UNITED STATES PARTMENT OF OMMERCE

UBLICATION NBS TECHNICAL NOTE 614

U.S.

PARTMENT

OF

OMMERCE

National

n

3 f

Longitudinal Mapetoresistance

in Polar Semiconductors
NATIONAL BUREAU OF STANDARDS

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ONM. BUREAU OF STAKDJ^BS

JUN 8 1972 UNITED STATES DEPARTMENT OF COMMERCE

Peter G. Peterson, Secretary NATIONAL BUREAU OF STANDARDS • Lewis M. Branscomb, Director

'Q-

•M

a

NBS

>

TECHNICAL NOTE 614

ISSUED APRIL 1972

Nat. Bur. Stand. (U.S.), Tech. Note 614, 47 pages (April 1972) CODEN: NBTNA

Longitudinal Magnetoresistance

in Polar Semiconductors

Robert L. Peterson Quantum Electronics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide

a means for making available scientific data that are

of transient or limited interest. Technical Notes may

be listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

(Order by SD Catalog No. 013.46:614), Price 50 cents

Contents

Page

1. Introduction 1

2. Theoretical Basis 5

2.1 B = 7

2.1.1 Deformation potential, acoustic phonons . ... 11

2.1.2 Piezoelectric, acoustic phonons 12

2.1.3 Polar, optical phonons 13

2.1.4 Screened ionized impurities 15

2.2 Finite Magnetic Field 16

2.2.1 Deformation potential, acoustic phonons .... 21

2.2.2 Piezoelectric, acoustic phonons 22

2 . 2. 3 Polar, optical phonons 23

2 . 2. 4 Screened ionized impurities 24

3. Numerical Results 27

4. References 40

Longitudinal Magnetoresistance in Polar Semiconductors

Robert L. Peterson

The magnetophonon effect due to resonant

interactions of charge carriers with optical

phonons in nondegenerate polar semiconductors

is studied when several elastic scattering

mechanisms are simultaneously active also.

The simple model of the band structure, and

the displaced Maxwellian distribution function are used. Analytical expressions for drift

mobility are given for B = and the quantum

limit, and are compared with the results of

other techniques where possible. The numerical

calculations show that whenever magnetophonon

oscillations are seen in the ohmic regime, maxima

occur near the resonance magnetic fields for all

lattice temperature and strengths of scattering

mechanisms which compete with the optical

phonon scattering. This is in contrast with

typical experimental findings, in which minima

usually (but not always) appear near the resonance

fields for ohmic, longitudinal, conditions. The

explanation of the minima must therefore be sought

in the use of a distribution function in which carriercarrier scattering is less important.

Key words: Magnetophonon effect; magnetoresistance;

semiconductors; transport theory.

1. Introduction

This Technical Note is a back-up article for a much shorter

paper being published elsewhere. We here give a rather detailed

exposition of the calculations only indicated in that paper, and provide

some further discussion.
2, 3 In earlier articles, we developed the theory and showed

the results of numerical computations of longitudinal magnetoresistance

for the simple model of a semiconductor, using the displaced-Maxwellian distribution function for the carriers, and scattering of carriers by

2 optical phonons via the deformation potential ("nonpolar") and piezo3 electric ("polar") interactions. Scattering by acoustic phonons via

3 the deformation potential was also included in the latter. These

computations showed oscillations in magnetoresistance, electron temperature, and acoustic gain, due to the "magnetophonon effect." For

the longitudinal ohmic magnetoresistance, such oscillations were pre4

dieted theoretically in 1964, and have since been seen in abundance

5-16

experimentally. However, the experimental observations in the

longitudinal ohmic case have not appeared to bear out the theory in

more than a cursory way.

The experiments typically show minima in the ohmic longitudinal magnetoresistance near the resonance fields, i„ e. , the magnetic

fields for which the longitudinal optical (LO) phonon energy is an integral

multiple of the cyclotron energy of the carrier. However, in certain

materials neither the maxima nor the minima lie close to the

resonance points.

4

Gurevich and Firsov, by solving the Boltzmann equation

in an approximate way in the ohmic region, had shown that for pure

polar LO scattering the extrema of the longitudinal magnetoresistance

2
oscillations would be maxima near the resonance fields. Our computations based on the displaced-Maxwellian also showed this for both

2 3 3 the nonpolar and polar cases. Gurevich and Firsov further deduced

that when acoustic phonon scattering competed sufficiently strongly

with the LO scattering, these maxima would convert to minima. They

evaluated a parameter r which characterizes the relative strength of

acoustic to optical phonon scattering. When T is greater than about

4

unity, minima should be observed. Experiments showing minima

have been reported in which r has been estimated to be larger than

12

unity, supporting the Gurevich-Firsov theory. However, in other

experiments showing minima, it appears that p is substantially less

7 8 than unity ' (see also section 3).

Further, it is not evident just how the maxima for pure

optical phonon scattering will convert into minima with increasing

strength of competing scattering mechanisms: Do the magnetoresistance extrema shift to higher or lower magnetic field, as might be

10, 16

inferred from the more polar materials, or do the amplitudes

of the extrema simply reverse their signs with relatively little

shifting of the positions of the extrema?

These unresolved questions have prompted us to begin a

more detailed theoretical study of this situation. We report here the

results of an investigation in which the displaced-Maxwellian distribution function was used in a study of the longitudinal magnetore si stance

3
in polar semiconductors. Four scattering mechanisms were considered

simultaneously in varying strengths: the polar optical phonon scattering

mechanism, scattering of the carriers by acoustic phonons via both the

deformation potential and piezoelectric mechanisms, and also scattering due to screened ionized impurities. Computer studies were made

at several lattice temperatures. We have used the simple model of the

band structure, that is, the model in which there is but a single isotropic, parabolic conduction band centered at k= 0, and we do not include

Landau level broadening. This is the same model as used by Gurevich

and Firsov.

