EJ N&&7&AO7 +=E B523<E G4=(,<.=+-J N&W[&%O7 B+D1 211= 1R,1='J<D1-5 1RI-.31E G.3 ,B1 I+J, 51+3J ,. (+-(4-+,1 1-1(,3.=<( 2+=E J,34(,431J /<,B +31E4(1E (.?I4,+,<.=+- 1GG.3,L

*.=(13=<=C =14,3+- 1R(<,+,<.=J7 ,B1 S1,B1["+-I1,13 1T4+,<.= HS"FK /<,B<=bS;# <J ,B1 J,+,1'.G',B1'+3, +II3.R<?+,<.= N$%7 &9[A)OL Q=.,B13 I.JJ<2-1 I+,B,. +((1JJ =14,3+- 1R(<,+,<.=J7 (-.J13 ,. ,B1 JI<3<, .G `0#7 (.=J<J,J <= +(,4+--5 1RI.J'<=C ,B1 J5J,1? ,. + ,<?1'E1I1=E1=, 1R,13=+- I.,1=,<+- +=E (+-(4-+,<=C ,B1 1D.-4,<.=.G ,B1 E1=J<,5 <= ,<?1L #B<J 3.4,1 B+J 21(.?1 +((1JJ<2-1 ,B+=ZJ ,. ,B1 1R,1=J<.= .G`0# ,. <,J ,<?1'E1I1=E1=, G.3?4-+,<.= H#``0#K NA8[AWO7 I4, .= 3<C.3.4J 2+J<J 25

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

,B1 ]4=C1[c3.JJ ,B1.31? NA&OL S5 +=+-.C5 ,. ,B1 (+J1 .G (-+JJ<(+- ?1(B+=<(J7 .=1(+= <=,4<,<D1-5 4=E13J,+=E ,B+, <= #``0# ,B1 T4+=,4? ?1(B+=<(+- n,3+\1(,.35o .G,B1 J5J,1? 4=E13 ,B1 <=e41=(1 .G + ,<?1'E1I1=E1=, 1R,13=+- I.,1=,<+- <J G.4=E 25J1+3(B<=C G.3 ,B1 1R,31?+ .G += +(,<.=7 <=J,1+E .G ,B1 ?<=<?<f+,<.= .G + ,.,+- 1=13C57+J E.=1 G.3 ,B1 C3.4=E J,+,1L j=1 .2,+<=J B1=(1 ,B1 ,<?1'E1I1=E1=, k" 1T4+,<.=J +J+ C1=13+-<f+,<.= .G ,B1 J,+,<( (+J17 +=E G3.? ,B1J17 ,B1 31JI.=J1 G4=(,<.=J E1J(3<2<=C,B1 =14,3+- 1R(<,+,<.=J .G + J5J,1? NA6OL Q, ,B<J I.<=,7 ,B1 =.=,3<D<+- E<GV(4-,5 31J<E1J<= V=E<=C +II3.I3<+,1 +II3.R<?+,<.=J G.3 ,B1 ,<?1'E1I1=E1=, 1R(B+=C1[(.331-+,<.=I.,1=,<+- vxcNnOHr7 tKL >.,1 ,B+, ,B1 G4=(,<.=+- E1I1=E1=(1 <J .= ,B1 E1=J<,5 <= ,B1/B.-1 JI+(1 +=E +, +-- I+J, ,<?1JL M= I+3,<(4-+37 J<?I-1 +E<+2+,<( +II3.R<?+,<.=J2+J1E .= J,+,<( G4=(,<.=+-J I3.D1E ,. /.3Z I+3,<(4-+3-5 2+E G.3 J.-<EJ NA%OL

#B+=ZJ ,.?+=5 31(1=, E1D1-.I?1=,J .G <?I3.D1E -<=1+3 31JI.=J1 Z13=1-J E13<D1EG3.?bS;# NA9[WWO7 #``0# <J ,.E+5 + 31+- +-,13=+,<D1 ,.bS;# G.3 ,B1 (+-(4-+,<.=.G 1-1(,3.=<( JI1(,3+7 1D1= G.3 J.-<EJ7 1JI1(<+--5 /B1= ,B1 =4?213 .G +,.?J <= ,B14=<, (1--J <J -+3C1 +=E ,B1 J.-4,<.= .G ,B1 S"F 21(.?1J 4=G1+J<2-1L

M= ,B<J J1(,<.= /1 C<D1 + 23<1G 31D<1/ .G ,B1J1 ,B1.31,<(+- +II3.+(B1JL

/ /0 ?#<*"@9 !AB()*+"9 ("< )2# GW C::$+>*5()*+"

#B1 ,<?1'.3E131E .=1'I+3,<(-1 c311=lJ G4=(,<.= <J E1V=1E +J

GH8, $K = −i〈9)|T[

pψH8K pψ†H$K]

|9)〉, H$L8K

/B131 |9)〉 <J ,B1 ?+=5'2.E5 N'I+3,<(-1 C3.4=E J,+,17 pψH8KN pψ†H8KO <J ,B1 +==<B<'-+,<.= H(31+,<.=K .I13+,.3 .G += 1-1(,3.= <= ,B1 g1<J1=213C I<(,4317 T <J ,B1 ,<?1'.3E13<=C .I13+,.37 +=E 8 = {r8, t8, σ8} J,+=EJ G.3 ,B1 J1, .G ,B1 31+- JI+(1 +=E ,<?1(..3E<=+,1J I-4J ,B1 JI<= E1C311 .G G311E.?L #B1 c311=lJ G4=(,<.= G B+J I.-1J +,,B1 G31T41=(<1J .G 1-1(,3.= +EE<,<.= +=E 31?.D+- 1=13C<1JL

#B1 ,<?1 1D.-4,<.= .G G (.331JI.=EJ ,. ,B1 I3.I+C+,<.= .G += 1-1(,3.= H.3 + B.-1K+=E .215J

∂t8+

$− V1R,

GH8, $K = δH8, $K − i

E&vH8, &KG$H8, &, $, &+K, H$L$K

/B131 v <J ,B1 2+31 *.4-.?2 I.,1=,<+-7 V1R, <J += 1R,13=+- I.,1=,<+-7 +=E ,B1 ,/.'I+3,<(-1 c311=lJ G4=(,<.= G$ E1J(3<21J ,B1 (31+,<.= +=E +==<B<-+,<.= .G I+<3J .GI+3,<(-1JL #B1 1T4+,<.= .G ?.,<.= H$L$K ,1--J 4J B./ + ?.D<=C (B+3C1E I+3,<(-1I.-+3<f1J ,B1 J5J,1?7 (31+,<=C 1-1(,3.=[B.-1 I+<3J +-.=C <,J /+5L #B+, <J ,B1 31+J.=/B5 ,B1 ,/.'I+3,<(-1 T4+=,<,5 G$ <J <=D.-D1E <= H$L$KL M= .3E13 ,. E13<D1 ,B1 1T4+,<.=J.G ?.,<.= G.3 G$7 G&7 +=E B<CB13 .3E13 c311=lJ G4=(,<.=J ?4J, 21 (.=J<E131E+=+-.C.4J-5L #B1 B<13+3(B5 .G 1T4+,<.=J (.=,+<=<=C c311=lJ G4=(,<.=J .G <=(31+J<=C.3E13 (+= 21 1D+-4+,1E 4J<=C E<+C3+??+,<( 1RI+=J<.=J 2+J1E .= !<(ZlJ ,B1.31?LQ-,13=+,<D1-57 .=1 (+= 4J1 "(B/<=C13lJ G4=(,<.=+- E13<D+,<D1J ,. .2,+<= + J1, .G1T4+,<.=J G.3 G ,B+, (+= 21 G43,B13 +II3.R<?+,1EL g1317 /1 G.--./ ,B1 J1(.=E -<=1

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

,. C<D1 + 23<1G J4??+35 .G ,B1 GW G.3?+-<J?7 E1D1-.I1E <= <,J I31J1=, G.3? 25Y+3J g1E<= N$9OL

FD1= ,B.4CB G <J J,<-- 4=Z=./=7 H$L$K ,1--J 4J ,B+, <,J 1D.-4,<.= <=D.-D1J ,B1I.-+3<f+,<.= .G ,B1 J5J,1? H<L1L7 G$KL #B1 IB5J<(+- <E1+ +, ,B1 B1+3, .G "(B/<=C13lJG.3?+-<J? <J ,B+, I.-+3<f+,<.= (+= 21 (31+,1E +J + 31JI.=J1 ,. + ,<?1'E1I1=E1=,I13,432<=C 1R,13=+- I.,1=,<+- U. M= ,B1 JI<3<, .G ,B1 -<=1+3'31JI.=J1 ,B1.357 ,B1,<?1'E1I1=E1=, I+3, .G ,B1 I.,1=,<+- /<-- 21 ?+E1 ,. D+=<JB +, ,B1 1=E .G ,B1E13<D+,<.=7 J. ,B+, U → V1R,Hr K +, ± ∞L #. +II-5 ,B<J <E1+7 ,B1 .=1'I+3,<(-1 ,13? .G,B1 g+?<-,.=<+= <= H$L$K7 =+?1-57 h) = −∇$/$ + V1R,7 21(.?1J h) = −∇$/$ + U.

M, <J I.JJ<2-1 ,. JB./ ,B+, ,B1 D+3<+,<.= .G G /<,B 31JI1(, ,. U C<D1J

δGH8, $K

δU H&K= GH8, $KGH&, &+K − G$H8, &, $, &

+K. H$L&K

mJ<=C H$L&K7 ,B1 1T4+,<.= .G ?.,<.= H$L$K 31+EJ[

∂t8− h)H8K + i

E&vH8, &KGH&, &+K

GH8, $K

= δH8, $K + i

E&vH8+, &KδGH8, $K

δU H&K, H$LAK

/B131 ,B1 ,/.'I+3,<(-1 c311=lJ G4=(,<.= E.1J =., +II1+3 +=5?.31L QJ−iGH&7&+ K=ρH&K7 /B131 ρ <J ,B1 1-1(,3.=<( E1=J<,57 ,B1 ,13? −i∫ E&vH8, &KGH&, &+K <J =.,B<=C?.31 ,B+= ,B1g+3,311 I.,1=,<+-VgH8KL #B<J?1+=J ,B+, ,B1 1GG1(,J .G ,B1 I13,432+,<.=U .= ,B1 J5J,1? (+= 21 JI-<, <=,. + (-+JJ<(+- (.=,3<24,<.= H<L1L7 ,B1 g+3,311 ,13?K +=E,B1 (.=,3<24,<.= ,B+, 1=(.?I+JJ1J T4+=,4? 1GG1(,JL #B1 -+,,13 ,13? (+= 21 31/3<,,1=25 E1V=<=C ,B1 J1-G'1=13C5 6:

E&vH8+, &KδGH8, $K

δU H&K=

E&6H8, &KGH&, $K. H$LWK

#B1 J1-G'1=13C5 ,B131G.31 I-+5J ,B1 3.-1 .G + =.=-.(+- +=E E5=+?<(+- I.,1=,<+- ,B+,+((.4=,J G.3 +-- ,B1 1GG1(,J .G 1R(B+=C1 +=E (.331-+,<.= <= ,B1 J5J,1?L ;B5J<(+--57<, 31I31J1=,J ,B1 1GG1(,<D1 I.,1=,<+- ,B+, += 1R,3+ G13?<.= <= ,B1 J5J,1? G11-J G.3 ,B1I.-+3<f+,<.= ,B+, <,J I3.I+C+,<.= <=E4(1J E41 ,. 1R(B+=C1 1GG1(,JL mJ<=C H$LWK <=H$L$K .=1 .2,+<=J

∂t8− h)H8K − VgH8K

GH8, $K = δH8, $K +

E&6H8, &KGH&, $K. H$L6K

"1,,<=C 6 = ) <= H$L6K7 <, <J I.JJ<2-1 ,. E1V=1 ,B1 g+3,311 c311=lJ G4=(,<.=Gg7 /B<(B <J ,B1 31J.-D1=, .G ,B1 g+3,311 g+?<-,.=<+= (+-(4-+,1E /<,B ,B1 1R+(,1-1(,3.=<( E1=J<,5 ρ:

∂t8− h)H8K − VgH8K

GgH8, $K = δH8, $K. H$L%K

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

*.?2<=<=C H$L6K +=E H$L%K7 .=1 V=EJ ,B1 `5J.= 1T4+,<.= G.3 G:

GH8, $K = GgH8, $K +

E&AGgH8, &K6H&, AKGHA, $K. H$L9K

#B1 J1-G'1=13C5 <J 31JI.=J<2-1 G.3 ,B1 31=.3?+-<f+,<.= 1GG1(, ,B+, 5<1-EJ ,B1E<GG131=(1 21,/11= ,B1 I3.I+C+,<.= .G += <=E1I1=E1=, Hg+3,311K I+3,<(-1 +=E ,B1I3.I+C+,<.= .G += <=,13+(,<=C G13?<.=<( I+3,<(-1L

03.? H$LWK .=1 (+= ,B1= 1R,3+(, /<,B + -<,,-1 +-C123+ + E1V=<,<.= .G 6 .G I3+(,<(+-4J1:

6H8, $K = i

E&AGH8, AKŴHA, $q WKδV HWK

δU H&KvH&, 8+K, H$LUK

/B131 V H8K = U H8K + VgH8K <J ,B1 ,.,+- (-+JJ<(+- I.,1=,<+-7 +=E /B131 ,B1 <331'E4(<2-1 D13,1R G4=(,<.= B+J 211= <=,3.E4(1E:

ŴH8, $q &K = −δG−8H8, $K

δV H&K= δH8, $KδH8, &K +

δ6H8, $K

δV H&K. H$L8)K

#B1 E13<D+,<D1 δ6/δV <J ,B1= 4J4+--5 31I-+(1E 25 ,B1 (B+<= 34-1 Hδ6/δGK HδG/δV K7+=E H$L8)K <J ,3+=JG.3?1E <=,. += <=,1C3+- 1T4+,<.=:

ŴH8, $q &K = δH8, $KδH8, &K +

EAW6%δ6H8, $K

δGHA, WKGHA, 6K GH%, WKŴH6, %q &K.

H$L88K

0.--./<=C += +=+-.C.4J I3.(1E4317 .=1 +-J. .2,+<=J ,B1 1T4+,<.=J

6H8, $K = i

E&AGH8, AKŴHA, $q &KW H&, 8+K, H$L8$K

W H8, $K = vH8, $K +

E&AvH8, &KPH&, AKW HA, $K, H$L8&K

PH8, $K = −i

E&AGH$, &KGHA, $KŴH&, Aq 8K, H$L8AK

<= ,13?J .G ,B1 ,<?1'.3E131E I.-+3<f+,<.= .I13+,.3 PH87 $K +=E ,B1 E5=+?<(+-J(311=1E <=,13+(,<.= W H8, $K = δV H8K/δU H&KvH&, $K = ǫ−8H8, &KvH&, $KL FT4+,<.=JH$L9K7 H$L88K7 H$L8$K7 H$L8&K7 +=E H$L8AK G.3? ,B1 (-.J1E J1, .G VD1 1T4+,<.=J Z=./=+J g1E<=lJ 1T4+,<.=JL #B15 E1I1=E .= .=1 +=.,B13 +=E JB.4-E 21 J.-D1E <,13+,<D1-5G.--./<=C ,B1 J(B1?1 .G 0<C431 $L$L

#B1 nG+?.4Jo GW +II3.R<?+,<.= <J .2,+<=1E 25 E<J31C+3E<=C ,B1 J1(.=E ,13?.= ,B1 3<CB,'B+=E J<E1 .G H$L88K +=E <=J13,<=C Ŵ <= H$L8$K:

ŴH8, $q &K = δH8, $KδH8, &K, H$L8WK

6H8, $K = iGH8, $KW H$, 8K. H$L86K

c.<=C 215.=E ,B1 GW +II3.R<?+,<.= 31T4<31J ,B1 <=(-4J<.= .G D13,1R (.331(',<.=J7 25 J.-D<=C H$L88K /<,B <,J G.43'I.<=, Z13=1- E.?<=+,1E 25 δ6H8, $K/δGH&, AKL

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

Σ = G

G = G 0 + G 0

P = GGΓ

(δΣ/δ

Figure 2.2 Hedin’s pentagon connecting the Green’s function G, the self-energy 6, the vertexŴ, the polarizability P, and the screened Coulomb interaction W.

M=E11E7 H$L88K <J ,B1 ?+<= .2J,+(-1 ,. ,B1 (+-(4-+,<.= .G I.-+3<f+2<-<,<1J +=EJ1-G'1=13C<1J 215.=E ,B1 3+=E.? IB+J1 +II3.R<?+,<.= H];QK +=E ,B1 GW

+II3.R<?+,<.=L#B1 ?+<= I-4J .G ,B1 GW +II3.+(B <J ,B+, I.-+3<f+,<.= 1GG1(,J7 /B<(B J(311=

,B1 I3.I+C+,<.= .G += 1R,3+ I+3,<(-17 +31 1RI-<(<,-5 ,+Z1= <=,. +((.4=, ,B3.4CB +E5=+?<(+--5 J(311=1E J1-G'1=13C5L M=J,1+E7 <= g+3,311[0.(Z +II3.+(B ,B131 <J =.I.-+3<f+,<.= +=E =. 31-+R+,<.= .G ,B1 J5J,1? +=E ,B1 J1-G'1=13C5 <J J,+,<(L FD1= <GD13,1R (.331(,<.=J +31 =1C-1(,1E +=E ,B1 J1-G'1=13C5 <J 1D+-4+,1E /<,B<= ,B1 GW

+II3.R<?+,<.=7 ,B1 `5J.= 1T4+,<.= H$L9K <J J,<-- + J1-G'(.=J<J,1=, 1T4+,<.= <= G +=EJB.4-E 21 J.-D1E <,13+,<D1-5L M= ,B<J J1=J1 ,B1 J,+3,<=C I.<=, <J +32<,3+35L M= G+(,7 <,<J I.JJ<2-1 ,. J,+3, /<,B c311=lJ G4=(,<.=J .G +=5 =.=<=,13+(,<=C J5J,1? +=E <,13+,1J1-G'(.=J<J,1=,-5 HZ11I<=C ,B1 D13,1R Ŵ = 8KL

FD1= <G g1E<= .2,+<=1E ,B1 GW +II3.R<?+,<.= 25 <,13+,<=C g1E<=lJ 1T4+,<.=JJ,+3,<=C G3.? 6 = ) H<L1L7 G3.? ,B1 g+3,311 c311=lJ G4=(,<.=K7 ?.E13= (+-(4-+,<.=JJ,+3,<=C G3.? ,B1 /.3ZJ .G ",3<=+,< et alL NW6O7 g5213,J1= +=E Y.4<1 NW%7 W9O +=Ec.E25 et alL N$W7 WUO 4J1 3+,B13 + n21J, G7 21J, W o +II3.+(B N6)OL #B<J ?1+=J,B+, ,B1 <E1+ .G + J,3<(, <,13+,<D1 J.-4,<.= .G g1E<=lJ 1T4+,<.=J <J E<J?<JJ1E7 +=E,B1 GW J1-G'1=13C5 <J (.=J,34(,1E <=J,1+E 4J<=C ,B1 21J, ?1+='V1-E 31J4-,J ,B+,+31 +D+<-+2-1L #B1 J<?I-1J, (B.<(1 <J <= C1=13+- ,. 4J1 ,B1 k" 1<C1=D+-41J +=E1<C1=G4=(,<.=J (.?<=C G3.? + `0# (+-(4-+,<.= /<,B<= Y`Q .3 ccQL !1 31G13 ,.,B<J I3.(1E431 +J I13,432+,<D1 GW.

