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Elektronische Interferenz und Elektronentransport in ¨ Ubergangsmetall-Legierungen H. Solbrig und T. Schmidt am 5. Juli 2005 1. ¨ Ubergangsmetalle in Aluminium 2. Ausgangspunkt Spektraloperator 3. Atomare Netzwerke in Legierungen 4. Was wurde gezeigt ?

Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

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Page 1: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

Elektronische Interferenz und Elektronentransport

in Ubergangsmetall-Legierungen

H. Solbrig und T. Schmidt

am 5. Juli 2005

1. Ubergangsmetalle in Aluminium

2. Ausgangspunkt Spektraloperator

3. Atomare Netzwerke in Legierungen

4. Was wurde gezeigt ?

Page 2: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

1. Ubergangsmetalle in Aluminium: Interferenz

• Elektronen treiben die Struktur zu konstruktiver elektronischer Interferenz

Hume-Rothery

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JONES

FK

ki

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kf

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PSG

DOS

εF↑

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TM-Valenz

negativ

analog: Peierls, Jahn -Teller, martensitischer Ubergang

• Interferenz von Pfaden uber diskretes und kontinuierliches SpektrumFano

Re

Im

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Page 3: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

1. Ubergangsmetalle in Aluminium: Hybridisierung 1

•Amorphe Legierungen (Experimente, Gruppe Haussler)

• Sonderrolle der NFE-Legierungen mit 1.8 Elektronen/Atom

• Al1−x(3dTM)x zeigen Extrema von ρ4K, TK, |d ln(ρ(T))/dT|, ... bei kritischen

Konzentrationen xTMc

• Plasma-Resonanz deutet auf gefullte d-TM-Orbitale

−8

−6

−4

−2

0

2

0.0 0.2 0.4 0.6 0.8 1.0TM-Anteil

ZT

Meff

(x)

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........................

•Sc

•Ti

•V

•Cr

•Mn

•Fe

•Co

•Ni

• xTM

c

•Cu

• Interpretation

• aktive Fermi-Kugel in Resonanz

mit Schalenstruktur

Kpe = 2 kaktivF

• alle Al-p-Elektronen d-typisch machen

ZTMeff (x) = 1 − (1−x)Nat

xNat= 2 − 1

x

↑ von 4s1

• kritische Konzentrationen xTMc

ZTMeff (xTM

c ) = 1 − (10 − nTMd ) 1)

1) s. auch G. Trambly de Laissardiere et al. cond-mat/0410513v2

Page 4: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

1. Ubergangsmetalle in Aluminium: Neutrale Atomkugeln

Partielle Ladungen der Atomkugeln in 20 kristallinen und amorphen Legierungen

LMTO-ASA liefert: nTMd ≈ nfrei

d + 1frei 4s

⇒ Negative TM-Valenz bedeutet nicht das Fullen aller TM-d-Orbitale.

0

2

4

6

8

10

12

10 20 30

Kernladungszahl

•••••••••••••••••••••••••••••••••••••••••••

•••••••

••••••••••••••

•• ••

............................................................................................................................................................................................................................................................................................................. nfreis − 1

•••••••••••••••••••••••••••••••••••••••••••

•••

••••

•••••••••••• ••

••

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•••••••••••••••••••••••••••••

••••••••••••••

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Ele

ktr

onenzahl/

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s p d total

Page 5: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

1. Ubergangsmetalle in Aluminium: Neutrale Atomkugeln

Partielle Ladungen der Atomkugeln in kristallinen Legierungen (4d, 5d)

⇒ 4dTM und 5dTM zeigen Unterschiede im Vergleich zu den 3dTM.

0

2

4

6

8

10

12

40 50 60 70 80

Kernladungszahl

••• •

← nfreis

••• •

•••

← nfreid

•••

• ← nfreitot

Ele

ktr

onenzahl/

Ato

m

s p d total

Page 6: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

1. Ubergangsmetalle in Aluminium: Hybridisierung 2

• Bedingung fur maximal konstruktive Interferenz

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FERMI

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......................................................

JONES

1.81 NFE-Bandzustande je Atom besetzt

• Mit welcher effektiven TM-Valenz bleiben genau1.8 NFE-Bandzustande je Atom besetzt ?

(1 − x)Nat3 + xNat(1 + nTMd ) ≡ ZatNat Bandzust. total

(1 − x)Nat3 + xNatZTMeff (x) ≡ 1.8 Nat

ZTMeff (x) = 3 − 1.2

x

−8

−6

−4

−2

0

0.0 0.2 0.4 0.6 0.8 1.0

x (TM-Anteil)

ZT

Meff

(x)

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..............................................

