Electrical current through extended and localized states from the quantum Boltzmann equation

Physica 134A (1985) 249-264 North-Holland. Amsterdam

ELECTRICAL CURRENT THROUGH STATES FROM THE QUANTUM Carolyn

EXTENDED AND LOCALIZED BOLTZMANN EQUATION

M. VAN VLIET

Department of Elecm’cal and Computer Engineering, Florida Atlantic University, Boca Raton, Florida 33431, USA, and Centre de Recherches Mathematiques, Universite de Montreal, Montreal Que H3C 357 Canada Christiaan

G. VAN WEERT

Instituut voor Theoretische Fysica, Universiteit van Amsterdam, Amsterdam, The Netherlands

Alan H. MARSHAK

Department of Elecm’cal and Computer Engineering, Louisiana State University, Baton Rouge, Louisiana 70803, USA

Received

26 April

1985

The fundamental equation for current flow in isothermal inhomogeneous systems, such as in electron devices, is J(r) = (c/q)V,,((r), where ii = 5 - q@ is the electrochemical potential or quasi-Fermi level, & is the chemical potential, and @ is the electrical potential. The two parts lead to drift and diffusion current. In the present paper we consider current flow involving arbitrary one electron states Il), where 14’) may be an extended Bloch type state or a localized state, the total current being ponderomotive plus collisional current. Using the quantum mechanical Bohzmann equation derived previously by one of us, we show that the above thermodynamic result for J remains valid for any type of current when an exact second quantization formalism is used.

1. Introduction In isothermal systems (VT = 0) the general formula for position dependent is J(r) = (a/q)Ffi(r) where /1 is the electrochemical potential. This

current

result is well known from irreversible thermodynamics’). level it can be derived from the semiclassical k-space

On the microscopic Boltzmann transport

equation, see e.g. Van Vliet and Marshak*). The result remains valid even when the band structure is inhomogeneous, as in graded gap materials, see Marshak and Van Vlie?). For an appropriate an.alysis for the case that the E = ~(k, r), we must use an extended band structure depends on position, Wannier-Slater theorem4’5). Since ji = 5 - q@, where 5 is the chemical poten0378-4363/85/$03.30 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

250

C.M. VAN VLIET

et al.

tial and @ is the electrical potential, one easily shows that the general result involves at least two terms, viz (w/q)v[ = ~qD,o,,V(n or p); where D is the and diffusivity, and the drift term -qpflorp (n or p)V@; the diffusivities mobilities case states result

the

are related band

by generalized

structure

is position

Einstein dependent,

equations,

two other

effect and rigid band effect) also result from the general for J(r) ‘). The implications for devices were recently

extensive review paper’). In the present paper we want to give a very basic derivation for the above thermodynamic current result. current

see refs. 2 and 6. In

which

flows through

extended

states

which

terms

(density

of

thermodynamic discussed in an

statistical We want

mechanical to consider

may be of a Bloch

type

nature, as well as through localized states, such as is the case in a magnetic field (Landau states) or in amorphous materials or in heavily doped materials below the mobility edge (see Mott and Jones’)). In the latter case the current involves a many-body component, which we labeled previously as collisional current’), in addition to regular ponderomotive current. The derivation of the current result will employ the formalism of second quantization, and the quantum mechanical Boltzmann equation, derived previously by Charbonneau, Van Vliet and Vasilopoulos”). This equation is fully quantum mechanical, both for the streaming and collision terms. It can be made quasi-classical by using the Wigner formalism. For localized states conduction through Landau states or by hopping was considered by Vasilopoulos current

and Van Vliet”). The conductivity which derives from the collisional is given by a generalized Adams-Holstein type formula’2-‘4). In section

2 of this paper we derive a general result for the current density J(r) based on quantum field theory. We also present the quantum Boltzmann equation of refs. 9 and 10. Then in section 3 we derive the current result for conduction through extended states while in section 4 we obtain the appropriate result for localized states. As far as device applications are concerned, these results were surmised and implied in our review paper’). Since we now have derived the basic results for all types of current flow, the basis of the previous work by Marshak and Van Vliet is hereby

complete.