(If we had included nonparabolicity, which is an important

requirement for the much- studied n-type InSb, the displaced- Maxwellian

17

could not have been used. The same techniques used in deriving the

displaced-Maxwellian for a parabolic band result in a very much more

complicated distribution function in the nonparabolic case. )

Our reason for choosing this particular distribution function

is that it is the only one which at present seems to be amenable to

extensive numerical computation for polar semiconductors. Use of

the displaced-Maxwellian of course implies a good deal of carriercarrier scattering which establishes the distribution. Conditions for

1 8 the validity of this distribution are discussed by Conwell, who

references earlier work. Reference 2 contains further references to
work based on the displaced-Maxwellian. If it is felt that carriercarrier scattering is not predominant in a given material, then of

course the displaced-Maxwellian results should not be taken too

literally.

In the next section, we establish the basic formulas

corresponding to the model chosen. We derive formulas for ohmic

drift mobilities for the various scattering mechanisms for both B =

and the quantum limit, that is, the limit in which only the lowest

Landau level is considered to be occupied. These formulas are

compared with ones obtained by solution of the Boltzmann equation,

where possible. In section III we show several graphs of ohmic

magnetoresistance at various scattering strengths and temperatures.

We also exhibit some hot- electron magnetoresistance curves. Discussions of the results are given and some conclusions are drawn.

2. Theoretical Basis

The displaced-Maxwellian distribution function contains two

parameters, which are the carrier drift velocity v and the carrier

temperature T = l/kg . These are determined self- consistently

e e through two coupled equations which describe the constancy of the

2 carrier momentum and energy in the steady state, or in other words,

which balance the drift and scattering terms of the first and second

moments of the Boltzmann equation.
The scattering interactions for the three carrier-phonon

mechanisms considered here may all be written in the form

E

ic±'L

"q b e + Hermitian conjugate, (1)

where b is the phonon destruction operator for mode q, and r is q «• «•

the carrier position. The index a indicates a specific scattering

mechanism, and polarization indices have been suppressed. The

summation is over all phonon wave vectors q in one or more polarization branches. The coupling coefficient q is generally anisotropic,

and it is customary to substitute an averaged quantity for it.

The scattering interaction between a carrier and N

randomly distributed screened ionized impurities may be written

I 19

iq • (r-R)

p lmP ^-5 5— ' W

2-2 q

2

+r

~ 2

where, in SI (or MKS) units,

b= e 2

/x e„ . (3) o

Here K is the dielectric constant and e is the permittivity constant.

o The summation in eq (2) is over all impurity positions R and Fourier

components q. The screening length is r . An obvious extension of

19 the work of Argyres and Adams to the hot- electron regime yields

the expression
2 -2

q = r = be n , (4)

s s e c

where n is the carrier concentration.

c Subsequent expressions are written for electrons, of charge

-e with e > 0. Drift mobility is defined by

v (E, B)= -n (E, B) E. (5)

d

The magnetoresistance is

n (E, 0)u(E, 0) A p _ P(E, B) _j _

P Q

P(E, 0) n

c (E, B) U

(E, B) (6)

In this paper we shall ignore any E or B dependence of the carrier

concentration n , so that under ohmic conditions, the magnetoresistc ance is calculated from the ratio of drift mobilities. For nonohmic

constant- voltage conditions, the mobility ratio in eq (6) is calculated

from the ratio of drift velocities, while for nonohmic constant- current

(or constant drift velocity) the mobility ratio is given by E(B)/E(0),

where E(B) is the B-dependent electric field across the sample.

We consider the B = and B / cases separately, treating

the zero magnetic field case first.

2.. 1 B =

The complete Hamiltonian, with E in the z- direction, is

H = p /2m'" + H T + 13- V + eEz , (7)

L. a

a
where p is the electron momentum, m is its effective mass, H is the

*** L

lattice Hamiltonian, and the interactions V have been defined above. The

a

displaced- Maxwellian, normalized to unity is

3/2

-

1

2 * 2 — 2 *

p(k) = Q (2nBh/m ) exp [ - 8 h (k-k) /2m ] , (8)

where Q is the volume, and hk=m v . The steady- state equations

«* "*d 2 for electron momentum and energy are

momentum: eE = 4L1 S , (9) a m, a

energy: eE v , = Zj S . (10) d a e » a.

The left sides of these equations come from the drift term of the

Boltzmann equation, and the right sides are the scattering terms. The

summations are over all scattering mechanisms, and we shall not

consider possible correlations between the various scattering processes.

Also, we take the collisions with impurities to be perfectly elastic, so

that the impurity scattering makes no contribution in eq (10). The

2 scattering terms may be written

Se.a= (h2/ 2m*» k

E

k'<

k

' 2

- k2 » W

k

'akP<ii'- (12)
The transition probability per unit time for electron- phonon

scattering is

a

. = 2^ P (\ r_ q a q <*, q

W

k'k

= ^ P H nJ)wa (|n^} k', |njk) . (13)

The summation here is over all phonon numbers n and n ' in the

q q

sets fn 1 and f n' }. The lattice distribution function n({n }) yields

• q

q " q

the Planck factor for the number n of thermal phonons in mode q

q» a «. and branch n :

n = [exp(B h«) )-l] , (14)

q, a o q, a

where uu is the frequency of phonons of type &, and T = l/kR is

q» a L o

the lattiqe temperature. Also,

w -(2TT/h)| <{n' } k'l V |{n}k>| X q - a. q *-

6 [£(ln' jk')-M{n 1 k) ] . (15)

q — q «rf

The energy g, is given by

2 2 £({n }k)= ^4r + E h^ (n + I). (16)

q ~ 2m - q

q,a q 2'

For ionized impurity scattering, the transition probability is

simply
W1 /l imp

=(2 TT/^)|<k / |V. I k> | 2

6 [h 2

(k

2

-k'

2 )/2m* ] . (17)

iA. fa. The momentum and energy scattering terms for carrierphonon processes reduce to

1

I 2

m, a

*' a ' "»j a

S = A j dq q I q I J dx xF (q,x), (18)

*° 2

S = A f dq q uu

| q | f dx F (q, x), (19)

e, a «L q» a a J a

after integrating over carrier coordinates. Here

A= (2n3 m') 2 Q/4TT h , (20)

F (q,x)=(n + 1) e _Tla+

-Ti ^e^^' , (21) a q» a q, a

* 2 n r m (jo = -hr- 5- ± : 4'~ Tkx| . (22)

2m'

In these equations, x = cos 9 where 9 is the angle between q and E.