3-1.2x

≈ Experiment 2)

•Sc

•Ti

•V

•Cr

•Mn

•Fe

•Co

•Ni

•Cu

• xTM

c

i-AlCuFe Al3V AlRe

• Jedes 3d-TM hat sein xTMc

ZTMeff (xTM

c ) = 1 - (10 - nTMd )

• Wie viele non-NFE-Bandzust. entstehen bei xT Mc ?

Nnon−NFE = Nat (Zat − 1.8)

= xTMc Nat

1 + nTMd − ZTM

eff (xTMc )

= xTMc Nat 10

2) J. L. Barzola, Dissertation 2003 und M. Stiehler Diplomarbeit 2004 http://archiv.tu-chemnitz.de

Page 7: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

1. Ubergangsmetalle in Aluminium: Al-TM-Kopplung

• Al3V (6 Al, 2 V)Pearson’s Handbook, No. 139,

LMTO-ASA, special k-points, Kubo,

spez. Widerstand ρ(ε), Zustandsdichte n(ε),

Diffusivitat D(ε), z ist lange Seite

XCRYSDEN

Page 8: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

1. Ubergangsmetalle in Aluminium: Al-TM-Kopplung

Bandzustande unterhalb εF stark (V-d)-(Al-sp)-korreliert, oberhalb εF neuer Typ

Al3V, No. 139,

ASA-LMTO, spez. 5-set,

Anteile an Norm 1,

oben:

sp , Al , V

unten:

Al-sp , V-sp

Page 9: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

1. Ubergangsmetalle in Aluminium: Al-TM-Kopplung

• Cockayne-Modell (80 Al, 32 Cu, 16 Fe)LMTO-ASA, special k-points, Kubo,

spez. Widerstand ρ(ε), Zustandsdichte n(ε),

Diffusivitat D(ε)

< Psp >, 40-meV Mittelwert des sp-Anteils

XCRYSDEN

Page 10: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

1. Ubergangsmetalle in Aluminium: Al-TM-Kopplung

Bandzustande unterhalb εF nahezu fixierte (Fe-d)- und (Al-sp)-Anteile,

oberhalb εF lineare Al-sp-Zunahme, also anders zusammengesetzte Bandzustande

i-AlCuFeASA-LMTO,

(spez. 3-set),

Anteile an Norm 1,

oben:

sp , Al , Cu , Fe

unten:

Al-sp , Cu-sp , Fe-sp

Page 11: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

1. Ubergangsmetalle in Aluminium: Al-TM-Kopplung

• AlRe (2 Al, 2 Re)Pearson’s Handbook, No. 123,

LMTO-ASA, special k-points, Kubo,

spez. Widerstand ρ(ε), Zustandsdichte n(ε),

Diffusivitat D(ε), z ist lange Seite

XCRYSDEN

Page 12: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

1. Ubergangsmetalle in Aluminium: Al-TM-Kopplung

Uberall Re-d-dominiert, kein deutlicher Wandel der Zusammensetzung bei εF

AlRe, No. 123ASA-LMTO (okt 6-set)

Anteile an Norm 1,

oben:

sp , Al , Re

unten:

Al-sp , Re-sp

Page 13: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

1. Ubergangsmetalle in Aluminium: Was wurde gezeigt ?

Fur isotrope amorphe Legierungen von Al mit 3d-TM bei den kritischen

TM-Konzentrationen xTMc wurde eine Verteilung des von den freien Atomen

eingebrachten sp-Gewichts auf zwei Gruppen von Bandzustanden vorgeschlagen.

Die Gruppe der NFE-Bandzustande ist mit 1.8 besetzten Bandzustanden je Atom so

bemessen, daß Ubergange an der Oberflache der zugeordneten Fermikugel durch

konstruktive Interferenz vieler schwacher Streuwellen aus dem mittleren

Ordnungsbereich maximal unterstutzt werden (Hume-Rothery).

Die zweite Gruppe besteht aus 10 non-NFE-Bandzustanden je TM-Atom. Sie

enthalt das gesamte resonante TM-d-Gewicht. Das totale sp-Gewicht dieser

non-NFE-Bandzustande bildet die negative TM-d-Valenz.

Stochiometrische Trends einer Reihe von Meßgroßen (LIT 1,2) befinden sich im

Einklang mit diesem Konzept.

Die LMTO-Bandzustande einiger kristalliner Legierungen abseits der xTMc zeigen

trotz ihrer von xTMc abweichenden Konzentrationen deutliche Korrelationen der

Al-sp- und TM-d-Gewichte.