2. Quantum

field formalism

of the current and quantum

We consider an electron gas, in interaction with phonons the presence of an electric and possible magnetic field. written as

Boltzmann

equation

or impurities and in The Hamiltonian is

ELECTRICAL

CURRENT

FROM BOLTZMANN

EQUATION

251

.%, = &P-A-F(t)

/iv+ 2

= x0+

(ri - (ri)&F(t))

(2.1)

where 2 is the system Hamiltonian, composed of X,, for the electron gas and AV for the interactions, while -A *F(t) is an external field Hamiltonian; as shown in the second line of (2.1) F = -qE(t) is the external force (a c-number) and A = C (ri - (ri)J (with ri the position operator of the ith electron) is the conjugate many-body operator. The magnetic field, if present, is included in X,,. Let q and qt be the quantum field operators of the electron gas X,,, then the electrical current density operator is

J(r) = -

& {P+(r)Wr)

- [ v*+(r)1Wr)) + J,,(r) .

(2.2)

The first term is the regular ponderomotive current, (q/m) times the momentum operator associated with X0; (q is the absolute value of the electronic charge and J refers to electron current). The second term is due to the collisions caused by AV, and is a many-body effect, previously labeled as “collisional current”, see ref. 9, section 4. Its field operator form is not presently known, but below we will derive an expression for it by an indirect argument. The ponderomotive current is easily put in the occupation number form. Thus let {It)} b e a complete basis of one-electron states, let UC(r)= (r )l) be the wave-mechanical realization of these states, and let nr = cicJ be the occupation number operator, ct and c being the usual fermion creator and annihilation operators. Then, as is well known,

W) = C uJr)c17

(2.3a)

C u;(r)ci,,

(2.3b)

P+(r) =

t so that

?P+v??-[vl+b+]?P= 2 c~,ci[u~(r)Vz+(r)55

u,(r)Vu;(r)]

.

(2.4)

C.M. VAN

252

If the current

J

where

flow in a volume

VLIET

et al.

R is homogeneous,

then

homo pond

u = + = (h/im)V

is the one-particle

Hermitean

(2.5) is the second quantization form for JEZ’. form for J homoin a volume 0 with homogeneous (4.28), or ref. 10, eqs. (2.6), (2.8) and (3.8),

J homo

_ -

velocity

operator.

The full second flow was found

Eq.

quantization in ref. 9, eq.

-;(&A,A,)

V-6) Here

Ad is the reduced

Liouville

operator

or master

operator,

while A, is the

diagonal part of A in the representation of X,,. The Kronecker delta signifies that Ad destroys a nondiagonal part. If the lattice remains in equilibrium (no phonon drag) Adc~ciSi/ = Adnr can be replaced by a phonon gas average (or boson

average)

(A,n&,.

We showed

in refs. 9 and 10 that this average

reduces

to

where

nl is the occupation

is the Boltzmann

number,

I-y) = /{n,}) is the many-body

state,

and 3

operator,

(2.8) in which the w’s are the standard transition rates for the pertinent processes involved. Thus for electron-phonon interaction (ref. 10, section 4)

scattering

ELECTRICAL CURRENT FROM BOLTZMANN EQUATION

where

for phonon

absorption

253

processes

(2.10a)

while for phonon

emission

O(iY+ E 4) =

processes

$ IF(q)12(~le-iq”l~)~(~~-

F~-

hw,) ;

(2.10b)

here IF( depends on the model (deformation potential model, rigid ion model, etc.) and on the type of phonons involved (acoustical, optical, polar optical, etc.). The w’s are not generally reciprocal, though the Q’s are; one finds”),

wsr

Then

with /3 = l/kT,

=

wB

eP(S-EC).

from (2.6) and (2.7) we obtain

(2.11)

for the current

the “Boltzmann

form”

(2.12)

We notice that the first part is just (2.5). Thus, analogous to the transition from (2.4) to (2~9, we can conclude that for a nonhomogeneous system the local current

density

is given by

(2.13)

This is the complete result. We can now average over the nonequilibrium density by the inhomogeneous master equationg). We have

operator

p(t), as given

(2.14)

C.M. VAN VLIET

254

et al.

