Band Structure Calculations 1

Band Structure Calculations

1 Ener gy (eV) Wave vector k H N P -8 -6 -4 -2 0 2 4 6 8

Pseudopotentials

2 Question: How could it ever be true that electrons in a metal think they are moving freely in an empty box?

Pseudopotentials give conceptual answer.

☞ Restrict attention to single unit cell.

☞ Let

k

denote plane waves ei

k r_. ☞ Let c

denote core states.

k_ps k c c c k (L1) U k_ps U k c U c k c (L2) k_ps P2 2m U k_ps (L3) P2 2m U k c c k c (L4)

Pseudopotentials

3 h2k2 2m k U k c c c c k (L5) P2 2m U_ps k ps k (L6) U_ps U c c c c (L7) k_ps ps k (L8)

Empirical Pseudopotentials

4 0 Rc 0 Distance r U r U 0

Figure 1: The Ashcroft empty core pseu-dopotential is zero up to a critical radius R_c, and it equals a screened Coulomb potential U 0exp r d r thereafter.

First-Principles Pseudopotentials

5 0.0 2.0 0.0 r ( ˚A) nl r 5s ps 5s 4d ps 4d 5p ps 5p

Figure 2: Real and pseudo wave functions for the 5s, 5p, and 4d levels of silver. The 5p level is not much occupied in the ground state of silver, but it can be included in

First-Principles Pseudopotentials

6 0 2 4 6 8 r ( ˚A) -50 0 50 100 3d 2p 1s U ps r (eV) e2 r

First-Principles Pseudopotentials

7 Electron density n r i r

2 _{is spherically symmetrical in vicinity of nucleus}

Equation for radial functions is

h2 2m 1 r 2 r2r l l 1 r2 nl e2n r r r d r e2Z r xc n nl nl r 0 (L9) U_lps r h2 2m 1 r ps nl 2 r ps nl r2 l l 1 r2 e2nps r r r d r xc n ps nl (L10) r lm Y_lm lm r lm r d d sin Y lm r (L11)

Screening

8 Ups 4 Ze2 q2 2 (L12)

Result from later work....

1 Ups q 0 2 3 F (L13)

Linear Combination of Atomic Orbitals

9 k r 1 N Rl b_lei k R_aat l r R (L14) (L15) R R d raat r R aat r R ei k R R N b 2 _(L16) b 2 1 e i k (L17) and d r aat r a at r (L18) R R d raat r R h2 2m 2 U r aat r R ei k R R N b 2 (L19)

Linear Combination of Atomic Orbitals

10 R R d raat r R h2 2m 2 U at r R aat r R U r Uat r R aat r R ei k R R N b 2_(L20) d r R R ataat r R aat r R N e i k R R b2 d r R R aat r R U r Uat r R aat r R ei k R R N b 2 (L21) 1 e i k at e i k U at ei k (L22) where U at d r aat r U r Uat r aat r (L23) and at d r aat r U r Uat r a at r (L24)

Linear Combination of Atomic Orbitals

11 U ei k (L25)

Plane Waves

12 K l 1 b ₁ l2 b ₂ l3 b₃ (L26) N i 1 r i e_i e_i a₁ (L27) P2 2m U R (L28) h2K_max2 2m max K 0 k K max K K U K K n k K (L29) 1 dt h (L30)

Plane Waves

13 n 1 1 dt h n n 1 n dt 1 h n (L31)

Linear Augmented Plane Waves (LAPW)

14 k e i k r 1 2mh 2 2 k U r k k Y lm l r (L32) h 2 2mr2 rr 2 r l r U r h2l l 1 2mr2 l r l r (L33) k l 0 l m l A_lmY_lm r l r (L34)

Linear Augmented Plane Waves (LAPW)

15 ei k r 4 l 0 l m l il j_l kr Y_lm k Y_lm r (L35) k 4 l 0 l m l il j_l kRh Ylm k l R_h Y_lm r l r (L36) k K b k K k K (L37) (L38) 0 K q q K b q K (L39) where q q h2 q q 2m q q q q (L40)

Linear Augmented Plane Waves (LAPW)

16 q q 4 R 2 h h2 q q 2m j₁ q q R_h q q l 0 h2 2m 2l 1 P_l q q j_l qR_h j_l q R_h l Rh l R_h (L41)

Linearized Muffin Tin Orbitals (LMTO)

17 lm r ilY_lm r l r r R_h l _p l for r Rh ilY_lm r R_h r l 1 for r Rh (L42) l R_h pl 1 l Rh l p_l R_h l 1 R_h (L43) pl l 1 R_h l r R h l R_h l r R h l l 1 l l (L44a) with l R_h l R_h l r R h (L44b) r lm Bn k lm R ei k R lm r R (L45)

Linearized Muffin Tin Orbitals (LMTO)

18 0 lm Bn k lm p_l ilY_lm r r R_h l R 0 ilY_lm r R R_h r R l 1 ei k R (L46) R 0 ilY_lm r R R_h r R l 1 ei k R l m k ll mm 2 2l 1 r R_h l il Ym l r (L47) 0 lm Bn k lm p_l ilY_lm r r R_h l l m k ll mm 2 2l 1 r R_h l il Ym l r (L48) 0 lml m Bn k lm p_l ll mm k ll mm 2 2l 1 r R_h l il Ym l r (L49) 0 lm 2 2l 1 p_l ll mm k ll mm Bn k lm (L50)

Definition of Metals, Insulators, and

Semiconductors

19

Wilson’s theory:

☞ Insulator: All bands are either full or empty.

☞ Metal: At least one band is partially occupied.

☞ Semiconductor: Insulator where energy gap is less than around 1 eV.

Definition of Metals, Insulators, and

Semiconductors

20 Ener gy k Occupied States Fermi Surface Occupied States Wave vector k Ener gy k F Metal Insulator g Wave vector k

Noble Gases

21 exp g ionize L X W L Wave vector k Ener gy (eV) -4 -2 0 2 4 6 8 10 12 14

Nearly Free Electron Metals

22 Ener gy (eV) L X U Wave vector k -15 -10 -5 0 5 10 15

Nearly Free Electron Metals

23 Plane wave L L X W LMTO (B) Ener gy (eV) -10 -8 -6 -4 -2 0 2 4 (A) L L X W

Semiconductors

24

-12

-10

-8

-6

-4

-2

0

2

4

6

L

X

U

Ener

gy

(eV)

Wave vector k

Transition Metals

25 Ener gy (eV) Wave vector k H N P -8 -6 -4 -2 0 2 4 6 8