THE FERMI SURFACE FOR POINT INTERACTIONS

Abstract:

Raphael H¢egh-Krohn Helge Holden St<einar Johannesen Tore Hentzel-Larsen Matematisk institutt Universitetet i Oslo

Blindern, Oslo 3 Norge

Using a computer we study the Fermi surface for the one electron model of an infinite crystal in three dimensions with zero range interactions, i, e. vvi th so-called point interactions.

A computer program is available which has as input the cry- stal structure, the scattering length of the solid considered and the Fermi energy and as output a drawing of the corresponding Fermi surface inside its Brillouin zone.

1. Introduction.

'l'he notion of Fermi

surfac~

is ot great

irnport~ne~

;ln eQli4 state physics, and let us first recall what

vTe

me<::l.n

l::>y

a

f'el;;'m~

suface.

In the one electron model of an infinite

thr~~ di)TI~nsiQn~l

crystal we consider the Sc'hrodinger operai;Qr H:;::::

-6.+V

(in

1,111.i,t~

where

..fl

= 1 , m =

~)

on L2

q~^{3 )} wher~ A

is the

Lapl.~cian

a.nci V is periodic with periodicity A

whE=;~re /1.

is

iit

three

d~m~n$~"n~it,l

lattice in E

3 •

By standard techniquee t,his

implte~ th~t:.

where the dual group 'A ;::;:

J}³

/r can be identified

vJith

the··

Brillouin zone

B,

i.e. a Hign,er-SE:!!it,z eell of t.hEl

ortho9orv~1

(or dual) lattice r.

The band spectrum of

H

dissol.ves

into

discrete

e!l.qerw&;luEj!S

E1 (k) ;;; E

2

(k) ;;; . . . of H(k) and the Fermi sur:j:ace is the

~et

{k E BIEn(k) = EF for a n E

~}

whe:r-e

Ep

is

the F~mni en.~r;9~

which distinguish the occupied states frorn the nonoccupied oneS!

Instead of considering the fermi Sl,lrface inside the

Brillouin zone B, one can consider lt Qn

~³

by

e~iT~nqiM~

it periodically.

The actual computation of the Fermi s1,.1rfaoe is a

mi~ture

of theory and experiment with no unifying rigorous

theo~y~ S~e

[1)

for an extensive introduction

to

the sqbject,

Act1,1p,l,J.,y it se~rns

that the only Fermi surface

that hi:J,q ~een compt.rt,~.d ex;p~:l,.cit~,Y

starting with an

expli~it

potential ie

the

free

el~ctron &pprq~~~

mation, i.e. with no interaction.

In this paper we compute with the pid

Of a, COlT\flJ;ltet'

tl)e

Fermi surface explicitly starting from a non~trivial interaction, namely point interaction.

The study of Schrodinger operators with point interactions, i.e. zero range interactions or Fermi pseudo-potentials, was started by among others Fermi, Peierls, Breit, Thomas in the thirties in nuclear physics [2], and continued in the fifties by Huang, Yang, Lee, Luttinger and others in statistical mechanics

[ 3] .

In addition, and in this connection more interesting, we have the celebrated Kronig~Penney model [4], dating from 1933, which is a model of an infinite one-dimensional crystal with point interactions.

The non<-·trivial rigorous study of these operators especially in three dimensions was started in 1961 by Berezin, Faddeev and others

[s]

and made into a systematic theory by Grossmann, H¢egh- Krohn and Mebkhout [6], [7]. In particular, in [7] the periodic;

point interaction model is constructed, and its spectral proper'""

ties are determined

Thus the model vle study here is a three dimensional analo~ue

of the Kronig-Penney model.

More detailed properties of the spectrum when one removes some points with point interaction and thus destroys the periadi..,.

city are studies in [ 11] ,

Starting off from this there has been a thorough rigorous study of these operators and related operators with mo:r~ realis- tic shor-t- range potential.

this we refer to [12].

For an extensive exposition of all

He also note en _Eassant that in the beginning of this rigo- rous study non-standard analysis played an important part

[8].

parameter needed b::J specify the interac'cion completely is the scattering length.

Thus the equation v<::: derive for the Fermi suface contains only the scattering th of the one center problem, the Fermi energy and t.he lattice.

