A ])articularly intiiguiiig prosi)oct is th a t of using th e interaction between (Hfferent m aterials to create hybrid interfaces with toj)ological ])roperties. For instance depositing normal sc'uiiconductors on toj:) of three-dim ensional toi)ological insulators may result in a s tru c tu re th a t u n d er certain condi tions exhibits toi)ologically protected interface sta te s [114], A n even more a ttra c tiv e ])ros])ect is th a t of using this protocol for transferring to])ologically ])rotected s ta te s to graphene. Sincc' grai)hene-l)ased tra n s isto rs have l)een al ready d e m o n s tra te d [115]. one could tlu'n sp eculate on having grai)hene logic elem ents connected by topological-gra])hene interconnects, i.e.. on realizing an all grai>hene high-perform ance logic circuitry. A m a jo r ad vantage of such a stra te g y is that it is fu lh ’ comj)atible w ith tw o-dim ensional p attern in g , hi th e first section of this c h ap ter we proj)o,se and show t h a t a topological pha.se can be transferrt'd to gi'a])h('U(' by proxim ity w ith the three-dimensi(jual topo-
138 Topological tuiiiiig of tw o a n d t h r e e d im e n s io n a l D irac s e n h m e t a l s
logical in s u l a to r Bi^Se;}. B y u sin g d e n s it y functi<jnal a n d t ra n s ]) o r t th e o r y wo prove t h a t , a t t h e verge of th e che m ica l b o n d f o rm a tio n , a h y b r id s t a t e form s a t tlie graphene/B i^Sc;} interface. T h e s t a t e h a s D irac-cone-like dis p e rs io n a t th e r p o in t a n d a well-defined helical si)in -te x tu re . i n d ic a tin g its to p o lo g ica lly p r o t e c t e d n a tu r e .
M o ti v a te d l)v t h e tw o - d im e n sio n a l D irac-like s e m im e ta llic s t a t e in graj^hene. D irac s e n iim e ta ls in t h r e e d im e n s io n s w ere th e o r e ti c a l l y i)roposed re c e n tly [125]. Using Hrst-jjrinciples c a lc u la tio n s . W a n g a n d co-w orkers p r e d i c te d s o d iu m b i s n i u t h a t e (Na.jBi) a n d c a d m i u m a rs e n id e ( C d3A s2) to b e t h re e -d im e n s io n a l D irac s e m i m e t a l s [I2(i. 127]. T h e i r e xi)e rin ie n ta l re a liz a tio n wa,s not far b e h in d a n d t h e p r e d ic tio n ha.s b e e n verified by m e a n s of a n g le resolved p h o t o e m ission m e a s u r e m e n t s in a r e m a r k a b l y r a p i d flurry of a c ti v it y l)y a n u m b e r of grou])s [128. 129, 130, 131. 132]. In te re s tin g ly , a D ira c s e m i m e t a l s t a t e wa.s also fo u n d in zinc b le n d e c o m i)o u n d s [133]. A p a r t from h o s ti n g a l)ulk
D irac cone, b o t h Na.^Bi a n d C \l3A s2 also show a b a n d inversion at t h e c e n te r
of t h e B rillouin zone. T h i s m e a n s t h a t t h e y e x h ib it a su rfa c e D ira c s i)e c tru m w h e n confined to a sla b geo m e try , a n a lo g o u s t o c o n v e n tio n a l to])ological in s u la t o r s [14]. G ive n th e i r u n iq u e e le c tro n ic s t r u c t u r e t h e y 0])en u p a n e x c itin g p l a t f o r m t o s t u d y to p o lo g ica l p h a s e t r a n s it io n s , in te rw e a v in g tw o a n d th re tv d in ie n s io n a l D irac s ta t e s . In t h e s e co n d h a lf of t h is c h a p t e r we s t u d y t h e i n te r p la y (jf l)ulk a n d su rfa c e D ira c ferm ions in t h r e e - d im e n s io n a l D irac s e m i m e t a l s using first-p rin c ip le s c a lc u la tio n s . B y com l)ining d e n s ity fu n c tio n a l t h e o r y w ith t h e c o h e re n t p o t e n t i a l a p j^ ro x im a tio n we reveal a tojjolog- ical p h a s e t r a n s i t i o n in t h e alloy N a a B ii.^ - S b j, w h e re t h e s y s te m goes from a D ira c s e m im e ta l t o a triv ia l i n s u l a to r u p o n c h a n g in g t h e Sb c o n c e n t r a tion. T h i s t u n i n g of com ])osition allows o n e t o e n g in e e r t h e i)osition of t h e b u lk D ira c ])oints in t h e r e c ijn o c a l s p a ce . In te re s tin g ly , t h e p h a s e t r a n s i t i o n
C h a p t e r 8 139
coiiicideH w i t h a reve rsal of th e l)aiul o r d e r in g b e tw e e n t h e c o n d u c tio n a n d valence b a n d s .
8.1
P ro x im ity induced to p ological sta te in graphene
Se\'('ral pro])osals have l)een a lr e a d y b r o u g h t fo rw ard for m a k i n g grai)h en e a to p o lo g ic a l in s u la to r . In d ee d , o n e of th e first toj)ological i n s u l a to r m o d e l was b a s e d on a s ta g g e r e d h e x a g o n al la ttic e w it h h e lic ity -d e p e n d e n t comi)lex lioj)- p i n g ] )a ra m e te r, s im u l a t in g si)in-orbit i n te r a c tio n , i.e.. t h e K a n e -M e le m odel, w hich we have e n c o u n te r e d s('V('ral tim es in th is th e s is [27. 28]. However, since s p iu - o r b it coujjliug in gra])hene is tiny, a to p o lo g ica l pha.se m a y b e in d u c e d on ly by s tr o n g l y p e r t u r b i n g t h e g rap h e n (' e le c tr o s ta t ic p o t e n tia l, for in s ta n c e by a d so rl)in g h e a v y ions on to]) of t h e s h e e t [llG]. I m p o r ta n t ly , a lt h o u g h t h e o re tic a lly .sound, such i)ro])osal n'cjuires u l t r a - a c c u r a t e f a b r ic a tio n j^recision a n d a p p e a r s r a t h e r c ha llenging in i)ractice. H ere we s uggest a c o m p le te ly difl'erent ap])roach: we introduce' toi)ok)gically i)ro te c te d s t a t e s in gra])hene by ])roxiniity w i t h a la t ti c e - c o n n u e n s u r a te t h r e e - d im e n s io n a l to]K)logical in s u la to r. T h i s h a jjp e n s a t th e g r a i ) h e n e / B i2Se:j interface, a com ])osite which wa-s s y n th e s iz e d a b o u t tw o y e a rs ago [117. 118. 119], I)ut w hose e le c tro n ic s t r u c t u r e still r e m a in s unclear.