If elasticity is assumed for the acoustic phonon processes, the term

m iw /hq of eq (22) does not appear. However, in the numerical

q, a work the term causes no real complication, and so we have not

assumed elasticity there. Neither have we assumed equipartition,

but have used the full Planck factor, eq (19), in the numerical work.

(Equipartition becomes increasingly inaccurate with increasing E. )

10
We now consider the four mechanisms separately in more detail.

2.. 1. 1 Deformation potential, acoustic phonons.

20 The coupling coefficient can be written

^d,acl2=El h <l2

/ 2 P n %' (23)

where E is the deformation potential energy and p is the mass density.

Only longitudinally polarized acoustic phonons are involved, and we

use

0J q

=U^q, (24)

21 where u is the longitudinal speed of sound averaged over directions.

The E = B = drift mobility expression is readily found to be

1

2 4 2 3epu

t h

/ Zt? \

16 E \m k T /

1 J_i

This is obtained from the ohmic limit of eqs (9) and (18), using elasti22

city and equipartition, and has been derived previously by Sladek

23 and Paranjape and deAlba. The mobility as given by eq (25) is

smaller than that obtained by solving the Boltzmann equation for de24

formation potential acoustic phonon scattering alone, by the factor

9tt/32. The reason is that the displace d-Maxwellian implies carriercarrier scattering, which further limits the mobility.

11
2.1.2 Piezoelectric, acoustic phonons.

For a given direction of propagation q, the contribution of a

given polarization branch to |q _^| may be written

2 . ... 25 ac 1

K

2

bhu) /2nq

2

, (26)

q

where K is the dimensionless electromechanical coupling, and b is

given by eq (3). Both quantities are directional dependent. We use

an average value for the dielectric constant appearing in b. The expression (26) is summed over longitudinal and transverse modes, and

21 26

averaged over directions in the manner of Meijer and Polder. We

indicate the result symbolically here by

S,ac

|Z = Bp/Q" < 27 '

and give additional details regarding computation in section III.

The E = B = drift mobility is found to be

3eu h / 3 \ av / 2tt \ "o = ^i [—3 ) (28)

p \m kT L

/

22

in the elastic equipartition limit, a result given ea.rlier by Sladek.

As in the preceding case, the result (28) differs from the Boltzmann

equation result only in being smaller by the factor 9tt/32.

12
Here we use

2.1.3 Polar, optical phonons.

27

1

2

2 * 2

q = 2Tfh eE /Qm q . (29) ^op o n v

E has units of electric field and is given by

o m eWuJ /

s h W x y

' O

v ^° O /

E =^ — — - — » ( 30 ) o 4rre

where K and k are, respectively, the (directionally averaged) di- o co

electric constants for zero and infinite frequencies, and ^ is the

longitudinal optical mode (LO) frequency at k = 0. We make the usual

assumption that the LO frequencies at all k are equal to uj .

The phonon scattering expressions (18) and (19) are not the

simplest to use for optical phonon scattering. It is easier to integrate

first on the phonon modes q, then over the angular parts of k, resuiting in

2 .oc 2

Sm

2eE n/\2_Y v C _yx I v Y o o/y\ 1 J Yo - ' ,op

= v~S^J

e Jo dXe e ^

[xs+tn(x+s)]

(31)

X C(2yvs) + — [xs - ta(x+s)]C(2Yvx)k

13
/2h^ - Y x

o y \ -W «

' I —

J n e I ( o\ * / o J ^m tt / o

S = 4eE I

; ) n e I dx e x£n(x+s) X e , op o \

->: ^m tt

r

Yo

~ Y i |e sinh(2Yvs) sinh(2Yv x)

J

•• 2YvS " 2vvx J'

2 \ where s = (x +1) , and

The E= B= drift mobility is found to be

, 2 v /

3 / ^o \ sinh (V 2)

(32)

C(y ) = cosh y - y sinh y , (33)

* 2

2 v = [m v

d

/2ha)

Q

] , (34)

Y \ = fe ""-I]"

1

, (35)

Y = p hi« , Y= 3 huo (36)

o o o e o

/ TThuu o ^O-Wi—~

j) kTVz)^ ' (37)

o \ 2m Y / 1

Y o

7 o

28

first derived by Stratton. Properties of the modified Bessel function

K may be found in Ref. 29. The high temperature form of eq (37) is

9

TT /32 times smaller than that deduced from the Boltzmann equation,

as in the acoustic cases (equipartition assumed there is a "high"

14
temperature approximation). At lower temperatures (Y > 0. 36) it is

well known that the Boltzmann equation cannot be solved directly for

the polar optical interaction because of the breakdown of the relaxation

30 time approach. Howarth and Sondheimer used a variational technique,

which when corrected by the effective ionic charge in the manner of

31 Ehrenreich, gives precisely eq (37) in lowest order approximation.

2.1.4 Screened ionized impurities.

From eqs (2) - (4), (9), and (17), one can write

S

m, imp

2 ,„. .1

L -n b / 2B \ 2 2 —

I —2. I 16 v, 13 */ d ^rr m /

(38)

1

1

dx x""