Tr 2 [~(t)l{n~})({n~“}I~~n~l= C ({n,,}lp(t)l{nr.})~~n,= (Bp,), .

(2.15)

iq4

&-I Thus we find for (2.13)

c:.ci),[u,“.(r)vu,(r)+

(J(r)),~~(r,t)=-4(1C(

u,(r>v*u;(r)l

ii’

-

C (~;n,),uT(rb-~,(r)} .

(2.16)

We set res = 0, which means that the coordinate system must be chosen judiciously. For example, for box quantization involving plane waves ll)--+ similar in y and z (l/n”‘)exp(ik - r) the box runs from -LX/2 to L,/2; directions. Finally, in this section we recall the full quantum Boltzmann equation, ref. 10, eq. (6.17),

+

C [(Plr - r&“)- (ilr - r,,li>l[wnrU - (nirf),,> i”

(2.17) We shall be interested in the steady state results, valid for E(t) = E. We then replace the subscript f in the averaging brackets s, we write j(r, t)+ J,(r), and we note a(c:c,),/at+O. Again The steady state quantum Boltzmann equation then reads

t+ m, while by subscript

we choose

req= 0.

+ C I(5”lrl5”)- MI4)1Iw,& - (~rkJ + w,.,Cn,.LJ~~r) i” (2.18)

ELECTRICAL CURRENT FROM BOLTZMANN EQUATION

25.5

where in the last term to the left we changed the subscript “s” into “eq”, since this is essentially an equilibrium streaming term”); 3 is given by (2.8).

3. Current through extended states For extended states as occurring in crystalline materials, or in other materials, (for example, heavily doped materials above the mobility edge), there is only ponderomotive current, associated with the matrix elements ([‘lull). The other matrix elements (l[r1[) are zero for this case. We assume we have Bloch states or free electron states, characterized by k-space quantization. Thus I[) = Jk) = [k,, k,, k,). The matrix elements were calculated in the appendix of ref. 10:

(k’(vJk) = f Vkek(r)L3ksk = v$,,

.

(3.1)

We consider here the general case that the band structure may be positiondependent, as in the papers of Marshak and Van Vliet3,4*5,7).For the current density we have from (2.16)

J,(r)

= -4

Re

II

Z(k)Z(k’) d3k d3k’(c:.c,),uc(r)

A$

P;,(r) .

(3.2)

Here Z(k) = LJ/47r3is the density in k-space including spin, R being the volume of the sample, and m* is the effective mass. In order to describe local effects, while considering quantum mechanical quantization of the momentum it is necessary to introduce the one-particle Wigner function (ref. 10, part C). Thus let

(3.3) with inversion

(3.4) This holds a fortiori, when the subscript I is replaced by subscript “eq” or subscript “s”. Multiplying (2.18) by (0/87r3)eiU.r, integrating over u, setting

C.M. VAN VLIET

256 5’ = k -

(1/2)u,

[ = k + (1/2)u,

and noticing

et al.

(3.1) and

/3(n,>,,(l - We,) = -y,

(3.5) k

exp{ -

1-

,im

fi[&k-(112)u

-

Ek+(li2)ul}

=

1

(3.6)

’

P(Ek-(1,2)u - Fk+(l,2)u)

u-10

we obtain a(nk )eq d3U eiu’r ___ aE vkEk(r)6u,,,

_-.WL_ 8r3h3h

k

L!i 87r3h3h I

d3u e’“‘r(& k-(1/2)u

n = -___ E,

is

an effective

Now in the effective

EtWl/2)u

+

& k+(l/2)u

)&,,2)uc

k+(deq

d3u e’U’r.%k(nk)f,,o,

8ir’h” where

-

Ek+(l/2)u

electric

(3.7)

field, see below.

mass approximation 1 =

Qk 2m

-

*

(1/2)U12- lk + (1/2)u12}

=_Kk.u.