M, <J (-1+3 ,B+, + I13,432+,<D1 +II3.+(B <J \4J,<V1E /B1= ,B1 E1I+3,431 /+D1'G4=(,<.=J +31 +-31+E5 (-.J1 ,. ,B1 T4+J<'I+3,<(-1 .=1JL #B<J <J <=E11E ,B1 (+J1 G.3?+=5 J5J,1?J7 +=E ,B131G.31 I13,432+,<D1 GW B+J 211= 1R,31?1-5 J4((1JJG4- <=E1J(3<2<=C 1-1(,3.= +EE<,<.= +=E 31?.D+- 1=13C<1J G.3 ?1,+-J7 J1?<(.=E4(,.3J7 +=E

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

<=J4-+,.3J N68OL g./1D137 I13,432+,<D1 GW B+J +-J. I3.D1E ,. 21 <=J4GV(<1=, G.3?+=5 ,3+=J<,<.= ?1,+- (.?I.4=EJ N8A7 $U7 &87 6$[6WOL #B1 I3.2-1? <= ?+=5 (+J1J<J ,B+, ,B1 Y`Q/+D1G4=(,<.=J +31 =., -.(+-<f1E 1=.4CB7 -1+E<=C 1D1= ,. += +3,<V(<+-?1,+--<( (B+3+(,13 <= ?+,13<+-J ,B+, +31 <=J4-+,.3JL ;13,432+,<D1 GW ?+5 +-31+E5<?I3.D1 4I.= ,B<J J<,4+,<.= 25 .I1=<=C ,B1 C+I7 24, 4J4+--5 25 += <=J4GV(<1=,+?.4=,L

"1D13+- ?1,B.EJ B+D1 +II1+31E ,. J.-D1 ,B<J I3.2-1?L #B15 G.--./ ,/. ?+<=-<=1J:

8L Replacing the simple LDA with a better starting point and then performing

a single-shot GW calculation: "1D13+- +,,1?I,J ,. <?I3.D1 ,B1 J,+3,<=C I.<=,/<,B 31JI1(, ,. Y`Q (+= 21 G.4=E <= ,B1 -<,13+,4317 -<Z1 4J<=C ccQ N66O7 1R+(,1R(B+=C1 N6%O7 Y`QaccQ+ U N697 6UO7 .3 B523<E G4=(,<.=+-J N%)OL

$L Performing (restricted) self-consistent (sc) GW N$U[&8O: #B1 ,1(B=<T41 I3.I.J1E25 0+-11D et alL N&)O <J (+--1E T4+J<'I+3,<(-1 J(GW +=E B+J ,B1 +ED+=,+C1 .G 21<=C<=E1I1=E1=, .G ,B1 J,+3,<=C I.<=, H.2D<.4J-5 +, ,B1 I3<(1 .G + -+3C13 (.?I4,+,<.=+-(.?I-1R<,5KL

#B131 <J += +-,13=+,<D1 J1-G'(.=J<J,1=, I3.(1E431 ,B+, 5<1-EJ /+D1G4=(,<.=J1R,31?1-5 (-.J1 ,. ,B.J1 .2,+<=1E <= + T4+J<'I+3,<(-1 J(GW (+-(4-+,<.= +, + 31E4(1E(.?I4,+,<.=+- (.J,L #B<J ?1,B.E <J J(*jg"Fr7 +J 1RI-+<=1E 25 S34=1D+- et alLN$UOL #B1 E5=+?<(+- 1GG1(,J ,B+, +31 +2J1=, <= *jg"Fr (+-(4-+,<.=J (+= ,B1= 21+((.4=,1E G.3 25 I13G.3?<=C + V=+- I13,432+,<D1 GW J,1I J,+3,<=C G3.? ,B1 J(*jg'"Fr 31J4-,L J(*jg"Fr+ G)W) B+J +-31+E5 211= +II-<1E ,. ?+=5 ,3+=J<,<.= ?1,+-(.?I.4=EJ7 5<1-E<=C 1R(1--1=, 31J4-,J G.3 ,B1 2+=EC+IJ +=E ,B1 T4+J<'I+3,<(-1 2+=EJ,34(,431 N8A7 $U7 6$[6AOL >.,1 ,B+, /B1=1D13 (.?I+31E7 T4+J<'I+3,<(-1 J(GW +=EJ(*jg"Fr+ G)W) C+D1 D135 J<?<-+3 2+=EC+IJ +=E /+D1G4=(,<.=JL

/ / ?&D$*< 4B"8)*+"(.9

FD1= <= <,J I13,432+,<D1 G.3?7 GW <J 25 +-- ?1+J431J += 1RI1=J<D1 ,1(B=<T417 /<,B+ D135 4=G+D.3+2-1 J(+-<=C /<,B ,B1 =4?213 .G +,.?J <= ,B1 4=<, (1--L M, <J ,B131G.314=I3+(,<(+- G.3 ,B1 J,4E5 .G 2+=E J,34(,431J .G -+3C1 J5J,1?J +=E I3.B<2<,<D1 ,..2,+<= ,.,+- 1=13C<1J 1D1= G.3 J<?I-1 J5J,1?JL b4(B .G ,B1 (.?I4,+,<.=+- 1GG.3, <=GW (.?1J G3.? ,B1 E5=+?<(+--5 J(311=1E *.4-.?2 <=,13+(,<.= W. M, <J ,B131G.31<=,131J,<=C ,. 1RI-.31 ,. /B<(B 1R,1=, E5=+?<(+- 1GG1(,J +31 ?+=E+,.357 .3 /B1,B13=.=-.(+-<,5 <J ,B1 E.?<=+,<=C (B+3+(,13<J,<(L #B1 ?.D1 G3.? -.(+- k" I.,1=,<+-J ,.=.=-.(+- G4=(,<.=+-J B+J V3J, 211= I4JB1E G.3/+3E <= T4+=,4? (B1?<J,357 /B131,.E+5 ,B1 J.'(+--1E B523<E G4=(,<.=+-J +31 D135 I.I4-+3L #B1J1 G4=(,<.=+-J ?<R +G3+(,<.= α .G 0.(Z 1R(B+=C1 /<,B + (.?2<=+,<.= .G Y`Q +=E ccQ G4=(,<.=+-JL #B1+II-<(+,<.= .G B523<E G4=(,<.=+-J ,. ,B1 J.-<E J,+,1 B+E + J-./13 J,+3, N&W7 &6OL #B1J<,4+,<.= (B+=C1E 31(1=,-57 B1-I1E 25 ,B1 /<E13 +D+<-+2<-<,5 .G (.?I4,13 (.E1J ,B+,J4II.3, B523<EJ N%87 %$O +=E ,B1 J,1+E5 <=(31+J1 .G (.?I4,+,<.=+- I./13 (.D13<=C,B1 +EE<,<.=+- (.J, /<,B 31JI1(, ,. + -.(+- I.,1=,<+-L S1J<E1J 5<1-E<=C C..E J,34(,43+-

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

I3.I13,<1J N%&O7 B523<EJ B+D1 I3.D1E ,. (.331(, ,. + -+3C1 1R,1=, ,B1 2+=EC+I I3.2-1?N&W7 %AOL Q=.,B13 -+=E?+3Z (+?1/<,B ,B1 <=,3.E4(,<.= .G J(311=1E B523<EJ N%W7%6OL#B1J1 G4=(,<.=+-J -1+E ,. G+J,13 (+-(4-+,<.=J +=E <?I3.D1E 2+=EC+IJ7 1JI1(<+--5 G.3J?+-- 2+=EC+I J5J,1?JL 043,B13?.317 25 J(311=<=C ,B1 *.4-.?2 <=,13+(,<.= +, -+3C1E<J,+=(1J7 ,B15 +-J. C<D1 +((1JJ ,. ?1,+-JL

#B1 <=,4<,<.= -5<=C 21B<=E + B523<E G4=(,<.=+- <J 3+,B13 (-1+3L !B<-1 Y`Q .3ccQ (+-(4-+,<.=J J,3.=C-5 4=E131J,<?+,1 ,B1 C+I7 g+3,311[0.(Z (+-(4-+,<.=J .D13'1J,<?+,1 <, ,5I<(+--5 25 ?.31 ,B+= + G+(,.3 .G ,/.L S5 (B+=C<=C ,B1 ?<R<=C α G3.?) ,. 87 .=1 B+J + (.=,<=4.4J (B+=C1 21,/11= -.(+- k" +=E g+3,311[0.(Z7 +=E +=1JJ1=,<+--5 -<=1+3 D+3<+,<.= 21,/11= ,B1 31JI1(,<D1 C+IJL #B131G.317 ,. .2,+<= ,B11RI13<?1=,+- C+I7 .=1 J<?I-5 B+J ,. 4J1 ,B1 +II3.I3<+,1 ?<R<=C I+3+?1,13L #B<JD+-41 (+= 21 E1,13?<=1E G3.? + V, ,. + J13<1J .G J5J,1?J7 +=E <J .G,1= J1, ,. +3.4=Eα ∼ ).$ − ).&. #B<J (B.<(1 C<D1J D135 C..E 31J4-,J G.3 + -+3C1 (-+JJ .G J5J,1?J724, <, 4J4+--5 G+<-J /B1= ,B1 C+I <J D135 -+3C1 .3 D135 J?+--L #B<J G+(, <J 1+J5 ,.4=E13J,+=E <G /1 ?.D1 +/+5 G3.? + C1=13+-<f1E k" I<(,431 N&%O +=E (.=J<E13 ,B1B523<E +J += +II3.R<?+,<.= ,. ,B1 J1-G'1=13C5 6L M= ,B1 GW +II3.R<?+,<.=7 ,B1-+,,13 (+= 21 /3<,,1= +J

Hr, r ′qωK =∑

JrHr, r ′K +∑

31J,Hr ,r ′qωK, H$L8%K

/<,B∑

JrHr, r ′K 21<=C ,B1 J,+,<(+--5 J(311=1E'1R(B+=C1 HJrK ,13?7 +=E∑

31J,Hr, r ′qωK(.=,+<=<=C ,B1 J,+,<( *.4-.?2 B.-1 +=E E5=+?<(+- (.=,3<24,<.=JL MG ,B1 J(311=<=C<= ,B1 Jr ,13? <J 31I-+(1E 25 += 1GG1(,<D1 J,+,<( E<1-1(,3<( (.=J,+=, ǫ∞ = 8/α7 +=E∑

31J, <J ?.E1-1E 25 ,B1 J,+,<( +=E -.(+- I+3, .G ,B1 B523<E G4=(,<.=+- N%%O7 ,B1T4+J<'I+3,<(-1 1T4+,<.= B+J ,B1 J+?1 G.3? +J ,B1 C1=13+-<f1E k" 1T4+,<.= J.-D1EG.3 B523<E G4=(,<.=+-JL >.,1 ,B+, ,B<J <J J<?<-+3 ,. ,B1 Jr Y`Q ?1,B.E N&%7 %97%UO7 /B131 6 <J 31I-+(1E 25 + #B.?+J[013?<'J(311=1E 1R(B+=C1 I+3, +=E ,B1 Y`QR( G4=(,<.=+-L 03.? ,B1J1 +3C4?1=,J /1 (+= (.=(-4E1 ,B+, ,B1 IB5J<(+- D+-41 G.3,B1 ?<R<=C I+3+?1,13 α <J 31-+,1E ,. ,B1 <=D13J1 .G ,B1 E<1-1(,3<( (.=J,+=, .G ,B1?+,13<+- +, B+=E N8A7 9)7 98OL

Q, I31J1=,7 B523<E G4=(,<.=+- (+-(4-+,<.=J +31 ?.31 +=E ?.31 (.??.= <= ,B1'.31,<(+- J,4E<1J G.3 ;PJ +J ,B15 .GG13 + D135 C..E (.?I3.?<J1 21,/11= +((43+(5+=E 1GV(<1=(5L M= I+3,<(4-+37 2+=EC+IJ .G ,B1 .3E13 .G 8[$ 1P +31 <=(-4E1E <= ,B1<E1+- /<=E./ N98O G.3 31-<+2-1 2+=EC+I (+-(4-+,<.=J 4J<=C J(311=1E B523<EJ7 J4(B+J g"F)6L g./1D137 + ?+\.3 I3.2-1? J,<-- 31?+<=J7 31-+,1E ,. ,B1 4J1 .G .=1J<=C-1 ?<R<=C I+3+?1,13 G.3 +-- ?+,13<+-J7 31J4-,<=C <= J<C=<V(+=, 133.3J .G ,3+E<',<.=+- B523<E G4=(,<.=+-J G.3 -+3C1 +=E J?+-- C+I ?+,13<+-JL >1D13,B1-1JJ7 B523<EG4=(,<.=+-J 31I31J1=, G.3 ,B1 ,<?1 21<=C ,B1 .=-5 I.JJ<2-1 /+5 ,. J,4E5 1-1(,3.=<(1R(<,+,<.=J <= J5J,1?J /<,B -+3C1 4=<, (1--J HJ4(B +J J4I13(1--J (.=,+<=<=C B4='E31EJ .G +,.?J ,. E1J(3<21 E1G1(,JK7 /B<(B 1RI-+<= ,B1 <=,131J, G.3 V=E<=C <?I3.D1EG4=(,<.=+-J N98OL

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

/ /E F#)2#G-(.:#)#$ !AB()*+"

S+=E J,34(,431J +=E IB.,.1?<JJ<.= 1RI13<?1=,J +31 2+J1E .= + E1J(3<I,<.= <= ,13?J.G + .=1'I+3,<(-1 1GG1(,<D1 g+?<-,.=<+= G.3 .=1'I+3,<(-1 1R(<,+,<.=J H<L1L7 T4+J<'I+3,<(-1JKL M= (+J1 .G +2J.3I,<.= 1RI13<?1=,J7 <=J,1+E7 + .=1'I+3,<(-1 E1J(3<I,<.=<J =. -.=C13 +E1T4+,17 J<=(1 .=1 B+J ,. E1+- /<,B ,B1 J<?4-,+=1.4J (31+,<.= .G+ T4+J<'1-1(,3.= +=E + T4+J<'B.-1 ,B+, <=,13+(, <= ,B1 J5J,1?L M= ,B<J J1(,<.=7 /1,B131G.31 <=,3.E4(1 ,B1 1GG1(,<D1 ,/.'I+3,<(-1 g+?<-,.=<+= (+I+2-1 .G E1+-<=C /<,B,B1J1 =14,3+- 1R(<,+,<.=JL

#B1 S"F 4J1J ,B1 <=,4<,<D1 J(B1?1 .G bS;#7 /B<(B ?+Z1J ,B1 ,+JZ ,. <E1=',<G5 1GV(<1=, +II3.R<?+,<.=J 1+J<13L #B1 Z15 T4+=,<,5 <J ,B1 G.43'I.<=, 31E4(<2-1I.-+3<f+2<-<,5 L7 /B<(B (+= 21 1RI31JJ1E <= ,13?J .G ,/.'I+3,<(-1 c311=lJ G4=(,<.=G$ E1J(3<2<=C ,B1 I3.I+C+,<.= .G ,/. I+3,<(-1J HG.3 +2J.3I,<.=7 ,B1 31-1D+=, I+3,E1J(3<21J ,B1 I3.I+C+,<.= .G += 1-1(,3.= +=E + B.-1K:

LH8, $, &, AK = L)H8, $, &, AK − G$H8, $, &, AK, H$L89K

/B131 L) <J ,B1 E<J(.==1(,1E I+3, (.=J<J,<=C .G ,/. .=1'I+3,<(-1 c311=lJ G4=(,<.=JG:

L)H8, $, &, AK = iGH8, &KGHA, $K. H$L8UK

#B1 G4=(,<.= L J+,<JV1J + `5J.='-<Z1 J(311=<=C 1T4+,<.=7 Z=./= +J ,B1 S1,B1["+-I1,13 1T4+,<.=:

LH8, $, &, AK = L)H8, $, &, AK +

EW6%9 L)H8, $, W, 6KNυHW, %KδHW, 6KδH%, 9K

+ 4HW, 6, %, 9KOLH%, 9, &, AK. H$L$)K

M= .3E13 ,. J.-D1 H$L$)K7 .=1 B+J ,. +II3.R<?+,1 ,B1?+=5'2.E5 <=,13+(,<.= Z13=1-4L #B1 J,+=E+3E +II3.R<?+,<.= (.=J<J,J <= 4J<=C ,B1 GW J1-G'1=13C5 .G H$L86KL M=,B<J /+5 ,B1 J.-4,<.= .G ,B1 S"F (.331JI.=EJ ,. ,B1 <=(-4J<.= .G D13,1R (.331(,<.=J<= P ,B3.4CB + J1(.=E <,13+,<.= .G g1E<=lJ I1=,+C.= .G 1T4+,<.=JL !B1= ,B1 S"Z13=1- <J +II3.R<?+,1E ,. V3J, .3E13 <= ,B1 J(311=1E *.4-.?2 <=,13+(,<.= W7 ,B1S"F 31+EJ

L = L) + L)HAυ −A W KL . H$L$8K

M= H$L$)K7 /1 B+D1 E1V=1E ,B1 G.43'I.<=, 1R,1=J<.= .G ,B1 *.4-.?2 I.,1=',<+- AυH8, $, &, AK = δH8, $KδH&, AKυH8, &K7 /B131+J AW = δH8, &KδH$, AKW H8, $K <J,B1 G.43'I.<=, 1R,1=J<.= .G ,B1 J(311=1E *.4-.?2 I.,1=,<+-L #B1 V3J, ,13?7 ,B14=J(311=1E 1-1(,3.=[B.-1 1R(B+=C17 <J 31I4-J<D17 /B<-1 ,B1 J1(.=E ,13?7 ,B1J(311=1E 1-1(,3.=[B.-1 *.4-.?2 <=,13+(,<.=7 <J +,,3+(,<D1L >.,1 ,B+, ,B1 δ'G4=(,<.=J(.==1(, E<GG131=, <=E<(1J <= ,B1 ,/. (+J1J: ,B<J <J 21(+4J1 ,B1 G.3?13 J,1?J G3.?D+3<+,<.=J .G ,B1 g+3,311 I.,1=,<+-7 /B131+J ,B1 -+,,13 <J E41 ,. D+3<+,<.=J .G +=1R(B+=C1'-<Z1 J1-G'1=13C5 (.=,3<24,<.=L S1(+4J1 .G ,B1 /+5 ,B1 <=E<(1J +31 (.='=1(,1E <= ,B<J J1(.=E (+J17 ,B1 S"F (+==., 21 /3<,,1= <= + ,/.'I.<=, G.3?L #B<J

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

<JJ41 31I31J1=,J ,B1 31+- 2.,,-1=1(Z <= I3+(,<(+- (+-(4-+,<.=JL M=J,1+E7 ,B1?1+J43+2-131JI.=J1 G4=(,<.= <J .2,+<=1E D<+ + ,/.'I.<=, (.=,3+(,<.= .G L7 =+?1-57

P 31EH8, $K = −LH8, 8, $, $K. H$L$$K

03.? ,B1 31E4(1E I.-+3<f+2<-<,5 P31E ,B+, <J 4=E13J,..E ,. 21 ,<?1' H.3(.=,.43'K .3E131E7 ,B1 (+4J+- 31JI.=J1 G4=(,<.= χ (+= 21 <=G1331Eq ,B1 31-+,<.=21,/11= ,B1 ,<?1'.3E131E +=E (+4J+- 31JI.=J1J <J <= G+(, χ Hω > )K = P 31EHω > )KLQ=.,B13 4J1G4- T4+=,<,5 <J ,B1 <=E1I1=E1=, T4+J<'I+3,<(-1 I.-+3<f+2<-<,57 P)H8, $K =

−L)H8, 8, $, $KL >.,1 ,B+, ,B<J <J =., 1T4+- ,. χk" = −<Gk"Gk". #B1 1RI31JJ<.=-..ZJ J<?<-+37 24, ,B1 k" J,+,1J +=E 1<C1=D+-41J <= χk" +31 =./ 31I-+(1E 25 ,B1<3bS;# (.4=,13I+3,J7 /B<(B +317 +, ,B<J -1D1- .G +II3.R<?+,<.=7 E1,13?<=1E <= ,B1T4+J<'I+3,<(-1 GW +II3.R<?+,<.= N$9OL #. 21 I31(<J17 I3+(,<(+- (+-(4-+,<.=J 4J1GW 1<C1=D+-41J +=E k" /+D1G4=(,<.=J ,. 24<-E G.

0.3 J<?I-1 J1?<(.=E4(,.3J7 E5=+?<(+- 1GG1(,J <= ,B1 1-1(,3.=[B.-1 J(311=<=C<= W +=E <= ,B1 c311=lJ G4=(,<.= G ,1=E ,. (+=(1-q ,B131G.317 ,B15 +31 4J4+--5=1C-1(,1E <= 2.,B ,13?JL #B<J ?1+=J ,B+, <=J,1+E .G W H8, $K .=1 4J1J + J,+,<(+--5J(311=1E <=J,+=,+=1.4J <=,13+(,<.= +=E G.3 G ,B1 c311=lJ G4=(,<.= .2,+<=1E 25 +I13,432+,<D1 .3 J(GW (+-(4-+,<.=L M= I3+(,<(17 ,B1 S"F H$L$)K <J J.-D1E 25 E<+C'.=+-<f<=C + ,/.'I+3,<(-1 1R(<,.=<( g+?<-,.=<+=7 I3.D<E<=C <=G.3?+,<.= +2.4, ,B11R(<,.=<( 1<C1=J,+,1J +=E 1<C1=D+-41JL #B1 S"F 5<1-EJ 1R(1--1=, 31J4-,J G.3 ,B1.I,<(+- I3.I13,<1J .G J.-<EJ /<,B 2.,B (.=,<=44? +=E 2.4=E 1R(<,.=J7 <= 24-Z +=E=+=.J,34(,431E J5J,1?J N&UOL

/ /H ;+<#. I#$"#.9 =+$ 1JJ41

!B1=1D13 ,B1 J.-4,<.= .G ,B1 S"F <J 4=G1+J<2-17 <, <J B./1D13 I.JJ<2-1 ,. <E1=',<G5 1GV(<1=, /+5J ,. (+-(4-+,1 +2J.3I,<.= JI1(,3+7 C.<=C 215.=E ,B1 J<?I-1 J4?.D13 <=E1I1=E1=, ,3+=J<,<.=J 21,/11= k" J,+,1JL Q= +II1+-<=C <E1+ <J ,. 4J1 ,B1,<?1'E1I1=E1=, H#`K C1=13+-<f+,<.= .G `0#7 #``0# NA&7 9$O7 /B.J1 (1=,3+- T4+=',<,5 <J ,B1 1-1(,3.=<( E1=J<,5 ,B+, 31JI.=EJ ,. + ,<?1'E1I1=E1=, 1R,13=+- V1-EL#``0# 31,+<=J +-- ,B1 +ED+=,+C1J .G ,B1 `0# G.3?+-<J? <= ,13?J .G (.?I4,+',<.=+- 1GV(<1=(5L g./1D137 ,B1 -+(Z .G C..E +II3.R<?+,<.=J G.3 ,B1 ,<?1'E1I1=E1=,1R(B+=C1[(.331-+,<.= HR(K G4=(,<.=+-7 1JI1(<+--5 <= ,B1 (+J1 .G J.-<EJ N&U7 A%O7 <J<,J I3<=(<I+- -<?<,+,<.=L