This ies that the compu'cer program [ 14] has as input the lattice, the scatter

Fermi suface \vitJ"I the fol

(

.

^\

\ .1 i

and the energy and as output the ng four options:

ing Brillouin zone.

( ii) The surface'" over an arbitrary rectangle in the plane.

(iii) Contour 111aps of ( i ) ,

(iv) Contour maps of ii).

The Fen11:L ~rface is mathematically a multivalued, actually

infinite ·-:..lal_ued, fun.c

The r program however only able to draw single- valued functions, so we can see half the surface in (i) and a single sheet in i i . In this short paper we can only give some examples of Fenni surfaces for a small number of different lattices and values of the parameters.

However a specific Fermi surface with a particular lattice and values of tJ1e parameters can be ob-tained from the authors on request.

One may argue that a point interaction is not a realistic interaction. Hovvever one llirtue of the point interaction is that one can actually compute a non-trivial Fermi surface starting from a potential. In addition it is a possible starting point for a more general approach.

(actually in norm res lve ~ sense by S inger operators with

2 . Point interac o~s a He s·tud

in three dimen·::

w11ere a.~ ^l a.h: a.

t L

here ti1e case lattice poln':.

actions ^wi~ch st.renq integral kernel (-!1 -E) ~I

a

where Im/E

(Recall that and

TC

0 a:nd

0

\1Te will return to this in

e-e ectron nodel of an infinite crystal po t l teract1ons.

~ aravais lattice in R3 Le.

=

( ] )

1nd vectorso He consider

t1 ^;:):tJe c~i:c;rn for eac~h Bravais

C·:)r:cesponding to point inter- .::.t. ea._ ^L 1n A has resolvent with

( 2 )

( 3 ) 0

s the integral kernel of ^{(-!1-E) -I} on

~ k' -th element of the

inverse of the mat x [ ] on 12(A).

We briefly :;ec::s her to give some insight into the defini- tion of H • Formally we are interested in the operator

0:

H

=

-~-

7

_'-' ~6(•-k)

AEA

( 4)

where

o

^{is Dicac 's} delta function and v > 0 which is not a well-defined self~adjoint operator on L2 _q~3 ) ,

By making a Fouri,sr~transform we obtain the operator ( 5 )

where

q,A. (p) - ( 6)

and the operator ^....

I

^{f> <gj} is defined to be Sh = f ( g, h) , ( ( g , h ) is the in product on ^T .LJ 2 q~3 ⁾ ) and p2 is considered, as multiplication operator, i.e. _H0

=

p 2 means _{(H 0}f)(p)

=

p2f(p).

To make trd .. s following way:

rator well-defined we modify H in the

:3. ce ^(~)"'

{\ with \oThere

( 7)

where is the characteristic function of a ball with radius w, i.e.

X ( p)

₌ I PI

^{.:;: w} _{( 8)}

w 0 I r J

11-' w

and let v be ^w~ ing, v

=

^{v ( w) •}

By choosing

v ( w) ( w -1

( 9 )

=

+a:

where o: E B- is ar!:Ji trary one can show that Hw where

( 1 0)

will converge in strong resolvent sl;:!nse as w ^{~ co}

tor ^-/::,

(X

.

Note that the coupling constant v (w) tend$ to zero

as ^{(J..l + 00 •} For more details see

[ 6]

¹

[ 7],

[ I 2) ,

The constant a can be interpreted as related to the !$Qat..,.

tering length in the sense that a= l/4na is the scattering length of the one particle system with a single point inter~

i'iCtion.

\Je now return to the opera to:r- under

A

we can write

-/::,

a

-/:;, _a

. Using

the invarianoe

( I I )

where the

dual group

k =

~³ /r

(r

is the orthogonal lattice, where a . • b . ;::: 2-n: 6 . . )

,). ], .;t.

J

^{can 'be}

identified with the Higner-Seitz cell of tl").e orthogonal l~t.tice, i , e. the Brillouin zone B and where -b. a (k)

is a

^S~S),f-ad joint ope:rq.tor on Y.²

(r)

with integral kernel

( l :2 )

where Im/E > 0 and

gE(k)

= I G

^(A)e-iA•k

AEA E (I 3)

See [7] for more oetails.