1 [£n(l+x)-x/(l+x)]e

~ 5"C[2r(gx) 2 ] ,

where n = N /q is the impurity concentration,

g = 3 e /4 , (39)

e s

e = h q

2/2rrT=h 8 n b/2m"~, (40)

e c

r 2

= 8 m*v

2

/2 , (41)

e d

and C(y) is given by eq (33). In the ohmic region, the integral in eq (38)

32 can be expressed in terms of the exponential integral E ..(g):

15
Vs) - J"

• gx -

"• E1 (g)= i! dx e x . (42)

The E=B=0 drift mobility is

3. 3_ 3

U

/2tt k T \ r o-^i—i^-) nb x m / K>u

e ° E^g )-I ,(43)

where g is the E -» form of g. Equation (43) is apparently new.

o 19

Solution of the Boltzmann equation in the ohmic limit yields a somewhat different integral than that to which eq (38) reduces, which cannot

be expressed in terms of exponential integrals. However, for g

<<: 1,

which is typical in nondegenerate semiconductors, eq (43) may be simplified by using E (g ) « In (l/g ). Using g << 1 in the Boltzmann

19 equation result, one obtains the same expression as that to which eq

(43) reduces, except that the drift mobility in the displaced Maxwellian

formulation is 3rr/32 times smaller.

2.2 Finite Magnetic Field

In the familiar "Landau gauge, " the Hamiltonian for E and B

both in the z- direction may be written

2 2 * 2 * V*

H = [p + P +(p +m uj x) ]/2m + H _ + 2-jV +eE z , (44) X Z y C J_j rv Ct

where

0) = eB/m (45)

c

lb
is the cyclotron frequency. The carrier energy in the plane wave

states k , k , and Landau level n = 0, 1, 2, ... is thus

y z

£ (k ) = h

2k2/2m'"

: + huu (n+ i) . (46) n z z c 2

We have associated the energy degeneracy with the y- direction, with

N degrees of freedom. The displaced-Maxwellian may be written as y

the product p n (k , k ), where

n y z 1

nf* V 1 2 ^ /

Beh Y ^'e^1 ^ ^)2] ,(47)

y z \2rtm / 2m

= (l-exp(-B h^ )) exp(-nghu) ) . (48)

e c e c

The distribution function is normalized such that

CO £ E ( dk p p(k,k) = 1, 1= n k J-„ z n y z (49)

n=0 k

y

and L is the length of the sample in the z- direction.

in • i» The matrix elements of e A «•% needed for both phonon and

impurity scattering, are

iq« r

'tiK k >= o, v z I y y y- z z z <n'k 'k 'le Ink k >=.fi. , , 6 . / . J , (q ,k +q ,k ), (50)

y z I ' y z k^'-k^q k^ -k^, q^ n n

nx y y y

17
where

lq x x

J /„(q , k +q , k ) = f dx e u / x+ k +q \u (x + k nnxyyy ), (51)

«' n. '

!

y y J n \ y / m uu /h m 'uo /h

c c

and u (z) is the harmonic oscillator wave function n

_i n * 4 2

i H (z) /m (i) i H (z) /m uo \ -z /2

(2 n!)

and H (z) is the Hermite polynominal of order n. Now J , is a n n 'n function only of \ , where

2 *

2 2

and q = q +q_ „ One finds \

I J , ] 2

= e"

X

f ,(X) , (54) n n n n

where

r n s

"1 2

f , (X)= n!nM Xn

'- n I E ,,["V - ., .(nfen). (55) n'n L s=o s ! (n- s)! (n -n+s)! J

If equipartition and elastic collisions are assumed for the deformation potential acoustic phonon scattering, the greatly simplifying

result

18
CO 00 CO h (2flm*tti ) _1

J dq

J dq |J | = f dXe \ (X) =1 (56) ''-od xJ-o= y n'n1 J n'n

2 can be used, as it can be with the nonpolar optical phonon scattering.

However, for the other scattering mechanisms considered here, this

result cannot be used, and so, as with B = 0, we have not assumed

equipartition or elastic collisions for the acoustic phonons, as no

significant computational time-saving is thereby achieved.

The steady- state equations for carrier momentum and energy

can be written in the forms eqs (9) and (10), respectively. For

carrier-phonon interactions, the scattering terms are

1 2 Sm,a < B > = " A

J

dVXl J

dq

|>qjL l^ a

| \<V q,). (57)

CO - 1

oo S

eja (B),-A j dqjqj / dq

i(

« U

|-iI o

| 2 %a R

a

(qiif qB

). (58)

where A is given by eq (20), and

R (q , q )= ^ n p e f , (X) a U nz n, n '= n ' n'n

i i

(n +1) e a + -n e j , (59)

q, a q, a

. R h I q m'uJ(n-n') m (l) J 2 = l£_l^ + £ ± , q' a T *j . a i o * 2 hq„ i^q. 2m hqz

(60)

19
For screened ionized impurities, the momentum scattering

term may be written

, 2

2 - r .. °° °° -n bhui Be f 2 f _

|B >' ~ 3/ 2

6 J dye J„ s dXe [VU' yl m, unp

4tt

(61)

+ R

imp( ^- y)J

'

where

co n

R. (\,y)= 2- p | E f ,(l)c (X, y) sinh (2ry)

-1=0 (62)

imp n= n L „_ nn, n-£

+ Sfn)n^(X)exp(-,, Rehc )c^(X,y)sinh( 2ry^J ,

y = (y Uphffl ) 2

, (63)

C e c

c I

.. - 2

(\,y) = y^'V^ y)[ DU) + (y^+y)

2] . (64)

D ( X) = 8 h«J X + 4g , (65)

e c

and g and r are given in eqs (39) and (41), respectively.

We now look at some results in the quantum limit, which we

define here to mean that the term e e c is negligible compared to

unity, so that all carriers are in the lowest (n= 0) Landau level.