#B1 1R(B+=C1[(.331-+,<.= Z13=1- fr* .G #``0#7 /B<(B <J ,B1 D+3<+,<.= .G ,B11R(B+=C1[(.331-+,<.= I.,1=,<+- /<,B 31JI1(, ,. ,B1 1-1(,3.=<( E1=J<,57 <J ?.J, .G,1=+II3.R<?+,1E 25 H<K fr* = )7 -1+E<=C ,. ,B1 ];Q7 .3 H<<K fr* = δυY`Q

r* /δρ7 /B<(BC<D1J ,B1 +E<+2+,<( -.(+- E1=J<,5 +II3.R<?+,<.= HQY`QK NAW7 9$OL >.,1 ,B+, ,B1];Q E<GG13J G3.? += <=E1I1=E1=,'I+3,<(-1 (+-(4-+,<.=7 1D1= <G ,B1 R( Z13=1- <J f13.721(+4J1 <, +((.4=,J G.3 D+3<+,<.=J .G ,B1 g+3,311 I.,1=,<+- 4I.= 1R(<,+,<.=L #B<J<J + (-+JJ<(+- (.=,3<24,<.=7 /B<(B <= ,B1 .I,<(+- JI1(,3+ (.331JI.=EJ ,. -.(+- V1-E.3 E1I.-+3<f+,<.= 1GG1(,J7 I+3,<(4-+3-5 <?I.3,+=, /B1= (B+3C1 <=B.?.C1=1<,<1J +31I3.=.4=(1EL

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

*.=(13=<=C ,B1 (+-(4-+,<.= .G +2J.3I,<.= JI1(,3+ .G J1?<(.=E4(,.3J7 =1<,B13 ,B1];Q =.3 ,B1 QY`Q <J J4GV(<1=, ,. 5<1-E 31J4-,J <= T4+=,<,+,<D1 H+=E J.?1,<?1J =.,1D1= <= T4+-<,+,<D1K +C311?1=, /<,B 1RI13<?1=,JL M= I+3,<(4-+37 ,B1 -+(Z .G + -.=C'3+=C1 H8arK E1(+5 .G ,B1 QY`Q -<=1+3 31JI.=J1 Z13=1- fR( (+= (+4J1 -+3C1 133.3J <=,B1 (+-(4-+,<.= .G +2J.3I,<.= JI1(,3+ .G 1R,1=E1E J5J,1?J N&U7 A%7 9&OL >1D13,B1-1JJ7<, B+J 211= JB./= NA%7 W8O ,B+, <= J.?1 (+J1J += +C311?1=, /<,B 1RI13<?1=,J (+=21 31(.D131E 25 31,+<=<=C .=-5 ,B1 -.=C'3+=C1 (.=,3<24,<.= HY]*K ,B+, ,B1 1R+(,fR( JB.4-E B+D1 <= ,B1 +J5?I,.,<( -<?<,L M= ,B1 G.--./<=C /1 /<-- J11 += 1R+?I-1.G 31J4-,<=C ?.E1- Z13=1-JL S.,,< et alL NAU7 9AO +((.4=, G.3 (.=,<=44? 1R(<,.=<(1GG1(,J7 C<D<=C D135 C..E .I,<(+- I3.I13,<1J G.3 J?+--'?1E<4? C+I J1?<(.=E4(,.3J7/<,B + (.?I4,+,<.=+- 243E1= (.?I+3+2-1 ,. + `0# +II3.+(BL

j=1 JB.4-E 31?1?213 B./1D13 ,B+, *Mc"?+,13<+-J I.JJ1JJ 2.4=E 1R(<,.=J N9W[99O7 +=E ,B+, ,B1 #``0#[Y]* +II3.+(B B+J I3.D1E ,. /.3Z 21,,13 G.3 J5J,1?J /<,B(.=,<=44? 1R(<,.=JL #B+, <J /B5 ,B1 .=-5 G4--5 31-<+2-1 +II3.+(B <= ,B<J (+J1 <J ,B1J.-4,<.= .G ,B1 (.?I4,+,<.=+--5 E1?+=E<=C S"F .3 ,B1 4J1 .G + ?.31 J.IB<J,<(+,1E?.E1- Z13=1-7 /B<(B +((43+,1-5 E1J(3<21J ,B1 1-1(,3.=[B.-1 I+<3 E5=+?<(J NA97 W)7W$7 W&OL >1D13,B1-1JJ7 ,B1 J<?I-1 ?.E1- Z13=1-J .G,1= 31?+<= ,B1 21J, (.?I3.?<J121,/11= +((43+(5 +=E 1GV(<1=(5 /B1=1D13 .=1 /+=,J ,. E1+- /<,B -+3C1 J4I13(1--J,. J,4E5 <=,13G+(1J .3 E1G1(,JL

/E !>(5:.#9 += C::.*8()*+"9

#B1V3J,'I3<=(<I-1J +II3.+(B1J7 I31J1=,1E <= "1(,<.= $L$7 B+D1 31(1=,-5 211= +II-<1EG.3 ,B1 V3J, ,<?1 ,. (.?I-1R ?+,13<+-J .G ,1(B=.-.C<(+- <=,131J,L M= ,B<J (B+I,137 /1J1-1(, ,B311 <--4J,3+,<D1 +II-<(+,<.=J .G J1-G'(.=J<J,1=, GW <= ,B1 V1-E .G ;P 1=13C5(.=D13J<.=7 /B131 ,B1 4J1 .G +((43+,1 J,+,1'.G',B1'+3,?1,B.EJ B+J +--./1E ,. (-+3<G5(.=,3.D13J<+- 31J4-,J <= -<,13+,431L

j43 V3J, 1R+?I-1 (.=(13=J (.II13 <=E<4?aC+--<4? E<J1-1=<E1aE<J4-VE1 H*Mc"K(.?I.4=EJL #B1J1 ?+,13<+-J +31 E.I1E 25 =+,<D1 E1G1(,JL #B1J1 E1G1(,J I-+5 +=<?I.3,+=, 3.-1 <= ,B1 E1V=<,<.= .G ,B1 1R(1I,<.=+- J,+2<-<,5 4=E13 .I13+,<=C (.=E<',<.=J .G ,B1 1-1(,3.=<( I3.I13,<1J .G ,B1J1 (.?I.4=EJL !1 E<J(4JJ ?.31 <= E1,+<-B131 ,B1 1GG1(, .G *4 D+(+=(<1JL

#B1 J1(.=E 1R+?I-1 <J Z1J,13<,1 +=E J,+==<,1 *4$h="=H"7"1KA (.?I.4=EJL #B<JG+?<-5 .G T4+,13=+35 (B+-(.C1=<E1 <J 1?13C<=C +J += +-,13=+,<D1 ,. *Mc" ?+,13<+-J+=E JB+31J /<,B ,B1 I+31=, G+?<-5 1R,3+.3E<=+35 J,34(,43+- +=E 1-1(,3.=<( I3.I13,<1JL

#B1 ,B<3E 1R+?I-1 <J ,B1 G+?<-5 .G *4HQ-7M=7c+Kj$ E1-+G.JJ<,1 ,3+=JI+31=, (.='E4(,<D1 .R<E1J7 /B<(B I.JJ1JJ + (B1?<(+- (.?I.J<,<.= J<?<-+3 ,. *Mc" (.?I.4=EJLQ -.=C'J,+=E<=C E12+,1 .= ,B1 E<31(, +=E <=E<31(, G4=E+?1=,+- C+IJ .G ,B1J1 ?+,1'3<+-J B+J <=,3<C41E 1RI13<?1=,+-<J,J +=E ,B1.31,<(<+=J G.3 51+3JL

*.=(13=<=C ,B1 (+-(4-+,<.=J E<J(4JJ1E <= ,B<J J1(,<.=7 ,B1 J1, .G ?1,B.EJ 4J1E<=(-4E1J `0# <= ,B1 J,+=E+3E Y`Q +=E ccQ7 Y`QaccQ+ U7 B523<E G4=(,<.='+-J +=E GW +II3.+(B1JL #B1 31I.3,1E J,+=E+3E `0# +=E GW (+-(4-+,<.=J /131

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

I13G.3?1E /<,B<= ,B1 I-+=1'/+D1 J(B1?1 <?I-1?1=,1E <= QSM>M# N9UO7 4J<=C=.3?'(.=J13D<=C IJ14E.I.,1=,<+-J NU)O +=E <=(-4E<=C J1?<'(.31 J,+,1J <= ,B1D+-1=(1L g523<E G4=(,<.=+- +=E Y`QaccQ+ U (+-(4-+,<.=J /131 I13G.3?1E /<,B,B1 P<1==+ ab initio J<?4-+,<.= I+(Z+C1 HPQ";K N%87%$7 U87U$OL #``0# +=E S"FJI1(,3+ /131 .2,+<=1E 4J<=C ,B1 (.E1 @QbSj NU&OL b.31 E1,+<-J .= (+-(4-+,<.=J(+= 21 .2,+<=1E <= ,B1 (<,1E 31G131=(1JL

/E/0 KB3F(9#< 12*"34*.5 CD9+$D#$9

#13=+35 (B+-(.I53<,1J (35J,+-J +31 .2,+<=1E G3.? ,B1 f<=(2-1=E1 J,34(,431 25 31I-+('<=C ,B1 h= (+,<.=J +-,13=+,<D1-5 /<,B *4 +=E M= .3 c+L *4M=H"7"1K$ (.?I.4=EJ +31(.??.=-5 31G1331E ,. +J *M"7 +=E *4c+H"7"1K$ (.?I.4=EJ +31 Z=./= +J *c"L#B1 ,/. G+?<-<1J ,.C1,B13 +31 ,B1 J.'(+--1E *Mc"?+,13<+-JL M= ,13=+35 (B+-(.I53<,1J1+(B +=<.= H"1 .3 "K <J (..3E<=+,1E 25 ,/. M= .3 c+ +=E ,/. *4 +,.?J7 /B<-1 1+(B(+,<.= <J ,1,3+B1E3+--5 (..3E<=+,1E 25 G.43 +=<.=JL #B1 1R<J,1=(1 .G ,/. E<GG131=,(+,<.=J 31J4-,J <= ,/. E<GG131=, 2.=E<=C -1=C,BJ RH<=7c+K[H"7"1K +=E R*4[H"7"1K7 -1+E<=C,. ,/. J,34(,43+- +=.?+-<1J NUAO: H<K #B1 ,1,3+C.=+- (1-- 1RB<2<,J + E<J,.3,<.=7 E1V=1E25 ,B1 I+3+?1,13 η = c/$a 6= 87 ,B1 3+,<. 21,/11= ,B1 -+,,<(1 (.=J,+=,J a +=E cL H<<K#B1 <E1+- f<=(2-1=E1 J<,1 G.3 ,B1 +=<.= <J I13,4321E7 5<1-E<=C + E1G.3?+,<.= .G ,B1+=<.= ,1,3+B1E3.=7 /B<(B <J ?1+J431E 25 ,B1 +=<.= E<JI-+(1?1=, I+3+?1,13

R$*4−H","1K − R$

HM=,c+K−H","1K

a$ 6=8

A. H$L$&K

S.,B J,34(,43+- +=.?+-<1J +31 J?+--7 24, =., =1C-<C<2-1L #B1 I31(<J1 1RI13<?1=,+-E1,13?<=+,<.= .G ,B1 +=<.= E<JI-+(1?1=, u <J ?.31 E<GV(4-, ,B+= ,B1 ?1+J431?1=,.G ,B1 -+,,<(1 (.=J,+=,J a +=E ( E41 ,. ,B1 <=B.?.C1=1<,5 .G ,B1 J+?I-1JL M=E11E7 ,B1E<JI13J<.= .G ,B1 ?1+J431E D+-41J <J D135 J?+-- G.3 ,B1 -+,,<(1 I+3+?1,13J H4J4+--5<8XK7 /B<-1 <, <J J<C=<V(+=, H+J -+3C1 +J %XK G.3 ,B1 +=<.= E<JI-+(1?1=, NUA7 UWOL

b13<=. et alL NUWO 31I.3,1E + 31-+,<D1-5 =+33./ JI31+E <= ,B1 2+=EC+I H≈8)XK +J +G4=(,<.= .G ,B1 E<JI13J<.= .G ,B1 +=<.= E<JI-+(1?1=, u. #B<J V=E<=C <J (.B131=, /<,B,B1 .2J13D+,<.= .G += 1R,3+.3E<=+35 J,+2<-<,5 .G *Mc" +2J.3213J 4=E13 .I13+,<=C(.=E<,<.=JL m=G.3,4=+,1-57 1RI13<?1=,J +-.=1 +31 =., (.=(-4J<D1 E41 ,. ,B1 -+(Z.G <=G.3?+,<.= .= ,B1 (.?I.J<,<.= +=E 4=<G.3?<,5 .G ,B1 J+?I-1JL M= G+(,7 <= 31+-J+?I-1J ,B131 <J + E<JI13J<.= .G D+-41J .G u. M, <J ,B131G.31 <=,131J,<=C ,. (+-(4-+,1,B1 1D.-4,<.= .G ,B1 C+IJ +J + G4=(,<.= .G u. #B<J /+J V3J, +EE31JJ1E ,B1.31,<(+--54J<=C `0# NU6O7 <= J.?1 (+J1J <=(-4E<=C 1?I<3<(+- (.331(,<.=J ,. C1, (-.J13 ,. ,B11RI13<?1=,+- C+I NUA7 U%OL Q= 4=(.??.=-5 -+3C1 D+3<+,<.= /+J G.4=E7 <= +II+31=,(.=,3+E<(,<.= /<,B ,B1 1RI13<?1=,+--5 I3.D1E <=J1=J<,<D<,5 .G ,B1 *Mc" C+I ,. +=5Z<=E .G E+?+C1 .3 I13,432+,<.=L

M= ,B1 G.--./<=C /1 /<-- G.(4J .= *M" ?+,13<+-J7 <= I+3,<(4-+3 4J<=C *4M="$ +J +,1J, (+J1L "<?<-+3 31J4-,J /131 G.4=E G.3 *4M="1$7 *4c+"1$7 +=E*4c+"$ N8A7 U9OL QV3J, T41J,<.= ,. +=J/13 <J <G ,B1 `0# E+,+ (+= 21 ,34J,1E7 +J ,B15 /131 .2,+<=1E G3.?k" 1<C1=D+-41J 4J<=C Y`Q .3 ccQ G4=(,<.=+-JL M=E11E7 ,B1J1 +II3.R<?+,<.=J +31Z=./= ,. I13G.3? 2+E-5 <= *Mc" ?+,13<+-JL 0.3 1R+?I-17 <= *M" ,B1 k" 2+=EC+I <J

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

D+=<JB<=C7 <= (.=,3+J, ,. ,B1 1RI13<?1=,+- D+-41J .G 8LWA 1P +=E 8L)W 1P G.3 *4M="$

+=E *4M="1$ NUUO7 31JI1(,<D1-5L b.31.D137 ad hoc (.331(,<.=J7 J4(B +J ,B1 J(<JJ.3.I13+,.37 .G,1= 4J1E ,. +E\4J, ,B1 `0# 2+=EC+I ,. ,B1 1RI13<?1=,+- D+-417 +31 ,..J<?I-<J,<( ,. +=+-5f1 ,B1 E1I1=E1=(1 .G ,B1 2+=EC+I .= u.

M= JI<,1 .G ,B1 4=E131J,<?+,<.= .G ,B1 2+=EC+I7 Y`Q +=EccQ4J4+--5 5<1-E C..EJ,34(,43+- I+3+?1,13J G.3 J1?<(.=E4(,.3J +=E <=J4-+,.3JL m=G.3,4=+,1-57 G.3 *M"7,B1 ,B1.31,<(+- 3+=C1 .G +=<.= E<JI-+(1?1=,J .2,+<=1E /<,B<= ,B1J1 +II3.R<?+,<.=JH)L$8W < u < )L$$)K -<1J .4,J<E1 ,B1 1RI13<?1=,+- 3+=C1L M, <J (-1+37 ,B4J7 ,B+, ,.4=E13J,+=E ,B1 I+3+E.R .G ,B1 2+=EC+I E1I1=E1=(1 .= ,B1 4=<, (1-- E1G.3?+,<.=J<= *Mc" (B+-(.I53<,1J .=1 B+J ,. C. 215.=E J,+=E+3E `0#L

!1 B+D1 +-31+E5 J11= ,B+, J(GW +=E B523<E G4=(,<.=+- +31 ,B1 ?.J, I13,<=1=,+II3.+(B1J 215.=E J,+=E+3E `0#L #B1J1 +II3.+(B1J /131 +II-<1E 25 P<E+- et alLN8AO ,. J,4E5 ,B1 <=,13I-+5 21,/11= J,34(,43+- +=E 1-1(,3.=<( I3.I13,<1J <= *M"(.?I.4=EJL g1317 /1 +=+-5f1 ,B1 ?+<= V=E<=CJ .G ,B+, /.3Z <= (.?I+3<J.= /<,BI31D<.4J 1RI13<?1=,+- +=E ,B1.31,<(+- 31J4-,JL

MG ,B1 +,.?<( +33+=C1?1=, <= ,B1 4=<, (1-- <J 31-+R1E 4J<=C Y`Q .3 ccQ G4=(,<.='+-J ,B1 +=<.= E<JI-+(1?1=, u <J J5J,1?+,<(+--5 4=E131J,<?+,1E 25 W[8)X7 <= +C311'?1=, /<,B I31D<.4J 31J4-,J NU67 8))7 8)8OL b.31.D137 +-/+5J /<,B<= Y`QaccQ7 ,B1k" 2.,,.? (.=E4(,<.= 2+=E .D13-+IJ ,B1 k" ,.I D+-1=(1 2+=E7 5<1-E<=C =1C+,<D12+=EC+IJ N8)$OL j= ,B1 .,B13 B+=E7 D+-41J .G u <= +C311?1=, /<,B 1RI13<?1=,J (+=21 .2,+<=1E 4J<=C ,B1 g15E["(4J13<+[F3=f13B.G Hg"F)6K N%WO B523<E G4=(,<.=+-N8AOL M= G+(,7 ,B1 J,34(,43+- 31-+R+,<.= /<,B<= g"F)6 5<1-EJ u = ).$$6 Hu = ).$$UKG.3 ,B1 <E1+- ?.=.(35J,+- .G *4M="$ H*4M="1$KL g"F)6 C<D1J J-<CB,-5 4=E131J,<'?+,1E 2+=EC+IJ G.3 *Mc"L S+=EC+IJ <= 1R(1--1=, +C311?1=, /<,B 1RI13<?1=,J +31.2,+<=1E G3.? J(GW .=(1 ,B1 1RI13<?1=,+- C1.?1,35 H.3 ,B1 g"F)6 C1.?1,35K <JVR1EL m=G.3,4=+,1-57 J(GW (+==., 21 4J1E ,. I13G.3? J,34(,43+- 31-+R+,<.=JL

M= .3E13 ,. J,4E5 ,B1 D+3<+,<.=J .G ,B1 C+I 4I.= J,34(,43+- E<J,.3,<.=7 P<E+- et alLN8AO D+3<1E u <= ,B1 <=,13D+- )L$ < u < )L$W7 /B<(B 1=(.?I+JJ1J 2.,B 1RI13<?1=,+-+=E ,B1.31,<(+- 3+=C1J7 +=E 1D+-4+,1E ,B1 T4+J<'I+3,<(-1 2+=E J,34(,431J 4J<=C +=E(.?I+3<=C E<GG131=, ab initio J(B1?1JL #B15 +-J. D13<V1E ,B+, J1=J<2-1 D+3<+,<.=J.G ,B1 -+,,<(1 I+3+?1,13J a +=E c B+D1 =1C-<C<2-1 1GG1(,J .= ,B1 C+IL

#B1 E1I1=E1=(1 .= u .G ,B1 2+=EC+I .G *4M="$ <J JB./= <= 0<C431 $L& N8AOL !1(+= J11 ,B+, ,B1 k"[Y`Q (43D1 H?+C1=,+ /<,B V--1E (<3(-1JK B+J ,B1 J+?1 J-.I1 +J,B1 ,B1.31,<(+- (43D1 HE.,,1E -<=1K .2,+<=1E 25 <+=C +=E 01=C NU%O 4J<=C ,B1 Y`Q(.331(,1E 25 + J(<JJ.3 .I13+,.3L S5 +II-5<=C ,B1 I13,432+,<D1 G)W) +II3.+(B .=,.I .G `0# H2-41 -<=1 /<,B (3.JJ1JK7 2+=EJ +31 31.3E131E7 31D13J<=C ,B1 J<C= .G ,B12+=EC+I G.3 u > )L$8WL M= +=5 (+J17 G)W) C+IJ 31?+<= T4<,1 J?+--7 +=E ,B1 J-.I1 .G,B1 (43D1 E.1J =., (B+=C1 G.3 ?.J, .G ,B1 u 3+=C1 /B1= (.?I+31E ,. ,B1 k" 31J4-,JL#B1 J-.I1 .G ,B1 G)W) (+-(4-+,<.= <=(31+J1J /B1= u = )L$W7 21(.?<=C (.?I+3+2-1,. ,B1 J-.I1 .G ,B1 (43D1J .2,+<=1E +II-5<=C + J1-G'(.=J<J,1=, +II3.+(BL #B<J <J ,B1.=-5 (+J1 G.3 /B<(B ,B1 k" C+I <J +-31+E5 I.J<,<D1 +=E =. 2+=E 31.3E13<=C .((43JL

#B1 J(*jg"Fr I3.(1E431 H51--./ -<=1J /<,B J,+3JK -1+EJ ,. + D135 E<GG131=,31J4-,d,B1 C+I 1=-+3C1J H,. D+-41J J-<CB,-5 -+3C13 ,B+= ,B1 1RI13<?1=,+- .=1JK7 +=E

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

(a) (b)

0.22 0.24u

DFT–LDA

corrected DFT

sc COHSEX

sc COHSEX+G0W

0.2 0.22 0.24u

3sc COHSEX+G

HSE06+ε0

Figure 2.3 Photoemission bandgap versus the anion displacement u for CuInS2. The verticalshaded areas give the spread of experimental data for u, while the horizontal shaded areas givethe dispersion of measured bandgaps. (a) Calculations using DFT–LDA, G0W0, scCOHSEX, andscCOHSEX + G0W0. The dotted line refers to the work of Jiang and Feng [97]. (b) Comparisonof scCOHSEX + G0W0 and hybrid functional calculations, using HSE06 and a modified HSE06.(From [14].)