We .... ~;, ( k)

a

see from (12) that the negative part of the spectrum of consists of points where a- rn;

i/E -gE(k) ;:::

0,

Using this one can [7] explicitly compute the spectrum of

-IJ. :

0:

( 1 4)

where E1^{(o:) <} 0 provided o: < o:0 ^< 0 where o:0 is a suitable constant. E0 (o:),E 1 (a) will also depend on the lattice.

The equation for the Fermi surface is then

( 1 5)

where A, o:, E are input and the implicit function in k is output and we recall that

r

^{e i/Ej A.}

^I

A. t 0

GE

(A)

=1

^4njA.j

l

^{0 "A}

⁼

⁰

( 1 6 )

Hhen ve let E < 0 ( 1 5 ) can be written:

( 1 7 )

A

few words may be appropriate here to indicate how we solve this equation.

hTe sum all the terms in the infinite series 'd th

I

^{A.j < R}

for some fixed R, use the sy®~etry of the lattice (which implies that the program works for all Bravais lattices except for tric- linic) to simplify and obtain a polynomial equation which can be solved by standard techniques.

In general a n'th degree equation has n solutions.

However, adding more and more terms in (17) vhich increases the degree of the equation will not yield more and more different

solutions which is reasonable since the equation ( 17) with a

finite sum converges exponentially to the equation for a unique Fermi surface.

But as the computer only is able to draw single valued tuna~

tions, we usually end up with a smnll number of different draw- ings corresponding to different roots of the equation. To visua~

lize the Fermi surface one has to superimpose visually the 4i:f£e- rent drawings.

For example in fig ^o l •rle use an 'approximation which yields a third degree equation, and we obtain three drawings which howevel!' all are identical to the one in fig. 1.

However in fig. 2 we see an approximation which gives rise to a sixth degree equation, and we obtain six drawings. In this case there are only hvo with major differences namely fig.

2a

a,nd 2b.

In the next section we present some examples of Fermi sur.- faces with various values of the parameters and for some

lattices.

As mentioned ln the introduction, point interactions repre ...

sents a first approximation to more realistic short-range inter~

actions.

To be precise, let

( 1 8)

where V is a real-valued potential which is Rollnik (i.e.

f ^f

IV(x)V(y) II x-yj- 2 dxdy < ro)

R3 R3 with compact support and ~(~)

:;;:

is an analytic function with ~ (0)

=

1.

He assume that -L+V has a simple zero-energy resonance, i.e. the equation

(-6+V)<V

=

0

( 1 9)

has a simple sol uti on <V which does not he long to L2 q~³ ) • To be more specific, using the standard decomposition (E < 0)

where

and

=

^(-6-E) -I

~ 1

I 1 -2 I vj "2

u

=

^V sgn V; v =

(20)

( 2 1 )

( 22)

\·le see that eigenval :Jes of -6

+V

corresponds to non~t:i;i

vi<:\1

solutions of

<P+uG v<P

=

0,

E ( 23)

One can show, see [ 10], that if ^<1> is a solution of

(23)

then

(24)

is a solution of

(-6+V)<V =

^{E<V. (2S)}

So we assume that

( 26)

has a simple non-trivial solution ^<1> E L² (~³ ). Then one

can

s t i l l prove that

( 27)

is a solution of

(-6+V)<V = ⁰ (28)

now in the sense of distributions, and what we assume is tha.t

From this assumption we can prove Theorem 1 .

The operator H

E converges in norm resolvent sense

to the

operator ^-!:,.

a

a

=

given by (2) where a

is

given

according

to

jl ' ( 0 ) (V4> '4> )

I ( v'

4> )

I ~

^2.

Remark: The 4> in the definition of a is the <)! given l;>y (27).

Proof: See

[9].

Using the the same decomposition for H

€ as for ^{-t:,. ,}

i.e.

0:

(30)

we dissolve the bands of the spectrum into discrete eigerwa.l~es.

He can then prove the followin9 result.

Theorem 2.