20
Z . 2. 1 Deformation potential, acoustic phonons.

The two acoustic phonon mechanisms can be treated together

i— 1

Pa , up to a point, by writing | q

= B q /Q, where p = +1 for the a a a deformation potential interaction, and p = -1 for the piezoelectric a

interaction. The momentum scattering term (57) may then be written,

in the equipartition, elastic, ohmic, and quantum limits.

:< co p -i

-8AB m v ou f

, -y f -X a

Smj (QL) = a d_c dy e V dXe J _. ^ o •» o

(66)

1 e u

a

c

and

= [smV P Q

h

2 + 2m^c X/h

J

2

. ( 67

)

The ohmic drift mobility for deformation potential scattering is then

3,3 2

rr eh pu.

^QL =

2 ~ *5 i < 68)

Ej UJc (2m kT L

)

Comparing with the B = case, eq (25), we see that

U>QL

= 3hu)

c / 8kT

L • (69)

Apart from a possible variation of carrier concentration with magnetic

field, the longitudinal magnetoresistance, eq (6) thus increases

21
linearly with field in the ohmic, quantum limit. Equation (69) agrees

19 with the Boltzmann equation result without carrier- carrier scattering,

except that a 9 replaces the 8 in eq (69).

2.. 2. 2 Piezoelectric, acoustic phonons.

32

Using p = -1, and

a

i

dx e X

E (x) = a

" £n(l * a) (70)

in eqs (66) and (67), one finds

1

(" 3t 4h M'c-i euav I Li c-1

where

U QL B uu \_ *3 J In c ( 71 ) p c Zm

c = 3 hiu /4 . (72)

Comparing with eq (28), we have

^o 3 QL

c In c

2 c-l

(73)

Thus the ohmic longitudinal magnetoresistance increases logarithmically

with B at sufficiently high field, apart from any changes in carrier

concentration. Except for the numerical factor, eq (73) agrees with

22
that obtained from the Boltzmann equation without carrier- carrier

33

scattering, for c > > 1.

2 . 2. 3 Polar, optical phonons.

In the ohmic quantum limit, the momentum scattering term (57)

becomes

l

Li

where

I(TL

, B) = J Q

dx E^x) exp [-(c-1) x- yq

2/(16 ex)] , (75)

and c is given in eq (72). The ohmic drift mobility is thus

4 sinh ( Yo/2)

l, 3„ 3 . 2

WQL"V , * / Et„ K'i'.B) • (?6)

2m / o c Li

For c sufficiently greater than unity, it is evident that the term

? exp ["- V /(16cx)l in I(T , B) makes no essential contribution to

L

' o L

I(T , B) apart from a temperature- dependent multiplicative factor.

We thus write, using eq (7 0),

I(T_,B)« r( YJ (c-1)"

1 in c . (77)

23
That eq (77) is a good approximation can be seen from figure 1, where

the exact integral, evaluated numerically, and eq (77) are plotted

against c for three temperatures. The factor r(y ) is there taken

to be exp (-0.44 y ), chosen strictly empirically. Thus, to a good

approximation, we can write the drift mobility as

L, \ c-1 sinh (y /2) —} —,— o 76 a) - / c -In c — : r— » v ' 2m v E r (y ) o o

and the ratio of mobilities as, with, eq (37),

o 3r( Y ) cine (78)

o UQL y^^yJZ) c-1

The ohmic longitudinal magnetoresistance thus varies approximately

logarithmically at sufficiently large B, apart from any carrier concentration variation.

2 . 2. 4 Screened ionized impurities.

The momentum scattering term (61) reduces to

2 2 - r f 2 S (QL)=" n

ibe

| dy e" Y sinh(2ry)z"

1 eZE,(z) , (?9)

m, imp -r- J 2

5/£ o

in the quantum limit, where

24
c = Xo^c /4aJ

o

Figure 1. Comparison of the integral I( T , B) defined in eq (75'

(solid line) to the approximate form given in eq (77)

(dashed line).

Z5
z = 4(g+y )/b h<« , (80)

r is given by eq (41), and E (z) is the second-order exponential

32 integral. In the ohmic limit, the drift mobility is found to be

3

QL

8\2 J^~eeg (TTkT ) O J_i n

Ib

2m^[F(g

o )-F(go/c)]

c-1 (81)

where

F(x) = x e E^x) . (82)

With eq (43), the ratio of B = to quantum limit mobilities is

o 3c F(g

Q

) - F(g

Q

/c)

U QL 2(c-l) (l + g

o

) F(g

o )-go

(83)

The longitudinal magnetoresistance due to ionized impurities thus

saturates at sufficiently high magnetic field. The Boltzmann equation

19 result of Argyres and Adams, which omits carrier-carrier interaction, also shows this, but with a different saturation value. For

example, if g << 1, and c

>

1, then using E (x) « -y -£n x

o ~ IE

where y = 0.5772 ... is Euler's constant, one obtains from

eq(83)

26
M

3 y E

+ to g + (c-lf in c

1Q

to a good approximation. The result of Argyres and Adams reduces

to 2[tn(3/g

o )-l]

_1

(l + 3/c)"

1

,

3. Numerical Results

In this section we exhibit some results of our numerical v-ork.

For this, it is necessary of course to choose values for the parameters

such as m , oj , E,, etc. We have elected to use those which seem to o 1 be representative of n-InSb, which is probably the most extensively

studied material to date regarding magnetophonon effects. However,

it should be kept in mind that InSb has a very nonparabolic central

conduction band, whereas our model assumes a parabolic conduction

band. Further, the displaced-Maxwellian distribution function implies

a degree of carrier- carrier scattering which probably exceeds that

in most samples studied so far. Thus, the numerical work presented

here should not be understood as an attempt to describe n-InSb. Instead, this study was undertaken to determine the extent to which

temperature and elastic scattering could shift the magnetophonon

structure, within the displaced-Maxwellian formulation.