,B1 J-.I1 .G ,B1 (43D1 J<C=<V(+=,-5 <=(31+J1JL #B<J <J E41 ,. ,B1 I3.C31JJ<D1 ?.E'<V(+,<.= <= ,B1 J1-G'(.=J<J,1=, <,13+,<.=J .G ,B1 4=J+,<JG+(,.35 Y`Q J,+3,<=C I.<=,L!B1= ,B1 J(*jg"Fr 1<C1=J,+,1J +31 4J1E +J + J,+3,<=C I.<=, G.3 + I13,432+,<D1G)W) (+-(4-+,<.= H2-+(Z E+JB1E -<=1JK7 ,B1 2+=EC+I C1,J 31E4(1E 25 + (.=J,+=,D+-41 .G +2.4, )L& 1P7 /<,B.4, +GG1(,<=C ,B1 J-.I1 .G ,B1 E1I1=E1=(1 .= u. >.,1 ,B+,,B1 J(*jg"Fr+ G)W) 2+=EC+I <J <= +C311?1=, /<,B 1RI13<?1=,+- E+,+ G.3 ,B1+=<.= E<JI-+(1?1=, ?1+J431E G.3 ,B1 ?.=.(35J,+- N8)&OL

".?1,B<=C ?.31 (+= 21 -1+3=, G3.? g"F)6 (+-(4-+,<.=J G.3 ,B1 C+IJ H?+3..=-<=1 /<,B V--1E ,3<+=C-1JK ,B+, 5<1-E += <=,13?1E<+,1 J-.I1 21,/11= `0#[Y`Q +=EJ(*jg"Fr+ G)W)L #B1 1GG1(, .G ,B1 g"F)6 G4=(,<.=+- <J E1,13?<=1E 25 ,B1 VR1E+?.4=, .G g+3,311[0.(Z Hg0K 1R(B+=C1 <=(-4E1E:

Eg"F)6R( = EccQ

AEg0,J3

R −8

AEccQ,J3

R , H$L$AK

/B131 ,B1 1R(B+=C1 ,13? 8A

Eg0,J3 (+= 21 J11= +J += +II3.R<?+,1E (.=,3<24,<.= ,.,B1 J1-G'1=13C57 /B.J1 *.4-.?2 <=,13+(,<.= <J J(311=1E 25 ,B1 ?<R<=C I+3+?1,138A

+=E ,B1 JB.3,'3+=C1 HJ3K J(311=<=C G+(,.37 /B<-1 8A

EccQ,J3R <J + J(311=1E ccQ

1R(B+=C1L M= (.=,3+J,7 <= + GW (+-(4-+,<.= ,B1 1R(B+=C1 I+3, .G ,B1 J1-G'1=13C5 <JJ(311=1E 25 ,B1 <=D13J1 E<1-1(,3<( (.=J,+=,7 +J + V3J, +II3.R<?+,<.= 4J<=C ,B1 J,+,<(ε∞; ,B1 -+,,13 D+3<1J G.3 E<GG131=, D+-41J .G uL #. JB1E -<CB, .= ,B<J I.<=,7 P<E+-et alL N8AO I13G.3?1E + J13<1J .G g"F)6 (+-(4-+,<.=J 4J<=C 8

ε∞7 (.?I4,1E G.3 1+(B

D+-41 .G u7 ,. 31I-+(1 ,B1 ?<R<=C I+3+?1,13 <= G3.=, .G ,B1 J(311=1E g+3,311[0.(Z

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

1R(B+=C1L #B1 31J4-,<=C (43D1 HD<.-1, -<=1 /<,B .I1= ,3<+=C-1JK B+J ,B1 J+?1 J-.I1+J ,B1 J(*jg"Fr (43D17 I.<=,<=C ,. ,B1 G+(, ,B+, ,B1 1-1(,3.=<( J(311=<=C <J ,B11JJ1=,<+- <=C31E<1=, ,B+, (.=,3.-J ,B1 2+=EC+IL #B<J <E1+ /+J G43,B13 1RI-.<,1E 25b+3T41J et alL N98O ,. .2,+<= + =.=VR1E ?<R<=C I+3+?1,13 G.3 B523<E G4=(,<.=+-J7J1=J<,<D1 ,. J(311=<=CL

>.,1 ,B+, J(GW (+==., 21 +II-<1E +, ,B1 ?.?1=, ,. J4I13(1--J (.=,+<=<=C d

1-1(,3.=J <G ,B1 4=<, (1-- <J -+3C13 ,B+= $)[&) +,.?JL QJ -.=C +J -+3C13 (1--J +31.4, .G 31+(B G.3 GW (+-(4-+,<.=J7 B523<E G4=(,<.=+-J7 <= I+3,<(4-+3 /<,B <?I3.D1E+II3.R<?+,<.=J G.3 ,B1 ?<R<=C I+3+?1,137 I3.?<J1 ,. 21 + (.?I4,+,<.=+--5 1GV(<1=,+-,13=+,<D1 /<,B + J+,<JG+(,.35 E1C311 .G +((43+(5L

M= J4??+357 +-- ,B1.31,<(+- 31J4-,J7 <=E1I1=E1=,-5 .G ,B1 -1D1- .G J.IB<J,<(+,<.=731D1+- D+3<+,<.=J .G ,B1 2+=EC+I /1-- .4,J<E1 ,B1 1RI13<?1=,+- 3+=C1 .G D+-41JL M=G+(,7 (.=J<E13<=C +J += 4II13 -<?<, G.3 1u7 ,B1 3+=C1 /B131 2.,B ,B1.31,<(+- +=E1RI13<?1=,+- D+-41J +31 <=(-4E1E H1u ® ).)$K7 /1 E1E4(1 ,B+, ,B1 C+I D+3<+,<.=1Eg E41 ,. ,B1 +=<.= E<JI-+(1?1=, +-.=1 /.4-E 21 1Eg = &$.$ × 1u ≈ ).6W1PL#B<J J4CC1J,J ,B+, ,B1 I4ff-1 .G 1-1(,3.=<( J,+2<-<,5 .G *Mc" (.?I.4=EJ (+==., 21J.-D1E J<?I-5 25 31I-+(<=C Y`Q G.3 + ?.31 +((43+,1 T4+J<'I+3,<(-1 (+-(4-+,<.=L

Q B<=, .G ,B1 I.JJ<2-1 J.-4,<.= .G ,B<J +II+31=, I+3+E.R .G 2+=EC+I J,+2<-<,5 (+=21 G.4=E 25 +-J. (.=J<E13<=C ,B1 1GG1(, .G B<CB (.=(1=,3+,<.=J .G E1G1(,J N88OL g1317/1 (.=J<E13 +J += 1R+?I-1 ,B1 1GG1(, E41 ,. *4 D+(+=(<1J V*47 JB+--./ +((1I,.3E1G1(,J ,B+, +31 Z=./= ,. 21 I31J1=, <= +-- J+?I-1J7 (.=J<J,1=, /<,B ,B1 .2J13D+,<.=.G I',5I1 (.=E4(,<D<,5L M= I+3,<(4-+37 V*4 <J ,B.4CB, ,. B+D1 D135 -./ G.3?+,<.=1=13C<1J N8)AO (.?I+31E ,. .,B13 <=,3<=J<( E1G1(,JL j=1 JB.4-E =., G.3C1, ,B+, .,B13E1G1(,J +31 Z=./= ,. 21 I31J1=, <= -+3C1 (.=(1=,3+,<.=J7 <= I+3,<(4-+3 *4M= +=E M=*4

+=,<J<,1J7 /B<(B I+3,<(<I+,1 <= ,B1 G.3?+,<.= .G (.?I-1R1J .G E1G1(,J /<,B D135 -./G.3?+,<.= 1=13C5 N8)AOL M=E11E7 + (.?I-1,1 ?.E1- JB.4-E <=(-4E1 ,B1 1GG1(,J .G +--?.J, (.??.= E1G1(,JL

Q-31+E5 +, ,B1 k" -1D1-7 ,B1 I31J1=(1 .G *4 D+(+=(<1J V*4 .I1=J 4I ,B1 2+=EC+IN8)A7 8)WO7 /<,B + -+3C13 1GG1(, G.3 <=(31+J<=C (.=(1=,3+,<.=J .G V*47 .=(1 +C+<=<= (.=,3+E<(,<.= /<,B ,B1 .2J13D1E J,+2<-<,5 .G 1RI13<?1=,+- 2+=EC+IJ +C+<=J, J,.<'(B<.?1,35 E1D<+,<.=JL #B<J ,31=E <J (.=V3?1E 25 G)W) (+-(4-+,<.=J G.3 J4I13(1--JN8AO (.331JI.=E<=C ,. (.=(1=,3+,<.=J .G V*4 <= ,B1 4J4+- 1RI13<?1=,+- 3+=C1 .G.GG'J,.<(B<.?1,35 NUWOL #B1 (.=(1=,3+,<.= .G V*4 <J 31-+,1E ,. ,B1 G.3?+,<.= 1=13C51EG ,B3.4CB + S.-,f?+== E<J,3<24,<.=7 /B.J1 31-1D+=, ,1?I13+,431 +=E (B1?<(+-I.,1=,<+-J +31 J1, 25 ,B1 C3./,B (.=E<,<.=J HT ≈ W))[6)) ◦*7 (.II13 I..3 J+?'I-1JK +, /B<(B E1G1(, 1T4<-<23<4? .((43JL b.31.D137 ,B<J 1T4<-<23<4? <J +JJ4?1E,. 21 T41=(B1E E43<=C ,B1 3+I<E (..- E./= .G ,B1 J+?I-1JL g./1D137 I31D<.4J(+-(4-+,<.=J .G G.3?+,<.= 1=13C<1J /131 E.=1 +, ,B1 -1D1- .G `0# +=E J4GG131EG3.? ,B1 4=J+,<JG+(,.35 E1J(3<I,<.= .G ,B1 -.(+-<f1E J,+,1J (.=,3<24,<=C ,. ,B1 PSbLY+=5 +=E h4=C13 N&&O I3.I.J1E ,. 4J1 Y`Q+ U ,. (.331(, ,B1 G.3?+,<.= 1=13C5+J G.--./J:

1EG = 1EY`QG − 1EY`Q+U

v . H$L$WK

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

0.2 0.22 0.24u

−0.5

Figure 2.4 The shifts of the conduction band minimum (CBM) and valence band maximum(VBM) 1E c,v of CuInS2 with respect to the LDA band edge versus the anion displacement u.We compare results for scCOHSEX with self-consistency only in energy (dashed curve) andin both wavefunctions and energies (continuous curve). The blue horizontal lines represent1E c,v given by DFT + U [106], where U is set to 6 eV. (Modified from [14].)

P<E+- et alL N8AO 4J1E ,B<J <E1+7 1D+-4+,<=C ,B1 PSb JB<G, 1Ev /<,B<= J(GW. M=,B<J (+J17 1Ev E.1J =., (.331(, + JB.3,(.?<=C .G k" ,B1.357 24, <?I3.D1J .= ,B1Y`Q G4=(,<.=+-7 +J ,B1 1R+(, 1R(B+=C1[(.331-+,<.= G4=(,<.=+- /.4-E C<D1 ,B1 1R+(,PSb +-31+E5 /<,B<= `0#L M= 0<C431 $LA7 .=1 (+= V=E ,B1 2+=E 1EC1 (.331(,<.=J1Ec,v /<,B 31JI1(, ,. ,B1 k" PSb7 (+-(4-+,1E 4J<=C J(GW +=E JB./= G.3 E<GG131=,-1D1-J .G J1-G'(.=J<J,1=(5L #B1 PSb JB<G, B+J + E<JI13J<.= .G +3.4=E 8 1P <= ,B1(.=J<E131E 3+=C1 .G u. #B<J E<JI13J<.= <J ,.,+--5 +2J1=, <= `0# H+ UK (+-(4-+,<.=JN8)6O HB.3<f.=,+- 2-41 -<=1KL >.,1 ,B+, ,B1 JB<G,J +31 J<C=<V(+=,-5 E<GG131=, +((.3E<=C,. /B1,B13 2.,B /+D1G4=(,<.=J +=E 1<C1=D+-41J .3 .=-5 1<C1=D+-41J +31 4IE+,1E <=,B1 J1-G'(.=J<J,1=, I3.(1E431L 0<C431 $LA I3.D1J +J /1-- ,B+, ,B1 D+3<+,<.=J .G ,B12+=EC+I /<,B u +31 E41 ,. 2.,B D+3<+,<.=J .G ,B1 *Sb +=E ,B1 PSbL

0.--./<=C ,B1 -.C<( .G ,B1 G11E2+(Z -..I J(B1?1 .G 0<C431 $LW7 .=1 (+= 31+-<f1,B+, <G + E<J,.3,<.= .G ,B1 -+,,<(1 .((4331E G.3 +=5 31+J.= H1LCL7 1R,1=E1E E1G1(, +=E+R<+- J,3+<=K7 <, /.4-E +(,4+--5 B+D1 -<,,-1 <=e41=(1 .= ,B1 C+I D+-41 21(+4J1 .G ,/.(.?I1=J+,<=C 1GG1(,JL

∆u {∆VBM

∆Ef ∆[VCu]

Figure 2.5 Schematic representation of the feedback loop that stabilizes the bandgap.(From [14].)

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

Y1, +JJ4?1 ,B+, ,B1 2+=EC+I <J + G4=(,<.=+- .G u +=E NV*4O J. ,B+, EC = ECHu7NV*4OKL Q=5 D+3<+,<.= .G ,B1 2+=EC+I 1Eg (+= 21 1RI31JJ1E +J

1EC =∂ EC

∂u1u +

∂ EC

∂NV*4O1NV*4O, H$L$6K

/B131 u <J ,B1 +=<.= E<JI-+(1?1=, +=E NV*4O <J ,B1 (.=(1=,3+,<.= .G *4 D+(+=(<1JL#B1 ,/. D+3<+2-1J u +=E NV*4O +31 <= G+(, =., <=E1I1=E1=, 24, -<=Z1E ,B3.4CB ,B1D+3<+,<.= .G ,B1 PSb <=E4(1E 25 D+3<+,<.=J .G uL M= ,B1 (+J1 .G + JB+--./ +((1I,.37<,J G.3?+,<.= 1=13C5 JB.4-E 21 (.331(,1E 25 ,B1 (.331(, I.J<,<.= .G ,B1 PSb HJ111T4+,<.= H$L$WKK +J

1EGHuK = 1E`0#G − 1E J(GW

v HuK. H$L$%K

#B1 E13<D+,<D1J 1=,13<=C <= 1T4+,<.= H$L$6K /131 (+-(4-+,1E 25 V,,<=C ,B1 (43D1J<= 0<C431 $L6L

"B+--./ +((1I,.3J +31 J4II.J1E ,. 21B+D1 -<Z1 ,B1 PSb E41 ,. ,B1<3 I3.R<?<,5,. <,L #B<J B+J 211= ,B1 (.??.= J,3+,1C5 4J1E G.3 ,B1 I+J, $) 51+3J <= .3E13 ,.

−3 −2.5 −2 −1.5 −1

ln([VCu]/NCu)

Eg =1.6663+0.231∗ln([VCu]/NCu)

0.2 0.21 0.22 0.23 0.24 0.25u

Eg = −5.975+32.1∗u

0.2 0.21

0.22 0.23 0.24 0.25

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

∆EvscGW = 2.1335−11.06∗u

Figure 2.6 (a) Value of the bandgap of CuInS2 calculated in G0W0 versus the concentrationof Cu vacancies. (b) Value of the bandgap of CuInS2 calculated in scCOHSEX + G0W0 versusthe anion displacement. (c) 1E scGW

v of CuInS2 versus the anion displacement. (From [14].)

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

(.331(, G.3?+,<.= 1=13C<1J N&&OL #B1=7 <, <J I.JJ<2-1 ,. 31-+,1 ,B1 (.=(1=,3+,<.= .GV*4 ,. ,B1 G.3?+,<.= 1=13C5 ,B3.4CB + S.-,f?+== E<J,3<24,<.= +J

NV*4O = N*41−1EGZST , H$L$9K

/<,B T 21<=C ,B1 C3./,B ,1?I13+,431 G.3 /B<(B ,B1 E<GG4J<.= .G E1G1(,J <J T41=(B1EHT ≈ W))[6)) ◦*KL b.31.D137 B<CB (.=(1=,3+,<.=J .G V*4 (+= J<C=<V(+=,-5 1=-+3C1,B1 2+=EC+IL #B4J7 +=5 D+3<+,<.= .G u (+= (B+=C1 2.,B ,B1 2+=EC+I +=E ,B1 G.3?+,<.=1=13C5 .G V*47 +=E +J + (.=J1T41=(1 D+35 ,B1 (.=(1=,3+,<.= .G V*47 /B<(B /<--(B+=C1 ,B1 2+=EC+I +((.3E<=C-5L jD13+--7 ,B1 1GG1(,J .G u +=E V*4 (+=(1- 1+(B.,B13L

#B1 J+?1 E1I1=E1=(1 .G ,B1 C+I .= u /+J G.4=E G.3 *c" (.?I.4=EJ7 1R(1I,G.3 ,B1 G+(, ,B+, ,B1 ccQ D+-41J .G u +31 <=J<E1 ,B1 1RI13<?1=,+- /<=E./7 E41 ,.+ (.?I1=J+,<.= .G 133.3J <= ,B1 (+-(4-+,<.= .G 2.=E -1=C,B NU9OL Q=+-.C.4J 31J4-,J+31 +-J. G.4=E G.3 T4+,13=+35 *h#" (.?I.4=EJ7 /B.J1 1-1(,3.=<( I3.I13,<1J +311R,31?1-5 J<?<-+3 ,. ,B1 .=1J .G ,B1 I+31=, *Mc" G+?<-5L Y1, 4J +=+-5f1 ,B<JJ,+,1?1=, ?.31 <= E1,+<-L #B<J /<-- C<D1 4J ,B1 I.JJ<2<-<,5 ,. B+D1 + (-.J13 -..Z +,,B1 2+=E J,34(,431J .2,+<=1E 4J<=C ,B1 E<GG131=, -1D1-J .G +II3.R<?+,<.=L

M= -<,13+,431 ,B131 +31 .=-5 G1/ 31(1=, J,4E<1J ,B+, +EE31JJ ,B1 J,34(,43+-N8)%[88)O7 1-1(,3.=<( N8)97 88)7 888O7 +=E E1G1(, I3.I13,<1J N88$7 88&O .G *h#"L#B1 f<=(2-1=E1'E13<D1E Z1J,13<,1 J,34(,431 HIAK <J 31(.C=<f1E ,. 21 ,B1 ?.J, J,+2-1L#B1 1=13C5 E<GG131=(1 I13 +,.? /<,B 31JI1(, ,. ,B1 J,+==<,1 J,34(,431 HI sA$mK <J7B./1D137 .=-5 .G G1/ ?<--<1-1(,3.= D.-,J I13 +,.? N8)97 88)7 88AO7 I3.D<=C ,B+,Z1J,13<,1 +=E J,+==<,1 IB+J1J (+= (.1R<J, <= 1RI13<?1=,+- J+?I-1J7 +=E 1RI-+<=<=C,B1 31I.3,1E E<J.3E131E J,34(,431J N88WOL "<?<-+3-5 ,. (B+-(.I53<,1J *Mc"7 T4+,13'=+35 Z1J,13<,1 +=E J,+==<,1 *h#" +31 .2,+<=1E G3.? ,B1 f<=(2-1=E1 J,34(,431 2531I-+(<=C ,B1 h= (+,<.=J <= J4(B + /+5 ,B+, 1+(B +=<.= H"1 .3 "K <J (..3E<=+,1E25 .=1 h=7 .=1 "=7 +=E ,/. *4 +,.?JL #B1 1R<J,1=(1 .G ,B311 E<J,<=(, (+,<.=J31J4-,J <= ,B311 E<GG131=, (+,<.=[+=<.= 2.=E -1=C,BJ7 /B<(B <=E4(1 + E<JI-+(1?1=,.G ,B1 +=<.= G3.? <,J <E1+- f<=(2-1=E1 J<,1L #B+, E<J,.3,<.= <J ?1+J431E 25 ,B1 +=<.=E<JI-+(1?1=, I+3+?1,13J Hux , u y, uzK7 ,B+, <J7 ,B1 31-+,<D1 (..3E<=+,1J .G ,B1 +=<.=<= ,B1 (.=D1=,<.=+- 2.E5'(1=,131E ,1,3+C.=+- (1--L #B1 E<JI13J<.= .G 1RI13<?1=,+-E+,+ G.3 u <J Z=./= ,. 21 -+3C1 <= *Mc" (.?I.4=EJ NUA7 UWOL *.=(13=<=C *h#"7G1/13 J,34(,43+- ?1+J431?1=,J (+= 21 G.4=E <= -<,13+,431 +=E <= ?.J, (+J1J .=-5a +=E c +31 31I.3,1E N886[8$)OL *.=(13=<=C ,B1 ?1+J431E 2+=EC+IJ N8$8[8$%O71+3-5 J,4E<1J J4CC1J,1E + D+-41 .G +2.4, 8LA[8L6 1P 2.,B G.3 " +=E "1 (.?I.4=EJL#B<J /+J <= E<J+C311?1=, /<,B `0# (+-(4-+,<.=J N8)97 88)O7 4J<=C J1?<'-.(+- .3B523<E G4=(,<.=+-J7 /B<(B .2,+<=1E J5J,1?+,<(+--5 + J?+--13 C+I G.3 "1 (.?I.4=EJLb.31 31(1=, ?1+J431?1=,J N8$8O E1-<=1+,1 + C+I .G +2.4, - 1P G.3 *4$h="="1A71RI-+<=<=C ,B1 I31D<.4J .D131J,<?+,<.= /<,B ,B1 I31J1=(1 .G h="1 <= ,B1 J+?I-1L

M= 0<C431 $L%7 /1 E<JI-+5 2+=E J,34(,431J G.3 Z1J,13<,1 *4$h="="A7 .2,+<=1E4J<=C E<GG131=, ,B1.31,<(+- J(B1?1J +, ,B1 1RI13<?1=,+- C1.?1,35L M= 0<C431 $L%+7,B1 k" Y`Q 2+=E J,34(,431 <J (.?I+31E /<,B ,B1 J(GW 2+=EJL !1 .2J13D1 ,B+,

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

T Γ N−16

(a) (b)

Figure 2.7 Band structures for kesterite Cu2ZnSnS4 calculated using (a) DFT–LDA (dashedline) and scGW (continuous line), and (b) GGA + U (dashed line) and HSE06 (continuousline). Top valence bands are set to zero. (From [107].)