Let

an eigenvalue E

E

be an eigenvalue of ^{-!:,. (} k) • a for H (k)

E

Then

there exists when c: ⁺ 0 and E is analytic in e:. He have the following expansion

€

E = E0+~::E'+o(E)

E ( 31 )

where

E' == hA (A+E 0^k B) ( 32)

and

hk A

⁼

(2n)3[1BI

I

(jy+kj2-E)4 ^{1 ] -1} • yEr

( ~3 )

( IBI is the Lebesgue measure of the Brillouin ~one) and A,S are constants only depending on properties of -t:,.+V,

Remark: The explicit form of A and B are given in [9].

Proof: see

[9].

Remark: This means that the Fermi surface computed with point interactions represents a first aproximation to a Fermi surface with more general short-range interactions.

Using a scaling technique point interactions can also be related to another limit than the zero range limit.

Namely, let

H(E) =-~+~(E)

I

^V(•-k/E)

kEA

(34)

where V and ~ are as before. Then, using the unitary opera- tor U defined by

E

we see that

which implies that the eigenvalues H(E) respectively are related by

E

=

^E -2 ^E(E).

E

E

E and E(E) of H

E

(35)

( 36)

and

(37)

Looking at the operator H(E), we see that the limit E + 0 represents a situation where the centers (i.e. the points where each potential V(x-k/E), k E A, is concentrated) move apart. As usual we decompose H(E), i.e.,

(jJ

H(E)

=

jH(E,k)d3k A /\

and we have the following theorem

(38)

Theorem 3.

Let E(e,k) < 0 be an eigenva~ue of B(e,k) such that

lime -2 E(e,k) < 0. ₍₃₉₎

e-+-0

Then E(e,k) is analytic and has the following expansion

(40)

where E0 is an eigenvalue of ^{-6 ( k)} _a and

E'

is given by

( 3 2) •

3. Some Fermi surfaces.

Figure captions.

Fig. l : The simplest Fermi surface we include here is for a simple cubic crystal (SC or cubic P) with E = -1 and a= 0.12 and a= b

=

^c

=

¹ (for notation concerning the

lattices, see Kittel

[13])

inside the upper half of its Brillouin zone. Completely vertical or horizontal parts of the illustration are not parts of the Fermi surface.

Fig. 2a, 2b. The Fermi surface of a body centered crystal (BBC or cubic I) with E

=

^~1 and a= -0.14 inside the upper half of its Brillouin zone. The total surface within ~he

upper half of the Brillouin zone is the union of the two surfaces depitched above.

Fig. 3a, 3b: A contour plot of fig. 2a, 2b.

Fig. 4a, 4b: The Fermi surface of fig 2a, 2b extended periodi- cally. Again i t is difficult to vizualize the total surface in the sense ·that the total Fermi surface is the union of the two surfaces above extended periodically in the posi- tive and negative z-direction.

Fig. Sa, Sb:

Fig. 6a, 6b: A Fig. 7a, 7b:

fig. 4a,

A contour plot magnification A magnification

4b.

of fig 4a, 4b.

of part of the surface in fig. 4a, 4 of another part of the surface in Fig. Sa, b: The Fermi surface of a face centered cubic (FCC or cubic F) crystal with E

=

-1, a= -0.17 inside the upper half of the Brillunin zone.

Fig. 9a, b. Contour plot of fig. Ba, b.

Fig. 1 Oa, b: The Fermi surface of fig. 8 extended periodically.

Fig. 1 1 a, b: Contour plot of fig. 10.

Fig. 12a, b: A magnification of a part of fig. 1 Oa, b.

The next seven figures show how the Fermi surface inside the upper half of its Brillouin zone varies with E for an ortho- rhombic P crystal v1i th axes a

=

3, b

=

2, c

=

^{and with a} = 0 . Fig. 13: E

=

^{-1 .2,} The Fermi surface in homeomorphic to a

sphere around each print of the orthogonal lattice.

Fig. l 4. E = -1 . l

Fig. 15. E

=

-0.9. The Fermi surface 1s now connected in the x- direction.

Fig. l 6. E

=

^-0.425

Fig. 1 7. E

=

^-0.35, The Fermi surface is connected in the x- and y-directions.

Fig. l 8. E == -0 ^< 1 3 Fig. l 9. E

=

^{-0. 1}

The next three figures show a similar series for a tetrago- nal P crystal with axes a

=

b

=

2, c

=

1 and with a = 0.