27
Figures 2, 3, and 4 show curves of ohmic magnetoresistan.ee

versus magnetic field (expressed as oj /iw ) for three temperatures

corresponding to y = Q/T = h uj /kT = 14.0, 3.65, and 2. 60, O J_j O L-i respectively. With 9 = 281 K, these values correspond to lattice

temperatures of 20 K, 77 K, and 108 K. We have also used E = 470

V/cm, and m = 0. 0159 m , which is the value of the so-called ' o

11

magnetophonon mass in n-InSb. The value of the deformation

34

potential E has been uncertain, although Tsidilkovskii and Demchuk

have recently made a strong argument for a value of about 3 eV.

They reference much previous work, but omit the work of Nill and

McWhorter (4. 5 ev), Whalen and Westgate (16. 2 eV), and Martin

37 and Mead (16 eV or higher). In the figures we have used several

values for E . Also, we use the Nill and McWhorter value of 0. 06

coul/m for the piezoelectric constant e , which is close to that of

Whalen and Westgate. The piezoelectric coupling coefficient for

acoustic phonon scattering is obtained in the manner of Meijer and

2b Polder by considering all polarization modes and averaging over

directions. The result is

B = 2b

2

e 14

2h Q/pe 2

, (84)

where B appears in eq (27). The averaged quantity Q is

28
Po

0,5

:/w

o

4.0

Figure 2. Ohmic magnetoresistance at y = 14. (20 K for InSb).

Values of E (eV), e (coul/m ), impurity concentration

-3

(cm ), and T are, respectively: (a) 0, 0, 0, 0; (b) 4.5,

0. 0b, 4x10 3

, 0. 27; (c) lb. 2, 0, 0, 0. 01; (d) lb. 2, 0, 15

10 , 1.5b.

29
I I I I

| I I I

| 1 1 1 1 | / 1 i i i

1,5 — / —

7o =3.65 / . - v y -

- / */ -

1.0

// z^^^ ^^-

0.5

^^-

- ^y^\^^ - ^^>/'

-

i i i i I i i i i i i i i i i i i i i

:/uo

Figure 3. Ohmic magnetoresistance at y = 3. 65 (77 K for InSb).

Values of E (eV), e (coul/m ), impurity concentration

-3

(cm ), and T are, respectively: (a) 0, 0, 0, 0, ; (b) 4. 5,

0. 06, 4 x 10

13

, 0. 04; (c) 16.2, 0, 10

15

, 0.43; (d) 30, 0. 0b,

10

15

, 0. 58; (e) 0. 58; (e) 30, 0. 06, 4 x 1Q

13

, 0. 23.

30
Pn

i i i i i i i | I i i i | ii i i

1.5 — -

- V 2.60 /-

/ ~ - /~

- /

1,0

- -

- - s

<>^^^

-

0.5

b^^"

.//^a -

-

i i i I i i i 1 i i i i I i 1

i i

"c/^o

Figure 4. Ohmic magnetoresistance at y = 2. bO (108 K for InSb).

Values of E (eV), e (coul/m ), impurity concentration

_ 3 (cm ), and T are, respectively: (a) 0, 0, 0, 0; (b) 4. 5, 13 15

0.0b, 4x10 , 0. 03; (c) lb. 2, 0, 10 , 0. 31;(d)30, 0.0b,

10 , 0.51; (e) 30, 0.0b, 4x10 , 0.30.

31
(

3u

l 4u~ I /13 ' (85)

where u and u are longitudinal and transverse sound velocities for

propagation in the (1, 1, 1) and (1, 1, 0) directions, respectively,

given by

pU

l

2

= I

(C

11 + 2C12

+ 4C

44

+ 4e

i4

2^£ o

)

' (86)

PU

2

2

= C 44

+e

i4

2/Ke

o

• (8?)

The elastic constants are given B the values c, , = 6. 66, c = 3. 35,

11 12

10 ,2 and c= 3. 14x10 nt/m . We also chose K = 17.8 and D = 5. 78

44 ' M

3 gm/cm . The calculated values of u and u are 3900 and 2300 m/sec,

respectively, and the grand average u over directions and polarizations is 2800 m/sec, which is used in oj = u q. For the deformation

q av

potential scattering we used u = 3700 m/sec, which is the average

longitudinal sound velocity.

In evaluating the screening terms g and g , we have set the

carrier concentration equal to the impurity concentration. Finally, we

have summed over as many Landau levels as necessary to achieve

about 0. 1% accuracy. In some cases, as many as 14 levels were

included.

32
4 We have calculated the Gurevich-Firsov T (see the discussion

in the Introduction) for each of the curves in figures 2-4. Extending

their result to include all scattering mechanisms which compete with

the carrier- optical phonon scattering, and assuming that the respective

resistivities are simply additive, we obtain

(

h

) *E I IT uu kT T / a

\ o L /

r= T ^c n UJ kT / ~ P a

The three dimensionless resistivities p in the present case are

a

c

? L

2 * / 3 3 w 2 16E m / ha; k T \

(def.pot.,ac.)= 2 5~

V °

5 / > ( 89 > 3eE tl pu

V

TT o

M £

a ** 2 2 - 8m 'b e Q /h«J kT \

2 p(piezo. ,ac. )= — I V/ ' (90)

3e E h Du ' T1 o av

K

2

/ ,

* n_bb / hu)u) \ &o

P^P-^i2iE- I-TtS) [(1 + §o

)e W"11 ' (91

o \tt k T /

L

F- values were calculated at the first resonance point, that is at w =w

c o As is seen from the figure captions, F is typically below unity even

in the most favorable cases. Thus according to Gurevich-Firsov

theory, resonance point maxima, or perhaps simply very small

amplitudes of oscillation, are to be expected in n-InSb. However,

33
as pointed out in the introduction, magnetophonon oscillations in n-InSb

have always shown minima near the resonance points.