(3.8)

m” Substituting

this into the second

term to the left of (3.7)

as well as substituting

(3.4), we find for this term ih 2nd term Ihs = ___

8rr3m * I

Reversing i

hence

the order

d3U

elU’(r-“)

d3u ei”‘rk * u

of integration,

k - u = -8dk

1

d’r’ em’“‘r’p,,(k,r’)

(3.9)

we note

* Vc?(r - r’) ;

(3.10)

d3r’ peq(k, r’)8n3k - VcT(r- r’) = v, - Vp,,(k, r)

(3.11)

we find ____ h

8rr’m *

ELECTRICAL

CURRENT

FROM BOLTZMANN

EQUATION

257

The first term on the Ihs is written as qE,*v, a 8r3 G _/ d3U eiu’r / d3rr e-‘““‘p,(k, rr)&, . 1st term lhs = - ___

(3.12)

The Kronecker delta must be taken with a grain of salt: in order to arrive at a valid k, r description we must coarse grain p over a micro cell in phase space, Ao = A3qA3p = h3. Hence (3.12) becomes with u = p/h,

4&v, a -____

8r3ii3 ack

d3dd3m#, 4')= - qE; % $&,(k

A0

k

r),

(3.13)

where we notice that the phase cell is centered on position q’+ r. The same coarse graining procedure is to be applied to the collision term. Thus we find term on rhs = -Bkp,(k, r) .

(3.14)

We thus obtained the standard Boltzmann equation in (k, r) space*

-&

$ &,(k r) + vk * v&q(k>r>=-Bk&(k, r).

* Ok

(3.15)

k

In this section we further only consider the case that Sk is a linear operator, i.e., either the collisions are near elastic, or the electron gas is nondegenerate, so we can introduce a relaxation time. Then 9l:n, =

C (w,-n, - wunf).

(3.16)

f For later use we also introduce the operator

which is identical to (3.15) only if there is reciprocity, wti= Wan,;however, in view of (2.11) this is generally not the case. The solution of (3.15) can be written as** * Note that qEe. ok aphi = q(E.lb) - hp. ** Since S?i has an eigenvalue zero, associated with the equilibrium distribution (as follows from Boltzmann’s H-theorem), (9: + iti)-’ is a Green’s operator in the extended sense. We must omit the projector associated with the eigenvalue zero. This leads to the subtraction of pe4 in (3.18).

2.58

C.M. VAN VLIET et al.

1 = IimP ,,O!%A:+iw

(3.18)

I

of the Boltzmann operator. A relaxation where [%‘i + iw] -* is the resolvent time is usually introduced by writing 93~p,(k, r) = Ap,(k, r)/T(.ck). However, this Like in ref. 10 we better proceed as is generally a very poor approximation. follows.

We write

(3.19)

Now for a degenerate

OJ;

-aPEXj

Iask 1 vk

=

gas with peq = ePcfmEr)we have

_p

= -/!I

c z

[WkieP(fe~k)vk

C

wkie P(f-~k)(vk -

z

-

wik

vp)

eP(f-4vi]

=

$2

cpVk

)

where we used (2.11) in the 2nd term of the 2nd member and where the definition (3.17). We now introduce a relaxation time by writing

.93~Vk = c w,,-(v, - or) =

f

(3.20)

k

A-

we used

(3.21)

r(Ek>.

In ref. 10 we showed that this definition of T results in the usual expressions obtained for one-body or two-body collisions. Further, since T(E~) is usually a weak function of Ed, one has as a very reasonable approximation,

ok

and so on for (59$“v,.

For the second

;

term in (3.18) we have

(3.22)

ELECTRICAL CURRENT FROM BOLTZMANN EQUATION

259

(3.23)

Thus the same trick for the 3: operator applies. Hence (3.19) results in 1 --P $I+io

1

(3.24)

1/7(ck)+iW’

For (3.18) we now obtain

Wk r) = T(Ekhk

.$yqE,+V&5- VEk].

(3.25)

k

For a position-dependent

conduction band the Wannier Hamiltonian

is3,4) (3.26)

XN!= &k(r) - @j(r),

where @ is the electrical potential. The effective electric field is from Hamilton’s equation (3.27)

-qE,=@=-VXw=-VE,+qV@. Thus qE,+ VtJ- Lk=

Vf-qV@=

where /.i is the electrochemical becomes

A&

r) = T(&~)zJ~ 2.