Page 14: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

2. Ausgangspunkt Spektraloperator: Grundgleichungen

Spektraloperator ρ(ε) = 2X

p

|φp〉δ(ε − εp)〈φp| = − 2

πImGout(ε)

DOS n(ε) = 2X

p

〈φp|ρ(ε)|φp〉, Ne =

εFZ

dε n(ε)

Bandenergie Eband =

εFZ

dε ε n(ε)

Kubo σ(ε) = 2e2π~

Ω

X

p

〈φp|ρ(ε)vρ(ε)v|φp〉

Einstein σ(ε) =e2

Ωn(ε) D(ε)

Diffusionskonstante D(ε) = π~

X

p

〈vφp|ρ(ε)|vφp〉

Mott σ(ε) =e2π~

2 Ωn(ε)2

X

pq

|〈φp|v|φq〉|2ε

Große Leitfahigkeiten auf der Energieschale,

falls die Wellenfunktionen in Transportrichung verschiebbar sind.

Page 15: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

2. Ausgangspunkt Spektraloperator: bond order Θ12

|Λ〉 Zustandsraum des Systems, neue Basis im Unterraum (|Λ1〉, |Λ2〉):

|±〉 = (|Λ1〉 ± |Λ2〉)/√

2

n(ε) = 2

2

4X

Λ6=Λ1,Λ2

〈Λ|ρ(ε)|Λ〉 + ρ++(ε) + ρ−−(ε)

3

5

ρ++(ε) =1

2(ρ11(ε) + ρ22(ε)) + Re ρ12(ε)

ρ−−(ε) =1

2(ρ11(ε) + ρ22(ε)) − Re ρ12(ε)

1

2

`ρ++(ε) − ρ−−(ε)

´= Re ρ12(ε)

Θ12 =

εFZ

dε Re ρ12(ε)

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

ρ11 + ρ22

.............

.............

.............

.............

....................................... .............

..........................

.............

.............

.............

.......................... ............. ............. ...........

.. .......................................

.............

.............

.............

.............

.............

.............

....................................... ..

........... ..........................

.............

.............

.............

.............

.............

.................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Re ρ12

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

....................................................................................................................................................................................................................................................................

ρ++...............................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

ρ−−spektrale bond order Re ρ12(ε)

analysiertBeteiligung symm./antisymm.

Uberlagerungen

Page 16: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

2. Ausgangspunkt Spektraloperator: Atomkugel-Basis

n(ε) = 2X

Λ

〈Λ|ρ(ε)|Λ〉

|Λ〉 vollstandige, orthonormierte Basis des gesamten Systems

atomare Polyeder Atomkugeln Teilsysteme

|Λ〉 = |sλµ〉 |Λ〉 = |slm〉 |Λ〉 = |Netz〉, |Umgebung〉

••••

•• •

•••

••

••

• •

• •

..............................................................................................................................................................................................................................................................................................................

s′L

..............................................................................................................................................................................................................................................................................................................

sL..............................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................................................................................................

...................................................................................................................................................................................................................................................................................................

.......................................................................................................................................................

.....................................................................................................................................

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...............................................

.......

...

...

...

...

...

...

...

...

...

...

...

...

...

...

..

...

....

....

....

....

............................

ρ(ε) =X

sL,s′L′

|sL〉q

n0sL(ε) n0

s′L′(ε)“

δss′δLL′ + ThsL,s′L′

〈s′L′|

Page 17: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

2. Spektraloperator als Ausgangspunkt: Was wurde gezeigt ?

Bei Vorgabe des Spektraloperators (ρ(ε), hier fur T = 0) lassen sich alle

interessierenden elektronischen Eigenschaften des Systems berechnen.

Wir gehen davon aus, daß ρ(ε) durch eine Superzellen-Methode mit angemessener

k-Summation berechnet wurde und in Atomort-Drehimpuls-Darstellung vorliegt.

Partielle Zustandsdichten (Diagonalelemente) und bond-order

(Nichtdiagonalelemente) liefern eine atomgestutzte Analyse des Zustandsspektrums.

Der Vortrag geht einen Schritt weiter, indem das System in zwei Teile, Umgebung

und Netz, zerlegt wird. Die Umgebung ist vom NFE-Typ, wahrend im Netz

Kovalenz und TM-d-Hybride eine fuhrende Rolle spielen.

Es werden separate Eigendarstellungen des Spektraloperators in beiden Teilen

ermittelt und als neue Basis des Zustandsraumes benutzt. Es interessiert die

Zusammensetzung der Eigenvektoren und die Umgebung-Netz-Kooperation (U-N

bond-order).