J(GW (.331(,<.=J 4IJB<G, +-?.J, 3<C<E-5 ,B1 -./1J, (.=E4(,<.= J,+,1JL *.=(13=<=C,B1 D+-1=(1:

8L #B1 E<JI13J<.= .G ,B1 " p[*4 d +=,<2.=E<=C J,+,1J +, ,B1 ,.I .G ,B1 D+-1=(131?+<=J G+<3-5 4=+-,131E7 1D1= <G ,B1 .D13-+I .G Y`Q +=E T4+J<'I+3,<(-1 /+D1'G4=(,<.=J JB./J 31?+3Z+2-1 D+3<+,<.=J <= ,B<J 31C<.=L

$L #B1 2+=E/<E,B .G ,B1 " p[*4 d 2.=E<=C J,+,1J H-.(+,1E 21,/11= −&LW +=E−6L% 1PK J-<CB,-5 <=(31+J1JL

&L #B1 2+=EJ +JJ.(<+,1E ,. ,B1 Hh=7"=K[" 2.=E H21,/11= −9 +=E [8) 1PK +31<=D13,1E +=E E./=JB<G,1E 25 +2.4, $ 1P /<,B 31JI1(, ,. Y`QL

AL Q-J.7 ,B1 " s J,+,1J +31 ?.D1E E./= 25 +2.4, $ 1PL

M= 0<C431 $L%27 /1 JB./ ,B1 J+?1 2+=EJ +J .2,+<=1E G3.? ccQ+ U +=E g"F)6(+-(4-+,<.=JL M= ,B<J (+J1 g"F)6 2+=EJ +31 31?+3Z+2-5 J<?<-+3 ,. J(GW 2+=EJL #B<J<J =., I+3,<(4-+3-5 J43I3<J<=C +J ,B1 g+3,311[0.(Z ?<R<=C .G g"F)6 <J I+3,<(4-+3-5J4<,1E G.3 ?+,13<+-J /<,B C+IJ .G +2.4, 8[$ 1P N%A7 98OL QJ 1RI1(,1E7 ccQ+ U

JB<G,J E./= ,B1 J,+,1J /<,B *4 d (B+3+(,137 ,B13125 .I1=<=C ,B1 C+I ,. + 31+J.=+2-1D+-41L g./1D137 <, <J 1D<E1=, G3.? 0<C431 $L% ,B+, ,B1 .D13+-- E1J(3<I,<.= .G ,B12+=E E<JI13J<.=J <J T4<,1 <=+((43+,1L >.,1 ,B+, ,B1 +=+-5J<J .G ,B1 2+=E J,34(,431.G + *Mc" (.?I.4=E .3 +=.,B13 *h#" (.?I.4=E /.4-E B+D1 -1E ,. ,B1 J+?1(.=(-4J<.=J NU97 8)%OL

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

Cu2ZnSnS4

tannites

Cu2ZnSnSe4

T G N–6

1.64 eV

(a) (c)

(b) (d)

1.02 eV

0.86 eV1.33 eV

Figure 2.8 Band structures from scGW for (a) kesterite Cu2ZnSnS4, (b) stannite Cu2ZnSnS4,(c) kesterite Cu2ZnSnSe4, and (d) stannite Cu2ZnSnSe4. (From [107].)

M= 0<C431 $L97 /1 (+= J11 ,B1 J(GW 2+=E J,34(,431J .G ,B1 G.43 (.?I.4=EJ .G,B1 *h#" G+?<-5 <= ,B1 1=13C5 31C<.= (-.J1 ,. ,B1 G4=E+?1=,+- C+IL 0.3 Z1J,13<,1*4$h="="A +=E J,+==<,1 *4$h="="1A7 ,B1 1RI13<?1=,+- C1.?1,3<1J /131 4J1EN886[8$)OL 0.3 ,B1 31?+<=<=C (.?I.4=EJ7 ,B1 1RI13<?1=,+- +=<.= E<JI-+(1?1=,J/131 =., 31I.3,1E N886[8$)O7 +=E <, /+J ,B131G.31 =1(1JJ+35 ,. 31J.3, ,. ,B1 ,B1.'31,<(+- J,34(,431JL

#B1 (B.<(1 .G ,B1 ,B1.31,<(+- G3+?1/.3Z G.3 ,B1 C1.?1,35 .I,<?<f+,<.= <J 1JJ1=',<+-L M= G+(,7 ,B1 J,3.=C E1I1=E1=(1 .G ,B1 C+I .= +=<.= E<JI-+(1?1=,J <J +-J. I31J1=,<=*h#" (.?I.4=EJ N8)%OLQ-J. <= ,B<J (+J1 J-.I1J +31 J42J,+=,<+--5 -+3C13 G.3 J(GW

(+-(4-+,<.=J ,B+= G.3 `0#[Y`Q7 +=E 1D1= ,B+= G.3 g"F)6L j=(1 +C+<=7 ,B15 +31(.=,3.--1E 25 ,B1 J(311=<=C7 /B<(B <J 1JJ1=,<+- ,. <=(-4E1 <= + J1-G'(.=J<J,1=, /+57+J <, <J E.=1 <= + J(GW J(B1?1L S.,,< et alL N8)%O D13<V1E ,B+, Y`QaccQ'31-+R1EJ,34(,431J -1+E ,. 4=+((1I,+2-5 -+3C1 133.3J H4I ,. A)XK <= ,B1 J(GW 2+=EC+I71JJ1=,<+--5 31-+,1E ,. ,B1 133.3 <= u: *4'H"7"1K +=E h='H"7"1K 2.=E -1=C,BJ +31J-<CB,-5 ,.. J?+-- <= `0#[Y`Q7 /B<-1 "='H"7"1K 2.=E -1=C,BJ +31 ,.. -+3C1L Q-J.7,B<J ,<?1 ,B1 J.-4,<.= ,. ,B<J <JJ41 31-<1J .= ,B1 4J1 .G ,B1 g"F)6 G4=(,<.=+-7 /B<(B5<1-EJ 1R,31?1-5 +((43+,1 D+-41J G.3 u +=E ,B1 (1-- I+3+?1,13J7 ,B+=ZJ ,. + 21,,13E1J(3<I,<.= .G ,B1 -.(+-<f1E J,+,1J I+3,<(<I+,<=C <= 2.=E<=CL

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

#B1 ?+\.3 E<GG131=(1 +?.=C ,B1 G.43 2+=E J,34(,431J .G 0<C431 $L9 <J ,B1 /<E,B.G ,B1 2+=EC+IJ7 /B<(B +31 <= 1R(1--1=, +C311?1=, /<,B 1RI13<?1=,+- 31J4-,JL QJ <=,B1 *Mc" G+?<-57 "1 (.?I.4=EJ B+D1 + J?+--13 C+I /<,B 31JI1(, ,. " (.?I.4=EJL043,B13?.317 J,+==<,1J B+D1 (.=J<J,1=,-5 J?+--13 C+IJ ,B+= Z1J,13<,1JL

M= (.=(-4J<.=7 31?+3Z+2-1 J<?<-+3<,<1J /131 G.4=E <= ,B1 1-1(,3.=<( I3.I13,<1J .GJ,+==<,1J +=E Z1J,13<,1 *h#"7 +=E7 ,. + -+3C1 1R,1=,7 21,/11= *h#" +=E *Mc"(.?I.4=EJL #B<J G+(, J4CC1J,J .=(1 ?.31 ,B+, *h#" ?+,13<+-J +31 1R(1--1=,(+=E<E+,1J ,. 31I-+(1 ?.31 (.J,-5 *Mc" <= ,B1 +2J.32<=C -+513 .G ,B<='V-? ;PE1D<(1JL

!1 /+=, ,. (.=(-4E1 ,B<J J1(,<.= /<,B += 1R+?I-1 .G .I,<(+- JI1(,3+ (+-(4-+,1EG3.? J(GW 2+=E J,34(,431J7 /B<(B <J 31I.3,1E <= QC4<-13+ et alL NU9OL

",+3,<=C 1<,B13 G3.? ,B1 k" .3 J(GW 2+=E J,34(,431J .G *4c+"$7 QC4<-13+ et alLNU9O (+-(4-+,1E .I,<(+- +2J.3I,<.= JI1(,3+ /<,B<= -<=1+3 31JI.=J17 2.,B +II-5<=C#``0# +=E J.-D<=C ,B1 S"FL 0<C431J $LU +=E $L8) NU9O JB./ (+-(4-+,<.=J .G ,B1<?+C<=+35 I+3, .G ,B1 ?+(3.J(.I<( E<1-1(,3<( G4=(,<.= G.3 -<CB, I.-+3<f1E +-.=C,B1 c'+R<JL #B1 ,B1.31,<(+- (43D1J +31 +-J. (.?I+31E /<,B ,B1 1RI13<?1=,+- E+,+I42-<JB1E <= Y1D(1=Z. et alL N96O +=E ]<G1 et alL N8$9OL

#B1 1RI13<?1=,+- E<1-1(,3<( G4=(,<.=J .G Q-.=J. et alL NUUO /131 .2,+<=1E /<,B1--<IJ.?1,35 +, 3..? ,1?I13+,4317 /B131+J ,B1 JI1(,3+ .G Y1D(1=Z. et alL N96O /131

21 43 65

Energy (eV)

TDLDAGW-RPAexp. Levcenko

exp. Alonso

Figure 2.9 Imaginary part of the dielectric function for light polarized along the c-axis obtainedwith TDDFT using the ALDA kernel (dashed line), RPA using GW eigenvalues (dot-dashed line)compared to experimental data of Alonso et al. [99] (dots) and Levcenko et al. [86] (solid line;in arbitrary units, see text). (From [98].)

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

21 43 65

Energy (eV)

ε 2 (

TDDFT static LRCTDDFT dynamic LRC

GW-RPABSEexp. Levcenko

exp. Alonso

Figure 2.10 Imaginary part of the dielectric function for light polarized along the c-axisobtained with TDDFT using the static (dot-dashed line) and the dynamic (dashed line) LRCkernel, GW-RPA (gray solid line) and BSE (dotted line), compared to experimental measure-ments of Alonso et al. [99] (larger dots) and Levcenko et al. [86] (solid line; in arbitrary units).(From [98].)

1R,3+(,1E G3.? 31e1(,<.= JI1(,3+ ?1+J431E +, %%k7 4J<=C ,B1 k3+?13J[k3.=<C31-+,<.=JL S.,B 31J4-,J +31 G.3 J<=C-1'(35J,+- J+?I-1JL !1 (+= .2J13D1 ,B+, ,B1 ,/.(43D1J +31 3+,B13 E<GG131=,L M= I+3,<(4-+37 ,B1 +2J.3I,<.= JI1(,34? .G Y1D(1=Z.et alL N96O B+J + ?4(B J?+--13 <=,1=J<,5 ,B+= ,B1 1RI13<?1=,+- (43D1 .G Q-.=J.et alL NUUO +=E I31D<.4J 31J4-,J 25 ]<G1 et alL N8$9O7 I3.2+2-5 E41 ,. ,B1 T4+-<,5.G ,B1 31e1(,<=C J43G+(1 .G ,B1 J+?I-1JL M= D<1/ .G ,B1 +2.D17 ,B1 <=,1=J<,5 .G ,B1+2J.3I,<.= JI1(,34? .G Y1D(1=Z. et alL N96O /+J 31J(+-1EL Q=.,B13 E<J(31I+=(5+?.=C ,B1 +D+<-+2-1 1RI13<?1=,+- E+,+ (.=(13=J ,B1 21B+D<.3 .G ,B1 JI1(,3+ +,+3.4=E W 1PL !B<-1 ,B1 ?1+J431?1=,J .G Q-.=J. et alL NUUO G.4=E + I-+,1+4 <= ,B1+2J.3I,<.= +G,13 ,B1 I1+Z +, A 1P7 <= Y1D(1=Z. et alL N96O +=E ]<G1 et alL N8$9O + E11ID+--15 +II1+3J +G,13 A 1PL >.,1 ,B+, ,B131 +31 J42J,+=,<+- E<J(31I+=(<1J 21,/11= ,B1,/. 1RI13<?1=,J +=E ,B+, J4GV(<1=,-5 +((43+,1 ab initio (+-(4-+,<.=J +31 += 1JJ1=,<+-,..- ,. E<J(3<?<=+,1 21,/11= ,B1 ,/. J1,J .G E+,+L

M= 0<C431 $LU7 /1 (.?I+31 31J4-,J .2,+<=1E /<,B<= ,B1 ];Q 4J<=CJ(*jg"Fr+ G)W) 1<C1=D+-41J7 +=E QY`Q 4J<=C k" 1<C1=D+-41JL #B1 ];Q(43D1 .2,+<=1E 4J<=C k" 1<C1=D+-41J .D13-+IJ +-?.J, I13G1(,-5 /<,B ,B1 QY`Q(43D1L #B<J I3.D1J ,B+, ,B1 <=(-4J<.= .G ,B1 QY`Q Z13=1- E.1J =., B+D1 +=531-1D+=, 1GG1(, .= ,B1 +2J.3I,<.=7 +J ,B1 <?I.3,+=, (.=,3<24,<.=J E41 ,. 1-1(,3.=[1-1(,3.= +=E 1-1(,3.=[B.-1 <=,13+(,<.=J +31 =., +((.4=,1E G.3 25 ,B1 QY`Q Z13=1-

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

N&U7 AU7 8$UOL S.,B ,B1 ];Q +=E QY`Q JI1(,3+ 4J<=C k"'ccQ 1<C1=D+-41J +31E3+?+,<(+--5 31E'JB<G,1EL QJ + V3J, (.=(-4J<.=7 /1 (+= .2J13D1 ,B+, ,B1 +((43+(5 .G];Q .3 QY`Q ab initio (+-(4-+,<.=J +-.=1 /.4-E =., J4GV(<1=, ,. 1RI-+<= ,B1 -+3C1E<J(31I+=(<1J +?.=C 1RI13<?1=,+- .I,<(+- ?1+J431?1=,JL

MG ,B1 J(*jg"Fr+ G)W) 1<C1=D+-41J +31 4J1E G.3 += ];Q (+-(4-+,<.= HE.,'E+JB1E -<=1K7 ,B1 31J4-,<=C JI1(,3+ +31 2-41'JB<G,1E7 /B<-1 ,B1 -<=1 JB+I1 E.1J =.,(B+=C1L QJ + 31J4-,7 ,B1 +2J.3I,<.= I1+ZJ +31 =./ +, ,.. B<CB 1=13C<1JL M= G+(,7 <= ,B11=13C5 3+=C1 .G <=,131J, G.3 +2J.3I,<.= ,B1 GW (.331(,<.= <J 2+J<(+--5 1T4<D+-1=, ,.+II-5<=C + J(<JJ.3 .I13+,.3L

!B+, <J J,<-- ?<JJ<=C <J + I3.I13 +((.4=, .G 1R(<,.=<( 1GG1(,JL 0.3 ?+=5<=,13?1E<+,1'C+I J1?<(.=E4(,.3J7 <, /+J JB./= NAU7 W87 9AO ,B+, ,B1 Y]* ,. ,B11R(B+=C1[(.331-+,<.= Z13=1- /+J 1=.4CB ,. 5<1-E + C..E +C311?1=, /<,B 1RI13<'?1=,+- +2J.3I,<.= JI1(,3+L #B1 Y]* Z13=1- /.3ZJ I+3,<(4-+3-5 /1-- <= ,B1 (+J1 .G+ (.=,<=44? 1R(<,.=7 C<D<=C <= C1=13+- D135 C..E .I,<(+- I3.I13,<1J /<,B ,B1 1GV'(<1=(5 .G + #``0# (+-(4-+,<.=L M= 0<C431 $L8)7 /1 JB./ ,B1 JI1(,3+ .2,+<=1E4J<=C ,B1 J,+,<( NAU7 W8O +=E ,B1 E5=+?<( N9AO D13J<.=J .G ,B1 Y]* Z13=1-LJ(*jg"Fr+ G)W) 1<C1=D+-41J /131 4J1E <= ,B1J1 (+-(4-+,<.=JL

#B1 #``0#[Y]* JI1(,3+ JB./= <= 0<C431 $L8) I3.D<E1 + (-1+3 <?I3.D1?1=,/<,B 31JI1(, ,. ,B1 (43D1J E<JI-+51E <= 0<C431 $LUL #B1 <=,1=J<,5 .G ,B1 I1+ZJ +J/1-- +J ,B1 .=J1,J .G +2J.3I,<.= <J =./ (-.J13 ,. 1RI13<?1=,L g./1D137 ,B1 I1+Z +,+2.4, A 1P <J J,<-- -.(+,1E +, + B<CB13 1=13C5 ,B+= <,J 1RI13<?1=,+- I.J<,<.=L >.,1,B+, ,B1 1R(<,.=<( I1+Z +, ,B1 G4=E+?1=,+- 1EC1 <= *4c+"$ B+J ,B1 =+,431 .G +2.4=E 1R(<,.=7 1D1= <G ,B1 2<=E<=C 1=13C5 <J 3+,B13 J?+--L #B<J 1RI-+<=J /B5 ,B1Y]* ?.E1-J E. =., 31I3.E4(1 I13G1(,-5 ,B1 S"F 31J4-,L

MG .=1 /+=,J + I31(<J1 E1J(3<I,<.= .G ,B1 2.4=E 1R(<,.=<( I1+Z .G *4c+"$

N9W[99O7 <, <J =1(1JJ+35 ,. 31J.3, ,. ,B1 J.-4,<.= .G ,B1 S"FL M= 0<C431 $L8)7 /1J11 ,B+,d+-,B.4CB + JB<G,1E ?1JB .G 8))) k I.<=,J <J =., J4GV(<1=,-5 E1=J1 ,.J?..,B .4, ,B1 (43D1 E41 ,. ,B1 J,3.=C E<JI13J<.= .G ,B1 -./1J, (.=E4(,<.= J,+,1d,B1 1RI13<?1=,+- .=J1, +, +2.4, $LWW 1P +=E ,B1 I1+Z +, +2.4, &L9 1P +31 /1--31I3.E4(1E <= ,B1 S"F (+-(4-+,<.= NU9OL Q, ,B1 +2J.3I,<.= 1EC17 ,B1 1J,<?+,1 .G,B1 1R(<,.=<( 2<=E<=C 1=13C5 .G +2.4, )L8 1P <J +-J. (.=J<J,1=, /<,B 1RI13<?1=,+-E+,+ N8&)OL

g./1D137 .=1 (+==., G.3C1, ,B+, #``0# (+-(4-+,<.=J 4J<=C ?.E1- 1R(B+=C1[(.331-+,<.= Z13=1-J +31 ?4(B -1JJ <=D.-D1E ,B+= ,B1 J.-4,<.= .G ,B1 S"FL M= ?+=5(+J1J7 G.3 1R+?I-1 /B1= E1+-<=C /<,B J4I13(1--J /<,B E1G1(,J .3 E.I<=C7 ,B1 .=-5G1+J<2-1 +II3.+(B ,B+, C4+3+=,11J T4+-<,+,<D1-5 (.331(, 31J4-,J <J ,. I13G.3? #``0#(+-(4-+,<.=J 4J<=C ?.E1- Z13=1-J E13<D1E G3.? S"FL

/E/ J#.(=+99*)# 1$("9:($#") K+"<B8)*7# L>*<#9

#3+=JI+31=, (.=E4(,<D1 .R<E1J H#*jJK +31 /<E1 2+=EC+I J1?<(.=E4(,.3J (B+3+(',13<f1E 25 -+3C1 G311 (+33<13 E1=J<,<1JL #B1J1 (+33<13J +31 (31+,1E 25 1<,B13 <=,3<=J<(.3 1R,3<=J<( E.I<=C7 C<D<=C ,. #*jJ 2.,B -./ 31J<J,<D<,5 +=E ,3+=JI+31=(5 <= ,B1

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

D<J<2-1 1=13C5 /<=E./L #B1 ?.J, (.??.= 1R+?I-1J .G #*jJ +31 ='E.I1E J1?<'(.=E4(,.3J J4(B +J "=j$7 -=$j&7 +=E h=jL *+33<13 (.=(1=,3+,<.=J 4I ,. 8)$8 (?−&