(3.28)

VjZ, potential or quasi-Fermi

vi.

level. Eq. (3.25) now

(3.29)

k

We finally find the current. Since there is only a diagonal solution to the Boltzmann equation, we have (noticing J_, = 0) 6(k’ - k) (AC:+>, = A&k, Thus, from (3.2)

r)6k’k

=

A&

r>

Z(k’)

’

(3.30)

CM. VAN VLIET

260

J,(r) = where

5

1

Re d3kAp,(k r&T(r)

z+(r) = 0 -“%ik(r) reflects

on the reciprocal

$

et al.

(3.31)

V,(r),

the normalization

in a box 0. We expand

&(r)

lattice,

(3.32a)

Viik(r) = c i(g f k)A, g Now because

Re

2

of inversion

e’(g+k).r.

(3.32b)

symmetry

in the Brillouin

g(A, e’g’r) = Re x (-g)A

R

I: e lg.’

g

= -Re

c g(A, g

zone,

etg.r)* = -Re

x gAg elg’r, I(

(3.33)

where we used A_, = A:. Hence this sum is zero. Thus with this sum rule and with iiklm * = ok we find employing (3.29)

(3.34)

Now iik(r) is periodic large compared to

on an atomic the atomic

scale; since (3.26) holds over distances jr1 periodicity-as required by Wannier’s