Bei Verwendung des k-gemittelten Spektraloperators zur Berechnung der

Leitfahigkeit ist zu beachten, daß damit eine Brechung der Translationssymmetrie

einbezogen ist, was eher fur amorphe Phasen zutrifft.

Page 18: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

3. Ubergangsmetall-Netzwerk: Fe-Netz in i-AlCuFe

• 150-meV-Pseudogap entsteht im Fe-Netzwerk

alle 2000 Fe aus 16000-Atom-cluster i-AlCuFe

MT-SW-DOS eines Bergman-Fe-Atoms

0

2

4

6

8

−2 0 2

(ε − εF )/eV

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

......................................................................................................................................

.......................................................................................................................

MT-SW

state

s/eV

/ato

m

0

2

4

6

8

−2 0 2

(ε − εF )/eV

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................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LMTO

state

s/eV

/ato

m

Welche Rolle spielen die Al-Atome ?

Page 19: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

3. Ubergangsmetall-Netzwerk: Al-Brucken in d-AlNiCo

• Valenzdichten

(Modell nach

Mihalkovic)

ABINIT

Al-Brucken

TM-Al-TM

Kovalenz

bei Co

PGPLOT

Co ⇒

Co ⇒

Page 20: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

3. Ubergangsmetall-Netzwerk: Eigendarstellung 1

Umgebung und Netz separat diagonalisieren, |U〉, |N〉 neue Systembasis

ρ(ε) =

|sL〉

=

••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

|U〉 |N〉 ρ

〈U|

〈N|

l ↔

• Diagonalblocke: ρUU(ε), ρNN(ε) sind |U〉- bzw. |N〉-projizierte Zustandsdichten.

Interferenzcharakter (konstruktiv / destruktiv) in den Eigenvektoren ?

• Nichtdiagonalblocke: |U〉-|N〉 bond order

|±〉 =1

2(|U〉 ± |N〉)

Re ρUN(ε) =1

2

`ρ++(ε) − ρ−−(ε)

´> 0,

falls symmetrische Uberlagerungen |+〉 in den Eigenzustanden dominieren.

Page 21: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

3. Ubergangsmetall-Netzwerk: Eigenvektoren und Interferenz

• AlRe: große Eigenwerte durch konstruktive Interferenz, Σ Al-Re bond-order

zeigt: antisymmetrische |U〉-|N〉-Uberlagerungen binden

0

2

4

6

−10 −5 0

(ε − εF )/eV

state

s/eV

/Zelle

DOS

AlRe

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................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..........

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............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................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........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................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...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................

..............

............

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..............................

..............................

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

0.0

0.5

1.0

−10 −5 0

(ε − εF )/eV

state

s/eV

32 Eigenwerte

in der Umgebung

ρ32 > ρ31 > . . .

ρ32 ρ29

ρ25...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................

..

...

..

..

...

..

..

...............................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................

..

...

...

..

...

...

.......................................................

..................................................................................................................

...

..

...

...

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...

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...................................................................................................................................................................................................................................

...........................................................................................................................................................................................

........................................................................................................................................................................

..............................................................................................................

.

...

...

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...

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............................................................................................

...........................................................................................................................................................................................................

.........................................................................................

...........................................................................................................................................................................................................................................................................................................................................................................................................................

..............................................................................

.......................................................................................................................................................

................................................................................................................................................................................................................................................................................

............................................................

0.0

0.5

1.0

−10 −5 0

(ε − εF )/eV

state

s/eV

32 Eigenwerte

im Netz

ρ32 > ρ31 > . . .

ρ32 ρ29 ρ25

..........................................................................................................................................................................

...............................................................................................................................................................................................................................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...

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.............................................................................................................................................................................................

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......................................................................................................................................................................................

.....................................................................

−0.5

0.0

0.5

−10 −5 0

(ε − εF )/eV

state

s/eV

.................................................................................................................................

......................................................................................................................................................................................................................................................................................

...

..

...

...

..

...

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...

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....................................................

Interferenzanteil

...........................................................................................

..........................................................................................................................................................................................

........................................

............................................................................................

..........................................................................................................................................................................................................

..........................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................

..................................................................................................................

ohne Interferenzen

...........................................................................................................................................................

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......................................................

ρ32

−0.5

0.0

0.5

−10 −5 0

(ε − εF )/eV

state

s/eV

..............................................................................................................................................................................................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................................

...

..

...

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................................................................................................

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..............................................................................................

........................................................

Interferenzanteil

.......................................................................................................................................................................................

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...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................................................................................................................................................................

ohne Interferenzen

.........................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................

...............................................................................

......................................................................................

..................................................................................................