In each of figures 2-4, we show curves for pure optical phonon

scattering (F= 0), labelled (a), which show the magnetophonon

oscillations most clearly. To be noted are the facts that the resonance

4

extrema are maxima, as in the Gurevich-Firsov theory, but that the

maxima lie somewhat to the low- field side of the resonance points,

defined by uu = uu /N, N = 1, 2, . . . The shift is greatest at low

c o temperatures and low N- values. The pure polar optical curve at y =14

also shows the near- logarithmic behavior at large B, discussed in the

preceding section.

The principal result of adding other scattering mechanisms to

the polar optical scattering is to diminish and finally obliterate the

magnetophonon oscillations, as seen in the figures. In all cases we

have examined, the oscillations have vanished before T has reached

unity. One also sees from the figures that there is no significant

"phase shift, " due to these other mechanisms, moving the minima

toward the resonance points, which possibility was considered in the

introduction.

In figures 5 and 6, we exhibit the results of some hot-electron

calculations, obtained by solving simultaneously the carrier momentum

and energy steady-state equations. A " constant- current" (actually,

34
oW

d)uo

4->0)aaoH

-wODOs Co

•rH +Jduou

Oh

M 3Oo

*ooo

II

u

rH *> V N r-H -O o

II

x

W ii

<k

35
o

in

II CD o u

>- flo

+->

u

aj >^

CDo

4-> 3

CO .§

CO

0) . A n

Nj

a

CD i—i Pi jj OC o

cti

u

a .O s o o

f-i o

+ju II <—i

"tf 1

I—1 _ • CD <^ > 'a

<u o

-a 1 1 rd o ° o 1—1 I—' • T-*

ii

x

,3

H <*

O

W ii

-O

COu2

ao

<k

36
constant drift velocity) magnetoresistance curve is shown in each

figure. The ohmic magnetoresistance is also shown for comparison.

The dimensionless drift velocity v in the figures is defined in eq (34).

With the n-InSb parameters given earlier, figure 5 corresponds to a

5 temperature of 20 K, and v= 0. 015 corresponds to v =0. 11 xlO m/sec,

d or E = 0. 56 V/cm at B = 0. In figure 6, T = 77 K, and v= 0. 32 corres-

.Li 5 ponds to v =2. 34 x 1 m/sec, or E = 42 V/cm at B = 0.

The most noticeable features of figures 5 and 6 are the sharp

2, 3 cusps precisely at the resonance points. These are energy-transfer

effects superimposed on the background of momentum scattering which

comprises the ohmic curves. Nonparabolicity of the conduction band

and Landau level broadening would tend to round off the cusps. Figure

5, showing resonance maxima, is interesting in that it shows how the

magnetophonon structure, absent in the ohmic case, can be brought out

by application of nonohmic electric field or current. This effect has

38, 39 40

been exploited by Stradling and Wood, and by Aksel'rod et. al.

In figure 5 the scattering is dominated by ionized impurities, the

-3/2

resistivity of which varies essentially at T at both B =0 and the

quantum limit. Since the carrier temperature decreases to the lattice

2, 3 . temperature at the resonance points, the magnetoresistance increases. Figure 6 shows resonance cusps which are minima. At this

temperature, the optical phonon scattering dominates both the acoustic

37
phonon and impurity scattering. The minima are the result of the

polar optical phonon scattering itself.

Low temperature, hot- electron longitudinal magnetoresistance

38 41 42

in GaAs, CdTe, and InAs do show maxima, in agreement with

39, 40

figure 5. However, InSb is an exception. The reason for this

43

is not clear, although Yamada and Kurosawa argued that a deviation

from the Maxwellian distribution function could occur under hotelectron conditions and could account for the minima. Other, physical,

39 arguments can also be advanced , to explain, in part, the minima.

Further investigation is in order on this point.

In summary, the displaced-Maxwellian distribution function

together with the simple model of the band structure cannot explain the

observed longitudinal magnetoresistance resonance minima which

frequently oceur under ohmic conditions. This is in spite of the fact

that the B = and quantum limit mobilities largely agree, except for

numerical factors, whether computed from the displaced-Maxwellian,

or the Boltzmann equation without carrier- carrier scattering. It

does not seem likely that the simple band structure assumed (parabolic

and isotropic) is the principal reason for this failure, since apart

from InSb, the semiconductors studied do not deviate greatly from

the simple model (subsidiary valleys are surely not important at

ohmic conditions), and yet exhibit near-resonance minima, or in

38
some cases an ambiguous situation. More likely, the lack of agreement lies with the displaced-Maxwellian itself. Several factors point

4

to this. First of all, the original work of Gurevich and Firsov,

based on an approximate but direct solution of the Boltzmann equation,

implies this. The materials studied experimentally have usually been

15 -3

fairly pure ( - 10 cm impurity concentration), whereas the displaced-Maxwellian implies a carrier concentration usually greater

15 -3

than about 10 cm (this figure is quite uncertain; it is a function

of the strengths of all the carrier-phonon and carrier-impurity interactions, as well as of temperature, and electric and magnetic field

strengths ). Further, the work of Yamada and Kurosawa is a

further indication of the inadequacy of the displaced-Maxwellian for

the usual experimental situations. Nonohmic magnetophonon results

of the displaced-Maxwellian, however, can be qualitatively satisfactory,

3 as noted above and also previously.

39
4. References

1. R. L. Peterson, Phys. Rev. B15, (May, 1972).

2. R. L. Peterson, Phys. Rev. B2_, 4135 (1970).

3. R. L. Peterson, Bo Magnusson, and P. Weissglas, Phys. Stat.

Sol. 46, (1971).

4. V. L. Gurevich and Yu. A. Firsov, Zh. Eksperim. i Teor. Fiz. 47,

734 (1964) [ Soviet Phys. -JETP _20, 489 (1965)].

5. R. V. Parfen'ev, S. S. Shalyt, and V. M. Muzhdaba, Zh.

Eskperim. i Teor. Fiz.47, 444 (1 964) [ Soviet Phys. -JETP 20, 294

(1965)].