theorem4)-we must replace /$(r)l’ tensor as usual by u = qpn,,(r) with

4

P=-

4,(r)4m3

I

P

~~~(0)~~= 1. Defining

d”k T(F~)v~v, 2,

we note that (3.32) yields the required

J2.Q.

by

the

mobility

(3.35) k

result

(3.36)

ELECTRICAL

CURRENT

FROM BOLTZMANN

EQUATION

261

4. Current through localized states For localized

states,

This

example

is for

z-direction, direction.

the matrix the

elements

case

when

(ljrl[) we

are nonzero***.

apply

a magnetic

field

in the

E in the x-direction, and measure the current in the xthe Landau gauge A = (0, Bx, 0) we have the states IL) =

put For

IN, k,, k,) with

(4.1)

where & is the sample area, x,, = hk,/mw,, w. = qB/m * is the cyclotron frequency; & are harmonic oscillator wave functions. Now for the x-direction there are diagonal position matrix elements15),

(51~15’) = x,~,,.a,,,.+

(&,‘“I(N

+ 1)1’2s,:.+,

+ ~l~%,.-,]~~,~~

;

(4.2)

0

here x0 is the center of the Landau orbit and k represents matrix elements are nondiagonal only,

(fJv,l~‘) = i (2)“‘[-(N

It can be shown

that

(k,, k,). The velocity

+ 1)1’26N’,N+1+ N”*G,~,,_,]&.

the nondiagonal

ponderomotive

conductivity

is zero”).

Thus there is only collisional current, leading to the well-known results for transverse magneto conductivity, see ref. 11. For hopping effects due to states below the mobility edge we have a similar situation.

The

velocity

matrix

elements

are

zero,

but

the

position

matrix

elements are finite. It is possible to derive the appropriate conductivity formulas for nearest neighbor hopping and variable range hopping, see ref. 11. We now proceed to find again an expression for the local current density. For this current

J,(r) = -4

density

we have from (2.16)

c Mp,>P,(r*)+9~

From the quantum Boltzman equation (2.18), taking force experienced by the electrons, we have

(4.3)

t = 5’ and letting

F be the

*** Only for the Hall effect there is a nondiagonal ponderomotive current, see refs. 10 and 11.

262

The

C.M. VAN

wave

functions

atomic

sites,

setting

(llrj[) = R,,

for these

i.e., over

VLIET

localized

a volume

0,.

et al.

states

extend

Substituting

essentially

(4.4) into

over

a few

(4.3) we obtain,

J,(r) = Pq C bi>,,(l - (ni>,,)u*i(r>rW-) &““, spin

x

(R,. - R,)[ w,.(l

- (n&J

+ ws”&Jeq]

- F.

Note that in the sum corresponding to 5 we must include the case for the sum involving 3” since spin is conserved use the property of detailed balance:

(4.5) spin, while this is not in collisions. We now

We then find

C (r&(1 - (n,.S,,>w,,,u*i(r)rui(r)(R~- 4) *F.

J,(r) = pq

ii”, spin Interchanging into account

the summations Cspin = 2, we get

J,(r) = -Ps

over

4’ and I”, taking

(4.7)

half the sum, and taking

c bl>,,(l - (nr,,>,,)w~~[lus,,(r)l’ - Ius(r)l’lr(R5tc - Rc)*F.

(4.8)

CF

Again we write u((r) = 0 -1’2ii5(r)to take care of normalization. The function ii((r) goes to zero over a few atomic distances. Thus we may set ji$(r)l’r = R,.

(4.9)

We then find

J&r)= -?.F, 4

(4.10)

ELECTRICAL

CURRENT

FROM

BOLTZMANN

263

EQUATION

with

(4.11)

We must now indicate what the effective force is to be. Clearly, there is no Wannier Hamiltonian for this case. We know, however, that hopping involves both drift and diffusion, see, e.g., Nagel?). The number of carriers which participate in the absence of an electric field is n([)kT, i.e., only states with F( within kT from the chemical potential 5 are involved8’16). Thus instead of (3.26) the equivalent Hamiltonian should be (4.12) i.e., we have the sum of chemical and electrical work. Then by Hamilton’s equations, F=$=----

axquiv

_

a4

V(LJ-@)=-V/Ii.

(4.13)

Thus (4.10) yields the required result J,(r)=%-

Vi.

(4.14)

In (4.11) u is given by the extended Adams-Holstein’*) formula for the conductivity (derived by Argyres and Ross’~) for the case of Landau states). It has also been obtained for amorphous materials by C5pek17) and by Zvagin”). For hopping conductivity such as occurring in tail states for heavy doping, or in amorphous materials, u can be evaluated along the lines of ref. 11, section 3B.

Acknowledgment

This work was supported in part by NSERC, Ottawa, Grant A9522.

References 1) S.R. de Groot 1962).

and P. Mazur,

Non-equilibrium

Thermodynamics

(North-Holland,

Amsterdam,

264

2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17)

18)

CM.

VAN VLIET

et al.

K.M. Van Vliet and A.H. Marshak, Physica Stat. Sol. (b) 78 (1976) SOl. A.H. Marshak and K.M. Van Wet, Solid State Electr. 21 (1978) 417. C.M. Van Wet and A.H. Marshak, Phys. Rev. B 26 (1982) 6734. C.M. Van Vliet and A.H. Marshak, Phys. Rev. B 29 (1984) 5960. P.T. Landsberg and S.A. Hope, Solid State Electr. 16 (1977) 421. A.H. Marshak and C.M. Van Vliet, Proc. IEEE 72 (1984) 148. N.F. Mott and E.A. Davis, Electronic Processes in Noncrystalline Materials (Clarendon. Oxford, 1979). K.M. Van Wet, J. Math. Phys. 20 (1979) 2573. M. Charbonneau, K.M. Van Vliet and P. Vasilopoulos, J. Math. Phys. 23 (1982) 318. P. Vasilopoulos and C.M. Van Vliet, J. Math. Phys. 25 (1984) 1391. E.N. Adams and T.D. Holstein, J. Phys. Chem. Solids 10 (1959) 254. P.N. Argyres and L.M. Roth, J. Phys. Chem. Solids 12 (1959) 89. M. Charbonneau and K.M. Van Vliet, Phys. Stat. Sol. (b) 101 (1980) 509. A.H. Kahn and P.R. Frederickse. Solid State Phys. 9 (1959) 257. P. Nagels, in: Amorphous Semiconductors. M.H. Brodsky, cd. (Springer, New York, 1979), p. 125. V. Capek, J. Phys C8 (1975) 479. 1.P. Zvagin, Phys. Stat. Sol. (b) 101 (1980) 9.