.................................................................................................................................................................................................................................................

ρ25

−0.1

0.0

0.1

−10 −5 0

(ε − εF )/eV

state

s/eV

Σ Al-Re bond-order

....................................................

............................................................................................................................

.........................................................................................................................................................................................................................................................................................................

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..

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Page 22: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

3. Ubergangsmetall-Netzwerk: EVekt. und Ladungsdichten

Al3V: Elektronendichten zwischen den Atomkugeln (Flachen 0.04 e/A3).

tot. ValenzdichteABINIT

“Stabe“ entlang z

Transport bevorzugtentlang z imAluminium

partielle Dichte

|U〉-EVektormit maximal

konstruktiver Interferenz

⇐ ⇒mit ohne

Interferenz

partielle Dichte

|N〉-EVektormit maximal

konstruktiver Interferenz

⇐ ⇒mit ohne

Interferenz

XCRYSDEN

Page 23: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

3. Ubergangsmetall-Netzwerk: Eigendarstellungen 2

Kooperation Umgebung - Netz in der Kubo-Leitfahigkeit

σ ∝ Tr(ρ v ρ v) = Tr([RρR†] v [RρR†] v) = Tr(ρ [R†vR] ρ [R†vR])

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••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

ρ ρR† R†R Rv v

ll ↔↔

| z | z

• Achtung: k-summierter Spektraloperator bricht k-Erhaltung ( ≈ amorph) !

• Beitrage der Eigenvektoren mit stark konstruktiver / destruktiver Interferenz ?

• Beitrage der Pfade ohne / mit Ubergangen zwischen Umgebung und Netz ?

Page 24: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

3. Ubergangsmetall-Netzwerk: Eigenvektoren und Transport

• Al3V: Transportpfade an der Fermigrenze

−0.05

0.00

0.05

0.10

0.15

−10 −5 0

(ε - εF )/eV

σzz(ε

)/(µ

Ωcm

)−1

σzz

Al-AlV-VV-Al und Al-V

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−0.05

0.00

0.05

0.10

0.15

−10 −5 0

(ε - εF )/eV

σyy(ε

)/(µ

Ωcm

)−1

σyy

Al-AlV-VV-Al und Al-V

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• trotz Brechung der k-Erhaltung (≈ amorph)

• Haupttransportpfade im Al (s. auch partielle Dichten)

• Wechsel zu V kann behindern (negativer Beitrag)

• Anisotropie bleibt im V-d-Band erhalten

Page 25: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

3. Ubergangsmetall-Netzwerk: Eigenvektoren und Transport

• AlRe: Transportpfade an der Fermigrenze

−0.05

0.00

0.05

0.10

0.15

−10 −5 0

(ε - εF )/eV

σzz(ε

)/(µ

Ωcm

)−1

σzz

Al-AlRe-ReRe-Al und Al-Re

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−0.05

0.00

0.05

0.10

0.15

−10 −5 0

(ε - εF )/eV

σyy(ε

)/(µ

Ωcm

)−1

σyy

Al-AlRe-ReRe-Al und Al-Re

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• trotz Brechung der k-Erhaltung (≈ amorph)

• Haupttransportpfade im Al (s. auch partielle Dichten)

• Wechsel zu Re kann behindern (negativer Beitrag)

• Anisotropie bleibt im Re-d-Band erhalten

Page 26: Elektronische Interferenz und Elektronentransport in Ub ...sol/ebene3/hs-050705.pdf · 1. Ub ergangsmetalle in Aluminium: Hybridisierung 1 Amorphe Legierungen (Experimente, Gruppe

4. Was wurde gezeigt ?

• Amorphe Legierungen von Al mit 3d-TM haben bei den kritischen

Konzentrationen xTMc

• die optimale Fermikugel mit 1.8 besetzten NFE-Bandzustanden je Atom

• dazu 10 besetzte non-NFE-Bandzustande je TM-Atom

• eine negative TM-Valenz im Umfang des sp-Anteils dieser

non-NFE-Bandzustande.

• Die Untersuchung von Wechselbeziehungen zwischen Teilen eines Systems

(strukturell, chemisch) ist eine aktuelle Fragestellung. Eigendarstellungen der

ensprechenden Teile des Spektraloperators liefern geeignete Basissysteme des

Zustandsraumes des Gesamtsystems.

• Die Eigenvektoren selbst enthalten alle Interferenzen im Teilsystem.

• Die mit Eigenvektoren zweier Teile berechnete bond-order beschreibt die

Kooperation der Teile.

• Das Verfahren gestattet es, Transportpfade in einer Teilmenge von

Eigenvektoren zu isolieren.