+31 (.??.=-5 +(B<1D1E /<,B -+3C1 ?.2<-<,<1J .G &) (?$aPJ7 C<D<=C 3<J1 ,. (.=E4(',<D<,<1J .G 4I ,. &))) "a(? N8&8OL mI ,. =./ I',5I1 (.=E4(,<D<,5 B+J 211= ?4(BB+3E13 ,. +(B<1D1L #B1 31+J.= <J 31-+,1E ,. /1--'Z=./= E<GG131=(1J 21,/11= B.-1J+=E 1-1(,3.=J <= #*jJL 0<3J,7 B.-1J B+D1 + ?4(B B1+D<13 ?+JJ ,B+= 1-1(,3.=J +=E(.=J1T41=,-5 ?4(B -./13 ?.2<-<,5L "1(.=E7 E.I<=C 34-1J I.<=, ,. ,B1 E<GV(4-,5 .GI',5I1 E.I<=C <= .R<E1J N&AO 21(+4J1 .G ,B1 B<CB 1-1(,3.=1C+,<D<,5 +=E ,B1 I31J1=(1.G B.-1 Z<--13J7 J4(B +J V)7 /<,B 31-+,<D1-5 -./ G.3?+,<.= 1=13C5L

>1D13,B1-1JJ7 I',5I1 (.=E4(,<D<,5 /+J G.4=E <= E1-+G.JJ<,1 *4Q-j$ ,B<= V-?JN$)OL Q G1/ 51+3J -+,137 2<I.-+3 H2.,B =' +=E I',5I1JK (.=E4(,<D<,5 /+J E<J(.D131E<= *4M=j$ N$$7 8&$O7 +=.,B13 1-1?1=, .G ,B1 J+?1 E1-+G.JJ<,1 G+?<-5L g./1D137 <=I',5I1 #*jJ (+33<13 (.=(1=,3+,<.=J 31?+<= J?+--13 ,B+= 8)89(?−& +=E ?.2<-<,<1J-<?<,1E ,. µ < 8(?$/P J7 31J4-,<=C <= (.=E4(,<D<,<1J .G 8)) "a(? N8&8OL j,B13 I',5I1 .R<E1J 21-.=C<=C ,. E<GG131=, G+?<-<1J/131 G.4=E +G,13/+3EJ7 =+?1-5h=]B$jA

N8&&O +=E "=j N8&AOL FD1= ,B.4CB ,B15 B+D1 31-+,<D1-5 -./ +2J.3I,<.= .=J1,J H+,$L8 1P +=E $LU 1P7 31JI1(,<D1-5K +=E7 (.=J1T41=,-57 ,B15 (+= B+3E-5 21 (.=J<E131E,3+=JI+31=,7 ,B15 E<JI-+5 J<C=<V(+=,-5 B<CB13 I',5I1 (.=E4(,<D<,5 ,B+= E1-+G.JJ<,1JL#B1J1 JI1(,+(4-+3 V=E<=CJ .I1=1E ,B1 /+5 G.3 ,B1 G+23<(+,<.= .G #*j I[= \4=(,<.=JN8&WO7 +=E ,. ,B1 E1D1-.I?1=, .G + =1/ ,1(B=.-.C5 1=,<31-5 2+J1E .= n<=D<J<2-1(<3(4<,Jo N8&6[8&9OL b+=5 <==.D+,<D1 +II-<(+,<.=J 31T4<31 ,3+=JI+31=, 1-1(,3.=<(JJ4(B +J J,+(Z1E J.-+3 (1--J7 ,3+=JI+31=, J(311=J7 .3 G4=(,<.=+- /<=E./J ,B+, C1=13+,1J.-+3 1-1(,3<(<,5L

*4Q-j$ <J 25 G+3 ,B1 ?.J, J,4E<1E J5J,1? .G ,B1 G+?<-5 .G E1-+G.JJ<,1 #*jJ72.,B ,B1.31,<(+--5 +=E 1RI13<?1=,+--5L 0.3 + -.=C ,<?17 ,B131 B+J 211= =. +C311?1=,1<,B13 .= ,B1 .3<C<= .G ,B1 I',5I1 (.=E4(,<D<,5 .3 .= ,B1 1-1(,3.=<( 2+=EC+I .G ,B1I431 (35J,+-L b1+J431?1=,J .G ,B1 E<31(, optical 2+=EC+I HEE<3

C K .G *4Q-j$ G+--<= ,B1 3+=C1 G3.? $LU 1P ,. &LU 1P N8U7 $)7 8&67 8&U[8AUO7 /<,B ?.J, D+-41J <=,B1 <=,13D+- &LA[&L% 1PL #B1J1 1RI13<?1=,J +-J. 5<1-E + -+3C1 E<JI13J<.= .G <=E<31(,C+IJ HE <=E

C K7 G3.? 8L6W 1P ,. $L8 1P7 /<,B .=1 1RI13<?1=, ?1+J43<=C $LUU 1P N8A6OLm=G.3,4=+,1-57 ,B131 <J .=-5 .=1 IB.,.1?<JJ<.= 1RI13<?1=, N8A9O ,B+, C<D1J + D+-41.G &LW 1P G.3 ,B1 quasi-particle 2+=EC+IL >.,1 ,B+, ,B1 optical +=E quasi-particle

C+IJ E<GG13 25 ,B1 1R(<,.= 2<=E<=C 1=13C5L *.=(13=<=C *4M=j$7 .I,<(+- 1RI13<?1=,J?1+J431E EE<3

C 21,/11= &LU 1P +=E ALAW 1P N$$7 8W)7 8W8O7 /<,B .=-5 .=1 1J,<?+,<.=

.G E <=EC +, 8LAA 1P N8W)OL

03.? ,B1 ,B1.31,<(+- I13JI1(,<D17 ,B1 J<,4+,<.= <J +-J. T4<,1 (.?I-1R7 1D1= <G,B1 G4-- *4 &d JB1-- JB.4-E 1R(-4E1 ,B1 J,3.=C-5 (.331-+,1E 1-1(,3.= 31C<?1L #B1J1?+,13<+-J +31 4J4+--5 J,4E<1E <= -<,13+,431 /<,B<= `0#7 4J<=C ,B1 J,+=E+3E Y`Q .3ccQ N8&$7 8AA7 8A67 8A97 8W$[8WAOL !1 (+= 1RI1(, ,B+, ,B1 k" 2+=E J,34(,431J5<1-E 4=E131J,<?+,1E 2+=EC+IJ +=E ,B1 B523<E<f+,<.= 21,/11= ,B1 d 1-1(,3.=J .G,B1 ?1,+- +=E p 1-1(,3.=J .G .R5C1= <J ?<J31I31J1=,1E N8A7 6$OL

S.,B Y`Q/ccQ + U +=E B523<E G4=(,<.=+-J B+D1 211= +II-<1E ,. J,4E5 ?1?'213J .G ,B1 E1-+G.JJ<,1 G+?<-5 /<,B J.?1 J4((1JJ N6W7 8W&7 8WW[8W%OL >.,1 ,B+, ,B1J1

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

(+-(4-+,<.=J 5<1-E T4+J<'I+3,<(-1 C+IJ7 +=E =., .I,<(+- C+IJL #. 1D+-4+,1 ,B1J1 -+,,13T4+=,<,<1J7 .=1 ?.J,-5 31J.3,J ,. ,B1 J.-4,<.= .G ,B1 S"FL 0.3 E1-+G.JJ<,1 J,34(,431J,B131 <J .=1 J4(B (+-(4-+,<.= J,+3,<=C G3.? + ccQ + U 2+=E J,34(,431 N8W6OL M,5<1-EJ G.3 *4Q-j$ + D135 -+3C1 1R(<,.= 2<=E<=C 1=13C5 .G +2.4, )LW 1P G.3 ,B1V3J, E<31(, ,3+=J<,<.=L #B1 (B.<(1 .G ,B1 I+3+?1,13 U /+J G.4=E ,. B+D1 J<C=<V(+=,(.=J1T41=(1J .= ,B1 /<E,B .G ,B1 2+=EC+I7 24, <, E<E =., +GG1(, J<C=<V(+=,-5 ,B11R(<,.= 2<=E<=C 1=13C5L !1 (+= ,B4J +JJ4?1 ,B+, )LW 1P <J + 31+J.=+2-1 1J,<?+,1 .G,B<J -+,,13 T4+=,<,5L

M= ,B1 G.--./<=C7 /1 E<J(4JJ (+-(4-+,<.=J N6W7 8W%O .G ,B1 2+=E J,34(,431J .G*4Q-j$ +=E *4M=j$7 (.?I+3<=C ,B1 ?.J, +((43+,1 ,B1.31,<(+- ,..-J +D+<-+2-1 <=,B1 (.??4=<,5L #B1J1 <=(-4E1 J,+=E+3E Y`Q7 B523<E G4=(,<.=+-J H=+?1-5 S&Y@;N8W9O +=E ,/. e+D.3J .G g15E["(4J13<+[F3=f13B.G7 g"F)& +=E g"F)6 N%W7 %6OK7Y`Q+U 7 G)W)7 +=E J(GW. !1 1RI1(, J(GW ,. 21 ,B1 ?.J, +((43+,1 ab initio

+II3.+(BLM= 0<C431 $L887 /1 JB./ + J4??+35 .G (+-(4-+,1E E<31(, +=E <=E<31(, IB.'

,.1?<JJ<.= C+IJ .G *4Q-j$ N6WOL #B1 (.331JI.=E<=C 2+=E J,34(,431J +31 JB./=<= 0<C431 $L8$L #B1 ?<=<?4? EE<3

C .G *4Q-j$ <J +-/+5J G.4=E +, L7 /B131 ,B1E<I.-1 ,3+=J<,<.= 21,/11= ,B1 2+=E 1EC1 J,+,1J <J +--./1E N8&$OL Q-- (+-(4-+,<.=J71R(1I, J(GW 7 C<D1 + G4=E+?1=,+- E <=E

C 21,/11= ,B1 (.=E4(,<.= 2+=E ?<=<?4?+, Ŵ +=E ,B1 D+-1=(1 2+=E ?+R<?4? +-.=C ,B1 Ŵ−F -<=1L #B1 1RI13<?1=,+-E+,+ G.3 .I,<(+- C+IJ +31 +-J. I31J1=,1E /<,B += 133.3 2+3 ,B+, 31e1(,J ,B1 E<J'I13J<.= .G ,B1 ?.J, -<Z1-5 D+-41J G.4=E <= -<,13+,431L Y`Q 1RB<2<,J7 +J 1RI1(,1E7

LDA LDA+U HSE03 HSE06 G0W0 scGW scGW+P0

Egindirect

Egdirect

∆ =Egdirect

−Egindirect

Exp. direct gap

indirect

Figure 2.11 Bandgaps of CuAlO2 using LDA, LOA + U, hybrid, G0W0, scGW, and scGWincluding model polaronic corrections. The horizontal zones contain data extracted fromvarious optical experiments. (From [65].)

!"#$%&'()$ !"#$%&'*+,-./ 01234+35 67 $)8& 9:6 ;3<=,13 >+?1: @1, ,. *.?1 #3<?: $AA?? × 869??

Γ F L Z Γ Γ

–2.0

F L Z Γ Γ

F L Z Γ

Figure 2.12 Band structures for CuAl O2: comparison of LDA (red dashed lines) with scGW(a), HSE03 (b), and L DA + U (c). (From [65].)

,B1 J?+--1J, C+IJL S+J<(+--5 1D135 +II3.+(B 215.=E <, .I1=J 4I ,B1 C+I 25 E<G'G131=, +?.4=,J +=E ?.E<V1J ,B1 2+=E E<JI13J<.=JL #B1 E<31(, +=E <=E<31(, C+IJB+D1 J<?<-+3 21B+D<.3J <= ,B1 E<GG131=, ,B1.3<1J7 +=E 2.,B <=(31+J1 /B1= C.<=C G3.?Y`Q < G W < !"# $ < !"# % < &$'() < *+GW ,

'--./01 23 345 6/75+3 1289 :5 8-/03 -;3 3423 <-*3 -= 345 <534-6* 1/>5 75*;?3*

3423 275 :/34/0 345 5@857/<5032? 72015 :450 20 5@+/3-0 A/06/01 50571B -= 27-;06

,C 5D EFC%G /* +-0*/65756, H4/* /* 37;5 =-7 'IJ+U, G W 9 345 4BA7/6* !"# $9

206 !"# %, !-:5>579 =-7 *+GW 206 5>50 =-7 &$'()9 345 345-753/+2? 128 /* ?27157

AB 2A-;3 FKF,C 5D 3420 345 5@857/<5032? L06/01*,

M5 -A*57>5 3423 345 6/==5750+5 E6/71 − E /06

1 65+752*5* :/34 345 *-84/*3/+23/-0 -=

345 <534-69 752+4/01 0527?B N57- =-7 345 *+GW+2?+;?23/-0, H4/* /* 2 +-0*5O;50+5

-= 345 672*3/+ +42015 -= 345 +-06;+3/-0 A206 6/*857*/-09 :4/+4 6/*8?2+5* 345 +-0P

6;+3/-0 </0/<;< =7-< Ŵ 3- L :450 *+GW /* 288?/56 Q*55 R/1;75 S,FST, H45 ?-:5*3

+-06;+3/-0 A206* +7-** 2?-01 345 ?/05 A53:550 345 4/14P*B<<537B 8-/03* L 206 F9

:4/+4 75*;?3* /0 2 */10/L+203 +42015 /0 345 +4272+357 -= 345 +-06;+3/-0 A206 </0/P

<;<, J3 Ŵ 206 Z 345 A206 42* 20 E1 -= U; $d +4272+357 :/34 2 */10/L+203 2<-;03

-= s 206 p *3235* =7-< J?9 V9 206 U;, V0 345 -3457 42069 23 L 206 F9 345 A-33-<

+-06;+3/-0 A206 /* +-<8-*56 -= U; s 206 V p *3235*, H4575=-759 /3 /* 0-3 *;787/*/01

3423 GW +-775+3/-0 /* *- *37-01?B kP8-/03 658506503, R/1;75 S,F$ 6/*8?2B* 345

6/==5750+5 A53:550 345 5@85+323/-0 >2?;5 -= 345 !273755 8-3503/2? 206 345 5@+42015

8273 -= 345 *5?=P50571B9 +-0*37;+356 :/34 IRH :2>5=;0+3/-0* -7 :/34 *+UV!"#W

:2>5=;0+3/-0*, H45 5@+42015 *5?=P50571B +42015* ;8 3- X 5D9 :4/?5 345 !273755

357< /* 723457 +-0*3203, M5 -=350 0-35 3423 /0 *5</+-06;+3-7 84B*/+* -05 2**;<5*

3423 345 O;2*/P8273/+?5 +-775+3/-0* +20 A5 <-65?56 AB 2 7/1/6 *4/=3 Q345 *-P+2??56

YM"HSZ$P+ S YM"HSZ$PU23?-: R5A7;27B %9 S F$ [\% )7/0357 ]2<5\ (53 3- U-<5 H7/<\ SXX<< × F%[<<

Γ LF Z Γ

M=H hartree

Figure 2.13 (a) Kohn–Sham conduction band structure of CuAlO2. (b) Difference betweenthe expectation value of the operator M (M = HHar tr ee, 6x ) constructed with DFT wavefunc-tions or with scCOHSEX wavefunctions for the bottom conduction band. (From [157].)

*+/**-7 -85723-7T, R7-< 345*5 75*;?3* /3 =-??-:* 3423 -05 *4-;?6 75=72/0 =7-< ;*/01

34/* */<8?5 2887-@/<23/-0 =-7 65?2=-**/35 HUV*,

!BA7/6 <534-6* *;+4 2* !"# $9 !"# %9 206 &$'() -850 345 A206128* 206

756;+5 345 6/==5750+5 E6/71 − E /06

1 9 =-??-:/01 345 37506 -A*57>56 =-7 *+GW +2?+;?2P

3/-0*, H45 +-06;+3/-0 A206 </0/<;< QU&^T =7-< !"# $ /* *3/?? ?-+2356 23 Ŵ9 A;3

345 6/==5750+5 A53:550 345 U&^* 23 L 206 Ŵ 153* */10/L+203?B *<2??57, H45 <2/0

5==5+3 -= 'IJ+U /* 3- -850 345 'IJ 128 AB 20 2<-;03 3423 +20 A5 +-037-??56

AB 345 8272<5357 U. H45 6/==5750+5 E6/71 + E /06

1 /* /0+752*56 :/34/0 345 'IJ+ U

2887-@/<23/-09 6;5 3- 345 +42015 -= +4272+357 =-7 345 ?-:5*3 +-06;+3/-0 A206 2?-01

345 *B<<537B ?/05 A53:550 Ŵ 206 F, M5 +20 +-0+?;65 3423 4BA7/6 +2?+;?23/-0* 1/>5

2 A53357 65*+7/83/-0 Q/,5,9 +?-*57 3- *+GW T -= A206 6/*857*/-0* 3420 'IJ+U 9 5>50

/= 345 3:- 2887-2+45* B/5?6 */</?27 A206128*,

R-7 U;_0VS 345 *<2??5*3 E6/71 /* ?-+2356 23 Ŵ9 :4575 -83/+2? 3720*/3/-0* 275

=-7A/6650 EF$SG, J <520/01=;? +-<827/*-0 :/34 -83/+2? 5@857/<503* <;*3 3450

+-0*/657 345 128 23 ', &-34 37506* 206 O;203/323/>5 75*;?3* 275 202?-1-;* 3- 34-*5

=-7 U;J?VS, _0 8273/+;?279 *+GW 212/0 B/5?6* E6/71 ?27157 AB FKF,C 5D 3420 345

5@857/<5032? 72015 E%C9 FCZG,

M5 *375** 212/0 34239 3- 62359 *+GW /* 271;2A?B 345 A5*3 <534-6 2>2/?2A?5 3-

5*3/<235 A206128* -= :/65P128 *5</+-06;+3-7*9 206 3423 /3 1/>5* 5@+5??503 75*;?3*

=-7 +-<8-;06* ?/.5 U;SV 206 U;_0Q", "5TS EFX9 %SG, _3 /* ;0?/.5?B 3423 345 875*50+5

-= 65=5+3* +20 5@8?2/0 345 6/*+75820+B A53:550 5@857/<503* 206 *+GW 128*,

!-:5>579 34575 /* 20-3457 5==5+3 3423 42* A550 051?5+356 ;8 3- 0-:`345 +42015

-= *+7550/01 6;5 3- 345 8-?27/N23/-0 -= 345 ?233/+5, H45 8-?27-0 +-0*3203 65L056

2* α) = QF/ǫ∞ − F/ǫ TQ~/SmaS&ω'VT

F/S`:4575 a& /* 345 &-47 726/;*9 ω'V 345

?-01/3;6/02? -83/+2? =75O;50+B -= 345 4/145*3 E; 84-0-0 <-659 ǫ∞ 206 ǫ 345 ?-:

=75O;50+B 5?5+37-0/+ 206 345 *323/+ 6/5?5+37/+ +-0*3203*9 75*85+3/>5?B`<52*;75*

345 *3750134 -= 345 8-?27-0/+ 5==5+3, J++-76/01 3- 345 5@857/<5032? 6232 EFCaG9

;0=-73;0235?B 2>2/?2A?5 -0?B =-7 U;J?VS9 345 8-?27-0 +-0*3203 =-7 34/* *B*35< /*

?2715 Qα) ∼ FT9 /06/+23/01 2 0-0051?/1/A?5 +-037/A;3/-0 -= 345 ?233/+5 8-?27/N23/-0

3- 345 5?5+37-0/+ *+7550/01, _3 /* .0-:0 3423 /0 -3457 /-0/+ +-<8-;06* :/34 */</?27

8-?27-0 +-0*3203* 34/* +20 ?526 3- *47/0./01 -= 345 A206128 AB 2A-;3 F 5D EF% G, J

=;?? *+GW +2?+;?23/-0 /0+?;6/019 /0 20 ab initio =72<5:-7.9 345 5==5+3* -= 345 ?233/+5

8-?27/N23/-0 /* *3/?? A5B-06 752+4, !-:5>579 2 75?/2A?5 5*3/<235 :2* -A32/056 ;*/01

345 <-65? 87-8-*56 AB &5+4*3563 et al, EF% G9 :4/+4 1/>5* 2 *323/+ 75875*50323/-0

-= 345 +-037/A;3/-0 -= ?233/+5 8-?27/N23/-0 3- 345 *+7550/01 A2*56 -0 6/==5750+5 -=

5@857/<5032? *323/+ 6/5?5+37/+ +-0*3203*, &B 857=-7</01 2 8573;7A23/>5 GW *3589

/0+?;6/01 2 <-65? +-037/A;3/-0 3- 345 *+7550/01 6;5 3- ?233/+5 8-?27/N23/-09 -0 3-8

-= 345 *+UV!"#W9D/62? et al, E%CG =-;06 2 ;0/=-7< QkP/0658506503T 65+752*5 -= 345

A206128 AB F,S 5D, J* :5 +20 *55 /0 R/1;75 S,FF9 34/* +-775+3/-0 A7/01* 345 *+GW

75*;?3* =-7 E6/71 :5?? :/34/0 345 5@857/<5032? 72015 Q-0+5 345 5@+/3-0/+ +-775+3/-0

-= 2A-;3 ,C 5D /* 2?*- +-0*/65756T, J* /3 /* -A*57>56 /0 &5+4*3563 et al, EF% G9 345