Fig. 20. E = _{-1 ' l} Fig. 21

.

^E

=

^~0.6

Fig. 22. E

=

^~Oo4

The next two figures show the Fermi surface inside the upper half of its Brillouin zone for a tetragonal P crystal with axes a = b

=

2, c

=

3 and with a

=

0 for two values of E.

Fig. 23. E = -0.2417

Fig. 24. E

=

-0.2. The Fermi surface is now connected in the z- direction.

Fig. 25. The surface for a monoclinic C crystal with axes a= 2, b

=

l. 5, c

=

1 inside the upper half of its Brillouin zone. The angle between the axes

and E

=

^{-1 .}

-+ a and ⁺ c

Fig. 26, 27. Tfte Fermi surfaces of a trigonal crystal with a == 0 and E

=

^~1 for two different angles, 30 ⁰ and 70 , ⁰ bet~veen

the symmetry axis and each of the crystal axes.

Fig. 1

Fig. 2a

rJ

""

-G ~ ^(.(1

~

""

1'-'

Fig. 2b

Fig. 3a

f'ig. 3b

r! ^a>

~ ^<f1

lP

()'J

~,.:a

.,._-o

Fig.

4a

Fig. 4b

)

^{Fig. Sa}

; ,S ---._

) ⁰

^{Fig. Sb}

i

N

<C.

\

~ r

<fl

X-QX\.S

~

~-r---,r---.---,---.---,---~

't.S 4.0 !..S 5.0 2..'0 2..0 \.S

II

i

II I

i

II I I I I I I I I I I I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I

i

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I I

I

i

I I

I I

I I

I

I I I I I I I I

I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

I I I I I I I I I I I I I I I I

I I I I I I I I

I I I I I I I I

F'i g.

Fig.

7b

"'

~ ll)

;:>_

"'

^c,

-..

v.,

'\)

...

~

~ .;;

-$ "<f.-

Fig. Ba

"'

~ ^ll)

~

"'

^c,

-..

u.,

'\)

...

~ ^~

.;;

~ /

o+-

Fig. Bb

Fig. 9a

Fig. 9b

Fig. 10a

(

Fig. 10b

Fig. 11a

(

Fig. 11b

"!

g

^\'l'j

^Fig.

^12b

<1 _~

....

1-'i

1\;

'<::>

""' ^q,

~ /cf-'

-:):)

-2

Fig. 1 3

~~

:1 ^0.

'(

( '

I<) 1\)

....

~

'-'>

... :---+---"

-.:;,

Fig. 14

Fig. 15

Fig. 16

Fig. 17

Fig. 18

Fig. 1 9

Fig. 20

Fig. 21

Fig. 22

Fig. 23

Fig. 24

Fig. 25

I _I

I I

I I

I

^I

r/ ^{.J l}

~ ^{(f' l} I

~ ^i,{l

I

I

"'"

'

^I

""

^{l I} I

r-J I

I l I

0 I

I 1 I I 1 I 'I I I J J I 1

~, i

0+, I

"'

^I _I

I I I I I I I

Fig. 26 I ^I

) I II

I

II I I I

0

Fig. 27

References [ l

J c [ 2]

[ 3

J

[4]

[ 5]

[6]

[ 7

J

[8]

[ 9]

[ l 0

J

[ 11

J

Let

[12]

Albev

i rt qua nt tim Bo6k

:L le~/

1976.

i Vee lous compu t.er technical problems.

TI-:e Fe,:crni Surface.

''f '")

I -

~rt-~ -ticc;,l t\Iu_c~lea.r Pt1.ysics"

N.Y -He el Berlin, 1979.

Cll

s

s of Electrons in Crystal

'499~513 (1931).

of Nonrelativistic Two- and int Interactions. In Carg~se ics Ed. F. Lur9at.

A class of explicitly tonians for one-particle

nsions I.

The -:::n1e~particle theory

ingular perturbations and

The Short~Range Expansion to appear in Ann. Inst.

ior Hamiltonians Defined as

The spectrum of defect

21~~228~

Solvable models

~'=' olid Stai::.e sics.

e ter-B isbane-Toronto, 5th Edition

su~-:;ruut.ines from H1SL for the

C3E-·GS. =-.-F' fc the graphical part.