<-65? 5<8?-B56 3- 2++-;03 =-7 345 ?233/+5 8-?27/N23/-0 +20 -0?B ->575*3/<235 345

+-775+3/-0, J?? 345*5 75*;?3* 8-/03 3- 345 +-0+?;*/-0 3423 345 21755<503 -= 345 -3457

<534-6* :/34 5@857/<503 :2* =-73;/3-;* 206 6;5 3- 2 +20+5??23/-0 -= 577-7*,

_3 /* /03575*3/01 3- 0-35 3423 3- 65*/10 8P3B85 HUV* 345 ;*5 -= ?2B5756 *37;+3;75*

/* 75+-<<50656 /0 -7657 3- /0+752*5 345 >2?;5 -= 345 A206128, _065569 '2U;"V

EF%FG 206 &2U;Q"9"59H5TR EF%SG 6/*8?2B A206 *37;+3;75* >57B */</?27 3- 345 -05

-= U;J?VS, V05 +20 =-75*55 3423 2?*- /0 345*5 <2357/2?* 345 750-7<2?/N23/-0 -=

345 A206128 6;5 3- 345 ?233/+5 +-037/A;3/-0 3- 345 *+7550/01 +-;?6 3450 8?2B 20

/<8-73203 7-?5 206 */10/L+203?B 2==5+3 345 /0357875323/-0 -= 5@857/<503*,

H- +-0+?;659 :5 =-+;* 0-: -0 345 /06/75+3 128* -= R/1;75 S,FF, J02?-1-;*

+-0+?;*/-0* +-;?6 A5 -A32/056 =-7 U;_0VS -7 U;b2VS E%C9 FCZG9 A;3 :5 723457

=-+;* -0 U;J?VS 2* <-75 5@857/<5032? 6232 275 2>2/?2A?5 /0 ?/35723;75, J?? 4BA7/6*

206 GW +2?+;?23/-0* B/5?6 /06/75+3 128* <;+4 ?27157 3420 345 5@857/<5032? 72015

F,%CKS,F 5D9 5>50 32./01 /03- 2++-;03 20B 8-**/A?5 5@+/3-0/+ 206 8-?27-0/+ 5==5+3*,

^-75->579 *+GW9 345 A5*3 <534-6 ;*56 /0 34/* :-7.9 B/5?6* 345 4/145*3 E /061 23

27-;06 C 5D9 :4/?5 345 6/==5750+5 E6/71 − E /06

1 /* /0 150572? <;+4 *<2??57 3420 345

5@857/<5032? >2?;5 Q≈S 5DT9 206 5>50 >20/*4/01 =-7 *+GW +2?+;?23/-0*,

H45*5 275 >57B *37-01 271;<503* /0 =2>-7 -= c-A573*-0 et al, EFC$G :4- *;115*356

3423 345 5@857/<5032? d/06/75+3 128e 2A*-783/-0 :2* 6;5 3- 65=5+3*9 206 *4-;?6

0-3 A5 875*503 /0 345 65=5+3P=755 +-<8-;06, J?*-9 )5??/+57P)-775* et al, EFX%G

O;5*3/-056 345 /0357875323/-0 -= 345 ?-: 50571B 852.* 2* /06/75+3 3720*/3/-0*9 2*

345 2A*-783/-0 +-5=L+/503 /* <-75 3420 3:- -7657* -= <210/3;65 ?27157 3420 /0

3B8/+2? /06/75+3 2A*-783/-0 5615*, _3 42* A550 L7*3 2**57356 3423 345 65=5+3* 3423

+-;?6 A5 75*8-0*/A?5 275 -@B150 /0357*3/3/2?* V/9 2* IRH +2?+;?23/-0* :/34/0 345

'IJ 8756/+3 ?-: =-7<23/-0 50571/5* 206 345 /037-6;+3/-0 -= *3235* /0 345 128 23

,Z 5D 206 F,X 5D EF%$G, ^-75 75+503?B9 "+20?-0 206M23*-0 EF%XG 87-8-*569 A2*56

-0 *+755056 4BA7/6 +2?+;?23/-0*9 3423 203/*/35 U;P-0PJ? U;J? /* 75*8-0*/A?5 =-7 345

=523;75* /0 345 2A*-783/-0 *85+372 23 F,%CKS,F 5D,

_0 *;<<27B9 /3 /* +?527 3423 345 +-<8-;06* -= 345 65?2=-**/35 =2</?B 5@4/A/3

+-<8?5@ 206 ;0;*;2? A206128 84B*/+* 3423 +200-3 A5 +283;756 AB *3206276 345-P

753/+2? 2887-@/<23/-0*, _0 8273/+;?279 345 6/75+3 A206128 /* :5?? 7587-6;+56 AB 345

*3235P-=P345P273 <20BPA-6B 2887-2+45* /= 345 +-037/A;3/-0 -= ?233/+5 8-?27/N23/-0 3-

345 *+7550/01 /* 32.50 /03- 2++-;03, M5 +20 5@85+3 3423 34/* */3;23/-09 -= 2 ?2715 128

3423 /* 756;+56 *;A*3203/2??B AB 8-?27-0/+ 5==5+3*9 /* O;/35 150572? 206 /* 875*503 /0

<20B 8-?27 <2357/2?*, _0 =2+39 345 28827503 1--6 21755<503 A53:550 +2?+;?2356 128*

Q:/34 4BA7/6 =;0+3/-02?* -7 G W T 206 5@857/<5032? 128* =-7 <2357/2?* 2* */<8?5

206 :/65?B *3;6/56 2* '/R +20 A5 2++/65032?, H45 ;06575*3/<23/-0 -= 345 128 AB

345*5 <534-6* Q345 *+GW 128 /* /06556 S 5D ?27157 3420 345 5@857/<5032? 206 G W

128T /* +-<850*2356 AB 345 051?5+3 -= ?233/+5 8-?27/N23/-0 5==5+3*, R;73457<-759 345

<-6/L+23/-0* :/34 75*85+3 3- 345 'IJ f" A206* 275 *37-01?B kP6585065039 :4/+4

<2.5* O;5*3/-02A?5 345 +-<<-0 872+3/+5 -= ;*/01 2 *+/**-7 -85723-7, H45 A206

6/*857*/-0 -A32/056 AB 4BA7/6 =;0+3/-02? +2?+;?23/-0* /* /0 A53:550 345 'IJ 206

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*+GW +2?+;?23/-0* 7;?5 -;3 345 /0357875323/-0 -= 345 ?-: 50571B =523;75* /0 345

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206 35+40-?-1/+2? /03575*3 /* *3/?? ?2+./019 5*85+/2??B +-0+570/01 65=5+3* 206 /0357P

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<534-6*9 345 O;5*3 =-7 /<87->56 2887-@/<23/-0*9 2* :5?? 2* 345 *527+4 =-7 0->5?

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$, "N59 ", 206 ]19 f, QS ZT Physics of Semiconductor Devices9 M/?5BP&?2+.:5??,

X, )527*-09 b, QFaCZT U-0>57*/-0 -= *-?27 3- 5?5+37/+2? 50571B9 Am. J. Phys, /9 CaF,

C, g42-9 Y,9 M2019 J,9 b75509 ^, 206 R5772NN29 R, QFaa[T Fa,[k 5=L+/503 4-05B+-<A

35@3;756 <;?3/+7B*32??/05206 SX,Xk <-0-+7B*32??/05 */?/+-0 *-?27 +5??*9 Appl. Phys.

Lett, 019 FaaF,

%, M26/29 U,9 J?/>/*23-*9 J, 206 f2<<509 I, QS aT ^2357/2?* 2>2/?2A/?/3B 5@8206* 345

-88-73;0/3B =-7 ?2715P*+2?5 84-3->-?32/+* 658?-B<5039Environ. Sci. Technol, "19 S ZS,

Z, "4249 J,9 H-775*9 ),9 H*+4270579 c, et al. QFaaaT )4-3->-?32/+ 35+40-?-1B\ 345 +2*5 =-7

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L?< *-?27 +5??*9 Appl. Phys. A 549 F$FKFXZ,

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5?5+37-0/+ 87-8573/5* /0 U;_0Q"9 "5TS\ 2 L7*3P87/0+/8?5* *3;6B9 Phys. Rev. Lett, 36"9

C%X F,

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+-<<-0 /<8;7/3B9 Phys. Rev. Lett, 0/9 XXCSKXXCC,

Fa, &50.-9 R, 206 f-==BA5719 R, QFa[XT V83-P5?5+37-0/+ 87-8573/5* -= U;J_VS, J. Phys.

Chem. Solids, "/9 CZKCa,

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+-06;+3/01 U;b2VS 34/0 L?<* -0 *2884/75 Q FT *;A*37235* AB 8;?*56 ?2*57 658-*/3/-09

J. Appl. Phys, 249 FZa KFZa$,

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3720*827503 -@/65 *5</+-06;+3-7 U;_0VS :/34 65?2=-**/35 *37;+3;759 Appl. Phys. Lett,

029 FC[$KFC[C,

S$, !-450A5719 ), 206 f-409 M, QFa%XT _04-<-1505-;* 5?5+37-0 12*9 Phys. Rev, 3159

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S%, )5765:9 Y,),9 &;7.59 f, 206 #70N574-=9 ^, QFaa%T b50572?/N56 1726/503 2887-@/<23/-0

<265 */<8?59 Phys. Rev. Lett. 009 $[%CK$[%[, #7723;<\ /A/6, 029 F$a%Q#T QFaaZT,

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$F, >20 "+4/?=1227659 ^,9 f-320/9 H, 206 R2?55>9 ", QS %T o;2*/8273/+?5 *5?=P+-0*/*3503

bM 345-7B9 Phys. Rev. Lett, 459 SS%X S,

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g0V 206 b2J*9 Phys. Rev. B 029 S$CF X,

$X, g;01579 J, QS $T )72+3/+2? 6-8/01 *87/0+/8?5*9 Appl. Phys. Lett, 219 CZ,

$C, &7-3457*9 #,],9 _N<2B?->9 J,R,9 ]-7<2069 Y,V, et al. QS [T J++;7235 *-?/6P*3235

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658506503 ?/0527 75*8-0*59 Phys. Rev. Lett, //9 S[C KS[CS,

XS, b7-**9 #,f,i, 206 f-409 M, QFa[%T '-+2? 650*/3BP=;0+3/-02? 345-7B -= =75O;50+BP

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X$, c;0159 #, 206 b7-**9 #,f,i, QFa[XT I50*/3BP=;0+3/-02? 345-7B =-7 3/<5P658506503

*B*35<*9 Phys. Rev. Lett, / 9 aaZKF ,

XX, >20 '55;:509 c, QS FT f5B +-0+583* /0 3/<5P658506503 650*/3BP=;0+3/-02? 345-7B9

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3/<5P658506503 650*/3BP=;0+3/-02? 345-7B9 Phys. Rev. Lett, 059 FSFSKFSFC,

XZ, &-33/9 ",9 "+4/06?<2B79 J,9 I5? "-?59 c, 206 c5/0/019 ', QS ZT H/<5P658506503

650*/3B =;0+3/-02? 345-7B =-7 5@350656 *B*35<*9 Rep. Prog. Phys, 069 $CZ,

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+-775?23/-0 .5705? /0 5@350656 *B*35<*9 Phys. Rev. B 529 F%CF [,

Xa, &-33/9 ",9 "-33/?59 R,9 D2*39 ], et al. QS XT '-01P72015 +-037/A;3/-0 3- 345 5@+42015P

+-775?23/-0 .5705? -= 3/<5P658506503 650*/3B =;0+3/-02? 345-7B9 Phys. Rev. B 549

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CS, "427<29 ",9 I5:4;7*39 Y,f,9 "20029 J, 206 b7-**9 #,f,i, QS FFT &--3*3728 2887-@P

/<23/-0 =-7 345 5@+42015P+-775?23/-0 .5705? -= 3/<5P658506503 650*/3B =;0+3/-02?

345-7B9 Phys. Rev. Lett, 3609 F[%X F,

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75*8-0*5 =;0+3/-0* /0 3/<5P658506503 650*/3BP=;0+3/-02? 345-7B9 Phys. Rev. Lett, 439

C%X S,

CX, "3;A0579 c,9 H-.23?B9 _,D, 206)20.723->9V, QS XT #@+/3-0/+ 5==5+3* /0 3/<5P658506503

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C[, !BA573*509 ^,", 206 '-;/59 ",b, QFa[%T #?5+37-0 +-775?23/-0 /0 *5</+-06;+3-7* 206

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%$, b233/9^,9 &7;05>2?9 R,9 V?5>20-9 D, 206c5/0/019 ', QS ZTi0657*3206/01 +-775?23/-0*

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%X, f2<<57?206579 I,9 ^27O;5*9 ^, 206 &-33/9 ", QS FFT &206 *37;+3;75* -= U;Sg0"0"X206 U;Sg0"0"5X =7-< <20BPA-6B <534-6*9 Appl. Phys. Lett, 429 SXFaFC,

%C, D/62?9 Y,9 H720/9 R,9 &7;05>2?9 R, et al. QS F AT #==5+3* -= 5?5+37-0/+ 206 ?233/+5 8-?27P

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Lett, 36"9 F$%X F,

%%, '/9 Y,',9 c/10205*59 b,^,9 U42019 #f, et al. QS ST bM *3;6B -= 345 <532?P/0*;?23-7

3720*/3/-0 -= A++ 4B67-1509 Phys. Rev. B 559 $CF S,

%Z, c/0.59 ),9 o35/*49 J,9 ]5;15A2;579 Y, et al. QS CT R-+;* -0 84-3-5</**/-0 206 5?5+P

37-0/+ *37;+3;75 +-<A/0/01 1: +2?+;?23/-0* :/34 5@2+3P5@+42015 650*/3BP=;0+3/-02?

345-7B\ 20 202?B*/* -= >2?50+5PA206 84-3-5</**/-0 =-7 +-<8-;06 *5</+-06;+3-7*9

New J. Phys, 09 FS%,

%[, Y/2019!,9b-<5NPJA2?9 c,_,9 c/0.59 ), 206 "+45=p579^, QS aT '-+2?/N56 206 /3/057203

*3235* /0 ?203420/65 -@/65* ;0/356 AB bM'IJ+i9 Phys. Rev. Lett, 36 9 FS%X $,

%a, f/-;82./*9 #,9 g42019 ),9 U-4509 ^,', 206 '-;/59 ",b, QS [T bM O;2*/8273/+?5

+-775+3/-0* 3- 345 'IJ + imbbJ + i 5?5+37-0/+ *37;+3;75 -= A5+ 4B67-1509 Phys.

Rev. B 009 FCCFFX,

Z , R;+4*9 R, 206 &5+4*35639 R, QS [T _06/;<P-@/65 8-?B<-784* =7-< L7*3 87/0+/8?5*\

o;2*/8273/+?5 5?5+37-0/+ *3235*9 Phys. Rev. B 009 FCCF Z,

ZF, f75**59 b, 206 R;734<n;??579 Y, QFaa%2T #=L+/50+B -= 2AP/0/3/- 3-32? 50571B +2?+;?23/-0*

=-7 <532?* 206 *5</+-06;+3-7* ;*/01 2 8?205P:2>5 A2*/* *539 Comput. Mater. Sci.

59 FC,

ZS, f75**59 b, 206 R;734<n;??579 Y, QFaa%AT #=L+/503 /35723/>5 *+45<5* =-7 ab initio 3-32?P

50571B +2?+;?23/-0* ;*/01 2 8?205P:2>5 A2*/* *539 Phys. Rev. B /"9 FFF%aKFFF[%,

Z$, )2/579 Y,9 ^27*<209 ^, 206 f75**59 b, QS ZT, M4B 6-5* 345 &$'() !RmIRH 4BA7/6

=;0+3/-02? =2/? =-7 <532?*q9 J. Chem. Phys, 3 09 SXF $,

ZX, )2/579 Y,9 ^27*<209 ^, 206 f75**59 b, QS [T I/5?5+37/+ 87-8573/5* 206 5@+/3-0* =-7

5@350656 *B*35<* =7-< 4BA7/6 =;0+3/-02?*9 Phys. Rev. B 029 FSFS F,

ZC, !5B69 Y,9 "+;*57/29 b,#, 206 #70N574-=9 ^, QS $T !BA7/6 =;0+3/-02?* A2*56 -0 2

*+755056 +-;?-<A 8-3503/2?9 J. Chem. Phys, 3329 [S Z,

Z%, !5B69 Y,9 "+;*57/29 b,#, 206 #70N574-=9 ^, QS %T #7723;<\ 4BA7/6 =;0+3/-02?* A2*56

-0 2 *+755056 +-;?-<A 8-3503/2?9 J. Chem. Phys, 3 "9 SFaa %Q#T,

ZZ, bB1/9 R, 206 &2?6575*+4/9 J, QFa[aT o;2*/8273/+?5 50571/5* /0 *5</+-06;+3-7*\ "5?=P

50571B +-775+3/-0 3- 345 ?-+2?P650*/3B 2887-@/<23/-09 Phys. Rev. Lett, 5 9 SF% KSF%$,

Z[, &B?206579 I,^, 206 f?5/0<209 ', QFaa T b--6 *5</+-06;+3-7 A206 128* :/34 2

<-6/L56 ?-+2?P650*/3B 2887-@/<23/-09 Phys. Rev. B "39 Z[%[KZ[ZF,

Za, U?27.9 ", 206 c-A573*-09 Y, QS F T "+755056 5@+42015 650*/3B =;0+3/-02? 288?/56 3-

*-?/6*9 Phys. Rev. B 2 9 [CS [,

[ , J?.2;*.2*9 J,9 &7-O>/*39 ), 206 )2*O;275??-9 J, QS F T I5=5+3 ?5>5?* 347-;14 4BA7/6

650*/3B =;0+3/-02?*\ _0*/143* 206 288?/+23/-0*9 Phys. Status Solidi B "29 ZZC,

[F, ^27O;5*9 ^,J,',9 D/62?9 Y,9 V?/>5/729 ^,Y,H, et al. QS FFT I50*/3BPA2*56 </@/01

8272<5357 =-7 4BA7/6 =;0+3/-02?*9 Phys. Rev. B 219 $CFFa,

[S, b7-**9 #,f,i,9 I-A*-09 R,Y, 206 )5357*/?.29 ^, QFaa%T Topics in Current Chemistry9

>-?, F[F Q56, c,R, ]2?5:2j*./T9 "87/0157PD57?219 !5/65?A5719 8, [F,

[$, b-0N59 W,9 b4-*5N9 ), 206b-6AB9 c,M, QFaaCT I50*/3BP8-?27/N23/-0 =;0+3/-02? 345-7B

-= 345 75*8-0*5 -= 2 857/-6/+ /0*;?23/01 *-?/6 3- 20 5?5+37/+ L5?69 Phys. Rev. Lett, 0"9

X $CKX $[,

[X, &-33/9 ",9 R-;7752;9 J,9 ]1;B509 R, et al. QS CT #0571B 65850650+5 -= 345 5@+42015P

+-775?23/-0 .5705? -= 3/<5P658506503 650*/3B =;0+3/-02? 345-7B\ 2 */<8?5 <-65? =-7

*-?/6*9 Phys. Rev. B 0 9 FSCS $,

[C, &5??2A27A29 U,9 b-0Nr2?5N9 Y, 206 c/0+r-09 U, QFaa%T V83/+2?P2A*-783/-0 *85+37;< 0527

345 5@+/3-0 A206 5615 /0 U;b2"S 23 C f9 Phys. Rev. B /19 ZZaSKZZa%,

[%, '5>+50.-9 ",9 "B7A;9 ],],9 H5N?5>209 D,#, et al. QS ZT V83/+2? *85+372 206 50571B

A206 *37;+3;75 -= */01?5 +7B*32??/05 U;b2"S 206 U;_0"S9 J. Phys. Condens. Matter,

349 XC%SSS,

[Z, c/015/**509 Y,9 c51-?/0/9 Y,', 206 '5:-0+N;.9 ", QFaZ$T V83/+2? 87-8573/5* -= U;b2"S9

Surf. Sci. 109 ZZZKZ[C,

[[, H5??9 &, 206 f2*8579 !,^, QFaZ$T #@+/3-0* 206 345 *8/0P-7A/3 *8?/33/01 /0 U;b2"S9

Phys. Rev. B 09 ZX KZXS,

[a, b-0N59 W,9 c/10205*59 b,^,9 D57*3725359 ^, et al. QS CT J A7/5= /037-6;+3/-0 3- 345

J&_]_H *-=3:275 82+.2159 Z. Kristallogr, 69 CC[KC%S,

a , H7-;??/579 ], 206 ^273/0*9 Y,', QFaaFT #=L+/503 8*5;6-8-3503/2?* =-7 8?205P:2>5 +2?P

+;?23/-0*9 Phys. Rev. B "19 Faa$KS %,

aF, &?n-+4?9 ),#, QFaaXT )7-j5+3-7 2;1<50356P:2>5 <534-69 Phys. Rev. B /69 FZaC$K

FZaZa,

aS, f75**59 b, 206 Y-;A5739 I, QFaaaT R7-< ;?372*-=3 8*5;6-8-3503/2?* 3- 345 87-j5+3-7

2;1<50356P:2>5 <534-69 Phys. Rev. B /49 FZC[KFZZC,

a$, ^27/0/9 J,9 !-1209 U,9 b7/</019 ^, 206 D27*20-9 I, QS aT (2<A-\ J0 2A /0/3/- 3--?

=-7 5@+/356 *3235 +2?+;?23/-0*9 Comput. Phys. Commun, 3269 F$aSKFX $,

aX, Y2==59 Y,#, 206 g;01579 J, QFa[XT H45-7B -= 345 A206P128 20-<2?B /0 J&US +42?P

+-8B7/35 *5</+-06;+3-7*9 Phys. Rev. B 49 F[[S,

aC, ^57/0-9 Y,^,9 ^273rs0 65 D/62?5*9 Y,',9 ^24203B9 ", et al. QFaa%T U-<8-*/3/-0 5==5+3*

-0 345 +7B*32? *37;+3;75 -= U;_0"59 J. Appl. Phys, 269 C%F ,

a%, Y2==59 Y,#, 206 g;01579 J, QFa[$T J0/-0 6/*8?2+5<503 206 A206 128 20-<2?B /0 357027B

J&US +42?+-8B7/35 *5</+-06;+3-7*9 Phys. Rev. B 09 CFZ%,

aZ, Y/2019 R,I, 206 R5019 Y,(, QS [T #?5+37-0/+ <-6/L+23/-0 -= U;PA2*56 +42?+-8B7/35

*5</+-06;+3-7* /06;+56 AB ?233/+5 65=-7<23/-0 206 +-<8-*/3/-0 2?+45<B9 Semicond.

Sci. Technol. 19 SC F,

a[, J1;/?5729 _,9 D/62?9 Y,9 M240r-09 ), et al. QS FFT R/7*3P87/0+/8?5* *3;6B -= 345 A206

*37;+3;75 206 -83/+2? 2A*-783/-0 -= U;b2"S9 Phys. Rev. B 2"9 [CFXC,

aa, J?-0*-9 ^,_,9 M2./329 f,9 )2*+;2?9 Y, et al. QS FT V83/+2? =;0+3/-0* 206 5?5+37-0/+

*37;+3;75 -= U;_0"5S9 U;b2"5S9 U;_0"S9 206 U;b2"S9 Phys. Rev. B 519 ZCS $,

F , I-<2/09 U,9 '27/A/29 ",9 H2;0/5729 ", 206b;/??5<-?5*9 Y,R, QS $TJA /0/3/- +2?+;?23/-0

-= /037/0*/+ 8-/03 65=5+3* /0 U;_0"5S9 J. Phys. Chem. Solids 5"9 F%CZKF%%$,

F F, t62N5:*./9 Y,9 Y-+4B<9 ),H, 206 )27?/0*./9 f, QS ST &206 *37;+3;759 &-70 5==5+3/>5

+42715*9 206 ?233/+5 6B02</+* -= U;_0" =7-< 2A /0/3/- +2?+;?23/-0*9 J. Chem. Phys,

3309 SZS%,

F S, M5/9 ",!, 206 g;01579 J, QFa[[T c-?5 -= <532? 6 *3235* /0 __PD_ *5</+-06;+3-7*9 Phys.

Rev. B 109 [aC[K[a[F,

F $, JA7242<*9 ",U, 206 &570*35/09 Y,', QFaZ$T )/5N-5?5+37/+ 0-0?/0527 -83/+ U;b2"S 206

U;_0"S +7B*32? *37;+3;75\ ";A?233/+5 6/*3-73/-0 /0 JF&$ 206 JS&X 3B85 +42?+-8B7/35*9

J. Chem. Phys, /49 CXFCKCXSS,

F X, g42019 ",&,9 M5/9 ",9 g;01579 J, 206 f232B2<2P(-*4/629 !, QFaa[T I5=5+3 84B*/+*

-= 345 U;_0"5S +42?+-8B7/35 *5</+-06;+3-79 Phys. Rev. B /09 a%XS,

F C, W/2-9 !,g,9 (2019 ',U, 206 c-+.5339 J, QFaaXT "37;+3;72?9 -83/+2?9 206 5?5+37/P

+2? 87-8573/5* -= 58/32@/2? +42?+-8B7/35 U;_0$"5C L?<*9 J. Appl. Phys, 059 FC $K

F %, )57**-09 U, 206 g;01579 J, QS CT U-<8-*/3/-02??B /06;+56 >2?50+5PA206 -==*53 23

345 172/0 A-;0627B -= 8-?B+7B*32??/05 +42?+-8B7/35* +75235* 2 4-?5 A277/579 Appl. Phys.

Lett, 209 SFFa X,

F Z, &-33/9 ",9 f2<<57?206579 I, 206 ^27O;5*9 ^,J,', QS FFT &206 *37;+3;75* -=

U;Sg0"0"X 206 U;Sg0"0"5X =7-< <20BPA-6B <534-6*9 Appl. Phys. Lett, 429

SXFaFC,

F [, U4509 ",9 b-019 W,b,9 M2?*49 J, 206 M5/9 ",!, QS a2T U7B*32? 206 5?5+37-0/+ A206

*37;+3;75 -= U;Sg0"0WX QW = " 206 "5T 84-3->-?32/+ 2A*-7A57*\ L7*3P87/0+/8?5*

/0*/143*9 Appl. Phys. Lett, 4"9 XFa $,

F a, U4509 ",9 M2?*49 J,9 ';-9 (,9 (2019 Y,!,9 b-019 W,b 206 M5/9 ",!, QS F AT M;73N/35P

657/>56 8-?B3B85* -= .5*357/35 206 *3200/35 O;2357027B +42?+-150/65 *5</+-06;+3-7*9

Phys. Rev. B 2 9 FaCS $,

FF , )57**-09 U, QS F T #?5+37-0/+ 206 -83/+2? 87-8573/5* -= U;Sg0"0"X 206 U;Sg0"0"5X9

J. Appl. Phys, 3609 C$ZF ,

FFF, )2/579 Y,9 J*24/9 c,9 ]21-B29 J, 206 f75**59 b, QS aT U;Sg0"0"X 2* 2 8-3503/2?

84-3->-?32/+ <2357/2?\ 2 4BA7/6 !273755PR-+. 650*/3B =;0+3/-02? 345-7B *3;6B9 Phys.

Rev. B 049 FFCFS%,

FFS, U4509 ",9 b-019 W,b,9 M2?*49 J, 206M5/9 ",!, QS F 2T I5=5+3 84B*/+* -= 345 .5*357/35

34/0PL?< *-?27 +5?? 2A*-7A57 U;Sg0"0"X9 Appl. Phys. Lett, 459 SFa S,

FF$, ]21-B29 J,9 J*24/9 c,9 M24?9 c, 206 f75**59 b, QS F T I5=5+3 =-7<23/-0 206 842*5

*32A/?/3B -= U;Sg0"0"X 84-3->-?32/+ <2357/2?9 Phys. Rev. B 239 FF$S S,

FFX, U4509 ",9 b-019 W,b,9M2?*49 J, 206M5/9 ",!, QS aAT #?5+37-0/+ *37;+3;75 206 *32A/?P

/3B -= O;2357027B +42?+-150/65 *5</+-06;+3-7* 657/>56 =7-< +23/-0 +7-**P*;A*3/3;3/-0

-= __PD_ 206 _P___PD_S +-<8-;06*9 Phys. Rev. B 049 F%CSFF,

FFC, "+4-779 ", QS ZT "37;+3;72? 2*85+3* -= 262<203/05 ?/.5 <;?3/027B +42?+-150/65*9 Thin

Solid Films /3/9 Ca[C,

FF%, &2A;9 b,9 f;<279 (,9 &42*.279 ), 206 c2j29 D, QS [T #==5+3 -= 8-*3P658-*/3/-0

20052?/01 -0 345 17-:34 -= U;Sg0"0"5X 34/0 L?<* =-7 2 *-?27 +5?? 2A*-7A57 ?2B579

Semicond. Sci. Technol, 19 [C S$,

FFZ, !2409 !, 206 "+4;?N59 !, QFa%CT, niA57 O;23570n275 U42?.-150/65 65* b57<20/;<*

;06 g/00*9 Naturwiss / 9 XS%,

FF[, !2??9 ",9 "NB<20*./9 Y, 206 "35:2739 Y, QFaZ[T f5*357/35 U;SQg09R5T"0"X 206

*3200/35 U;SQR59g0T"0"X *37;+3;72??B */</?27 A;3 6/*3/0+3 </0572?*9 Can. Mineral,

359 F$F,

FFa, V?5.*5B;.9 I,9 b;?2B9 ',9 IB6+42.9 _, et al. QS ST "/01?5 +7B*32? 87582723/-0

206 +7B*32? *37;+3;75 -= 345 U;Sg0mU69!1m"0"5X +-<8-;06*9 J. Alloys Compd,

1"69 FXF,

FS , "+4-779 ",9 !-5A?579 !,Y, 206 H->279 ^, QS ZT J 05;37-0 6/==72+3/-0 *3;6B -= 345

*3200/35P.5*357/35 *-?/6 *-?;3/-0 *57/5*9 Eur. J. Mineral, 349 %C,

FSF, J409 ",9 Y;019 ",9 b:2.9 Y, et al. QS F T I5357</023/-0 -= A206 128 50571B -=

U;Sg0"0"5X 34/0 L?<*\ V0 345 6/*+75820+/5* -= 758-7356 A206 128 >2?;5*9 Appl.

Phys. Lett, 409 SFa C,

FSS, f2<-;09 ],9 &-;N-;/329 !, 206 c5N/19 &, QS ZT R2A7/+23/-0 206 +4272+357/N23/-0 -=

U;Sg0"0"X 34/0 L?<* 658-*/356 AB *872B 8B7-?B*/* 35+40/O;59 Thin Solid Films, /3/9

FS$, f2321/7/9 !,9 "2/3-49 f,9 M2*4/-9 H, et al. QS FT I5>5?-8<503 -= 34/0 L?< *-?27 +5??

A2*56 -0 U;Sg0"0"X 34/0 L?<*9 Sol. Energy Mater. Sol. Cells 5/9 FXF,

FSX, ]2.2B2<29 ], 206 _3-9 f, QFaa%T "872B56 L?<* -= *3200/35 U;Sg0"0"X9 Appl. Surf.

Sci, 4 9 FZF,

FSC, "5-?9 Y,",9 '559 ",(,9 '559 Y,U, et al. QS $T #?5+37/+2? 206 -83/+2? 87-8573/5* -=

U;Sg0"0"X 34/0 L?<* 87582756 AB 7= <210537-0 *8;3357/01 87-+5**9 Sol. Energy

Mater. Sol. Cells 0/9 FCC,

FS%, H202.29 H,9 ]2123-<-9 H,9 f2:2*2./9 I, et al. QS CT, )7582723/-0 -= U;Sg0"0"X 34/0

L?<* AB 4BA7/6 *8;3357/019 J. Phys. Chem. Solids 559 FaZ[,

FSZ, g42019 Y,9 "42-9 ',9 R;9 (, 206 W/59 #, QS %T U;Sg0"0"X 34/0 L?<* 87582756 AB *;?P

=;7/N23/-0 -= /-0 A52< *8;335756 875+;7*-7 206 345/7 5?5+37/+2? 206 -83/+2? 87-8573/5*9

Rare Met. /9 $FC,

FS[, c/=59 Y,U,9 I5@3579 c,],9 &7/650A2;149 ),^, 206 D52?9 &,M, QFaZZT V83/+2? 87-8573/5*

-= 345 +42?+-8B7/35 *5</+-06;+3-7* g0b5)S9 g0b5J*S9 U;b2"S9 U;J?"S9 U;_0"5S9

206 J1_0"5S9 Phys. Rev. B 359 XXaFKXC ,

FSa, b2>7/?50.-9 D, 206 &5+4*35639 R, QFaaZT V83/+2? =;0+3/-0* -= *5</+-06;+3-7* A5B-06

650*/3BP=;0+3/-02? 345-7B 206 7206-<P842*5 2887-@/<23/-09 Phys. Rev. B //9 X$X$K

F$ , "B7A;9 ],],9 H/1/0B20;9 _,^,9 ]5<5750+-9 ',', et al. QS CT #@+/3-0 *85+3729 >2?50+5

A206 *8?/33/019 206 50571B A206 *37;+3;75 -= U;b2X_0FX"S 206 U;b2X_0FX"5S +7B*32?*9

J. Phys. Chem. Solids 559 FaZXKFaZZ,

F$F, b/0?5B9 I, QS F T Handbook of Transparent Conductors9 "87/0157 D57?21,

F$S, ]/59 W,9 M5/9 ", 206 g42019 ", QS ST &/8-?27 6-8/01 206 A206P128 20-<2?/5* /0

65?2=-**/35 3720*827503 +-06;+3/>5 -@/65*9 Phys. Rev. Lett, 229 %%X C,

F$$, ^/N-1;+4/9 !,9 !/720-9 ^,9 R;j/3*;9 ", et al. QS ST g0c4SVX\ J 8P3B85 *5</+-0P

6;+3/01 -@/65 :/34 2 >2?50+5 A206 +-<8-*56 -= 2 ?-: *8/0 *3235 -= c4$+ /0 2 Xd%

+-0L1;723/-09 Appl. Phys. Lett, 269 FS Z,

F$X, V1-9 (,9 !/72<23*;9 !,9 ]-<;729 f, et al. QS [T 8P+42005? 34/0PL?< 3720*/*3-7 ;*/01

8P3B85 -@/65 *5</+-06;+3-79 "0V9 Appl. Phys. Lett, 419 $SFF$K $SFF$,

F$C, (2021/9 !,9 i5629 f,9 V4329 !, et al. QS ST R2A7/+23/-0 -= 2?? -@/65 3720*827503 8P0

4-<j;0+3/-0 ;*/01 A/8-?27 U;_0VS *5</+-06;+3/01 -@/65 :/34 65?2=-**/35 *37;+3;759

Solid State Comm, 3 39 FCKFZ,

F$%, &2057j559 J,9 b4-*49 U,9 I2*9 ", 206 U4233-8264B2B9 f, QS CT #?5+37-P-83/+2? +427P

2+357/*3/+* 206 L5?6P5</**/-0 87-8573/5* -= 752+3/>5 6+P*8;335756 8PU;J?VS+ @ 34/0

L?<*9 Physica B, 1069 S%XKSZ%,

F$Z, V4329 !,9 ]-<;729 f,9 !/72<23*;9 !, et al. QS $T R7-03/57 -= 3720*827503 -@/65

*5</+-06;+3-7*9 Solid State Electronics "09 SS%FKSS%Z,

F$[, H4-<2*9 b, QFaaZT ^2357/2?* *+/50+5\ _0>/*/A?5 +/7+;/3*9 Nature (London) 1249 a Z,

F$a, J?.-B9 #, 206f5??B9 ), QS CT H45 *37;+3;75 206 87-8573/5* -= +-8857 -@/65 206 +-8857

2?;</0/;< -@/65 +-23/01* 87582756 AB 8;?*56<210537-0 *8;3357/01 -= 8-:657 327153*9

Vacuum 049 SSFKS$ ,

FX , &2057j559 J, 206 U4233-8264B2B9 f, QS CT "/N5P658506503 -83/+2? 87-8573/5* -=

*8;3357P658-*/356 020-+7B*32??/05 8P3B85 3720*827503 U;J?VS 34/0 L?<*9 J. Appl. Phys,

409 [X$ [,

FXF, &2057j559 J,9 f;06--9 ", 206 U4233-8264B2B9 f, QS $T "B0345*/* 206 +4272+357/N2P

3/-0 -= 8P3B85 3720*827503 +-06;+3/01 U;J_VS 34/0 L?< AB 6+ *8;3357/019 Thin Solid

Films ""69 CKF ,

FXS, &2057j559 J,9 ^2/3B9 c, 206 U4233-8264B2B9 f, QS XT )7582723/-0 -= 8P3B85 3720*P

827503 +-06;+3/01 U;J_VS 34/0 L?<* AB 752+3/>5 6+ *8;3357/019 Mat. Lett, /29

F KF$,

FX$, I/337/+49 H,9 I?-+N/.9 ',9 b;</0*.2B29 H, et al. QS XT )4-3->-?3215 +4272+357/N23/-0

-= U;J?VS +7B*32??/35*9 Appl. Phys. Lett, 2/9 ZXSKZXX,

FXX, b/??/?2069 ",9 )5??/+57P)-775*9 Y,9 "51;729 J, et al. QS ZT #?5+37-0/+ *37;+3;75 -=

U;J?VS 206 U;"+iS 65?2=-**/35* ;0657 875**;759 Phys. Stat. Sol. (b) ""9 $ aK$FX,

FXC, V019 U, 206 b-019 !, QS $T #==5+3* -= 2?;</0;< -0 345 87-8573/5* -= 8P3B85 U;P

J?PV 3720*827503 -@/65 *5</+-06;+3-7 87582756 AB 752+3/>5 +-P*8;3357/019 Thin Solid

Films ""/9 SaaK$ $,

FX%, )5??/+57P)-775*9 Y,9 "51;729 J,9 b/??/?2069 J, et al. QS %T V0 345 A206 128 -= U;J?VS

65?2=-**/359 Appl. Phys. Lett, 229 F[Fa X,

FXZ, H2359 Y,9 Y;9 !,',9^--09 Y,U, et al. QS aT V7/1/0 -= pP3B85 +-06;+3/-0 /0 */01?5P+7B*32?

U;J?VS9 Phys. Rev. B 269 F%CS %,

FX[, (2021/9 !,9 _0-;59 ",9 i5629 f, et al. QS T #?5+37-0/+ *37;+3;75 206 -83-5?5+37-0/+

87-8573/5* -= 3720*827503 8P3B85 +-06;+3/01 U;J?VS9 J. Appl. Phys, 229 XFCaKXF%$,

FXa, (;9 c,",9 '/2019 ",U,9 ';9 U,Y, et al. QS ZT U4272+357/N23/-0 206 -83-5?5+37-0/+

87-8573/5* -= 8P3B85 0P6-856 U;J?VS L?<*9 Appl. Phys. Lett, 469 FaFFFZ,

FC , "2*2./9 ^, 206 "4/<-659 ^, QS $T R2A7/+23/-0 -= A/8-?27 U;_0VS :/34 65?2=-**/35

*37;+3;759 J. Phys. Chem. Solids 5"9 F%ZCKF%Za,

FCF, H58?/09 U,M,9 f2B620->29 H,9 (-;019 I,', et al. QS XT J */<8?5 <534-6 =-7 345

87582723/-0 -= 3720*827503 8P3B85 U2P6-856 U;_0VS L?<*\ );?*56P?2*57 658-*/3/-0

=7-< 2/7P*/035756 U2P6-856 U;S_0SVC 327153*9 Appl. Phys. Lett, 2/9 $Z[aK$ZaF,

FCS, _0172<9 &,Y,9 ^2*-09 H,V,9 J*24/9 c, et al. QS FT #?5+37-0/+ *37;+3;75 206 *<2??

8-?27-0 4-?5 3720*8-73 -= +-8857 2?;<rs02359 Phys. Rev. B 5"9 FCCFFX,

FC$, c-A573*-09 Y,9 )52+-+.9 ),9 H-:?579 ^, 206 ]556*9 c, QS ST #?5+37-0/+ *37;+3;75 -=

8P3B85 +-06;+3/01 3720*827503 -@/65*9 Thin Solid Films "339 a%KF ,

FCX, "4/9 ',9 R2019 g, 206 '/9 Y, QS [T R/7*3P87/0+/8?5* *3;6B -= 8P3B85 3720*827503 +-0P

6;+3/>5 -@/65* U;WVS QW = (9 "+9 206 J?T9 J. Appl. Phys, 36"9 Z$CSZ,

FCC, !;629 ^,],9 (209 (,9 M2?*49 J, et al, QS aT b7-;8P___J >57*;* ___& 65?2=-**/35*\

#?5+37-0/+ *37;+3;75 *3;6B9 Phys. Rev. B 269 $CS C,

FC%, '2*.-:*./9 c,9 U47/*350*509 ],#,9 &?2429 ), 206 )2?20/>5?9 &, QS aT "37-01 5@+/3-0/+

5==5+3* /0 U;J?VS 65?2=-**/35 3720*827503 +-06;+3/>5 -@/65*9 Phys. Rev. B 049 F%CS a,

FCZ, H720/9 R,9 D/62?9 Y,9 &-33/9 ", 206^27O;5*9^,J,', QS F T&206 *37;+3;75* -= 65?2=-**/35

3720*827503 +-06;+3/>5 -@/65* =7-< 2 *5?=P+-0*/*3503 bM 2887-2+49 Phys. Rev. B 2 9

[CFFC,

FC[, "358450*9 ),9 I5>?/09 R,Y,9 U42A2?-:*./9 U,R, 206 R7/*+49^,Y, QFaaXT JA /0/3/- +2?+;?2P

3/-0 -= >/A723/-02? 2A*-783/-0 206 +/7+;?27 6/+47-/*< *85+372 ;*/01 650*/3B =;0+3/-02?

=-7+5 L5?6*9 J. Phys. Chem. 429 FF%S$,

FCa, )5??/+57P)-775*9 Y,9 "51;729 J, 206 f/<9 I, QS aT c5=72+3/>5 /065@ -= 345 U;J?VS

65?2=-**/359 Semicond. Sci. Technol, "9 FC S,

F% , &5+4*35639 R,9 "5/0-9 f,9 !2409 ),!, 206 "+4</639 M,b, QS CT o;2*/8273/+?5 A206*

206 -83/+2? *85+372 -= 4/14?B /-0/+ +7B*32?*\ J?] 206 ]2U?9 Phys. Rev. B 0 9 SXCFFX,

F%F, i5629 f,9 _0-;59 ",9 !/7-*59 ", et al. QS T H720*827503 8P3B85 *5</+-06;+3-7\

'2U;V" ?2B5756 -@B*;?L659 Appl. Phys. Lett, 009 SZ F,

F%S, g2.;32B5>9 J,9 fB.B05*4/9 c,9 "+405/6579 b, et al, QS F T #?5+37-0/+ *37;+3;75 206

5@+/3-0/+ 2A*-783/-0 /0 &2U;U4R QU4="9 "59 206 H5T9 Phys. Rev. B 239 FCCF $,

F%$, !2<2629 _, 206 f232B2<2P(-*4/629 !, QS %T #057153/+* -= 023/>5 65=5+3* /0 U;J? S9

Physica B 10571009 [ [K[FF,

F%X, "+20?-09 I, 206 M23*-09 b, QS F T U-06;+3/>/3B ?/</3* /0 U;J?VS =7-< *+755056P

4BA7/6 650*/3B =;0+3/-02? 345-7B9 J. Phys. Chem. Lett, 39 $FaCK$Faa,

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