6. I. M. Tsidil'kovskii, M. M. Aksel'rod, and V. I. Sokolov, Fiz.

Tverd. Tela 7, 31 6 (1 965) [Soviet Phys. - Solid State 7, 253(1965)].

7. I. M. Tsidil'kovskii and M. M. Aksel'rod, Proc. 8th Int. Conf. on Phys. of Semicond. , Kyoto (J. Phys. Soc. Japan (Suppl. ) 21, 363

(1966)).

8. N. T. Sherwood and W. M. Becker, Phys. Letters 27A, 161 (1968).

9. P. I. Baranskii and O. P. Gorodnichii, Fiz. i Tekh. Polu. 2, 854

(1968) [ Soviet Phys. -Semicond. _2, 709 (1968)1.

10. A, L. Mears, R. A. Stradling, and E. K. Inall, J. Phys. C (Proc.

Phys. Soc. ) 1, 821 (1968).

11. R. A. Stradling and R. A. Wood, J. Phys. C (Proc. Phys. Soc.)

1_, 1711 (1968).

12. Kh. I. Amirkhanov, R. I. Bashirov, and Z. A. Ismailov, Proc. 9th

Int. Conf. on Phys. of Semicond. , Moscow (Nauka, Moscow, 1968)

p. 744.

13. S. Askenazy, J. Leotin, J. -P. Ulmet, A. Laurent, and L. Holan,

Phys. Letters 29A, 9 (1969).

40
14. L. Eaves, R. A. Stradling, and R. A. Wood, Proc. 1 Oth Int. Conf. on Phys. of Semicond. , Cambridge (U. S. Atomic En. Comm. , 1970) p. 816.

15. R. A. Stradling and R. A. Wood, J. Phys. C (Proc. Phys. Soc. ) 3, L 94 (1970).

16. L. Eaves, R. A. Stradling, S. Askenazy, J. Leotin, J. C. Portal,

and J. -P. Ulmet, J. Phys. C (Proc. Phys. Soc. ) 4, L, 42 (1971).

17. H. Frolich and B. V. Paranjape, Proc. Phys. Soc. (London) B69,

21 (1956).

18. E. M. Conwell, Solid State Physics, edited by F. Seitz, D.

i Turnbull and H. Ehrenreich (Academic, New York, 1967), Suppl.

9, p. 12ff.

19. P. N. Argyres and E. N. Adams, Phys. Rev. 104, 900(1956).

20. Ref. 18, p. 107.

21. The procedure to be used in averaging out the anisotropies of

acoustic phonon propagation is not unique. For example, for

deformation potential acoustic phonon scattering, and assuming

equipartition (n ~ l/s fl^ ), one should average the quantity

-2 q •

. ° _i9 u (q) in eq (18) and u (q) in eq (19). However, at large

values of E, equipartition becomes inaccurate. Further, u (q)

also appears in the exponents, eqs (21) and (22). The fact of near elasticity for acoustic phonon collisions minimizes this

complication. The "best" method would be to average the quantity n /u (q) in eq (18), and n in eq (19) for the defomation potential

q I q

case, and still different quantities for the piezoelectric case, but

this is extremely impractical computationally. We have adopted

the much simpler, and usual, procedure of using a directionally

averaged u (q) wherever it occurs with the deformation potential

'0

41
interaction, and a slightly more sophisticated procedure (see

sections II. 1 . 2 and III) for the piezoelectric case. Different

averaging procedures typically give results differing by at most

a few percent.

22. K. J. Sladek, Phys. Rev. 120, 1589 (i960).

23. V. V. Paranjape and E. de Alba, Proc. Phys. Soc. (London) 85,

945 (19b5).

24. W. Shockley, Bell Syst. Tech. J. J30, 990 (1951).

25. A. R. Hutson, J. Appl. Phys. _32, Suppl. , 2287 (1961).

26 o H. J. G. Meijer and D. Polder, Physical^, 255 (1966).

27. Ref. 18 p. 156.

28. R. Stratton, Proc. Roy. Soc. A246, 406 (1958).

29. Handbook of Mathematical Functions, edited by M. Abramovitz

and I. A. Stegun, National Bureau of Standards, Appl. Math.

Series 55 (U. S. GPO, Washington, D. C. , 1964), p. 376.

30. D. J. Howarth and E. H. Sondheimer, Proc. Roy. Soc. A219, 53

(1953).

31. H. Ehrenreich, J. Phys. Chem. Solids _2, 131 (1957).

32. Ref. 29, p. 228.

33. Mobility expressions for impurity and acoustic phonon scattering, as deduced from the Boltzmann equation or its equivalent, are given in the review article of L. M. Roth and P. N. Argyres, in

Semiconductors and Semimetals, Physics of III-V Compounds,

edited by R. K. Willardson and A. C. Beer (Academic Press,

New York,1966), Vol. 1, p. 159.

34. I. M. Tsidil'kovskii and K. M. Demchuk, Phys. Stat. Sol. (b)

44, 293, (1971).

35. K. W. Nill and A. L. McWhorter, Proc. 8th Int. Conf. on Phys. of

Semicond. , Kyoto (J. Phys. Soc. Japan (Suppl. ) 21 , 755(1966)).

42
36. J. J. Whalen and C. R. Westgate, Appl. Phys. Letters 1_5, 292

(1969).

37. J. P. Martin and J. R. Mead, Appl. Phys. Letters 17, 320(1970).

38. R. A. Stradling and R. A. Wood, Solid State Comm. 6, 701 (1968).

39. R. A. Stradling and R. A. Wood, J. Phys. C: Solid St. Phys. _3»_ 2425 (1970).

40. M. M. Aksel'rod, V. P. Lugovykh, R. V. Pomortsev, and J. M.

Tsidil'kovskii, Fiz. Tverd. Tela 11, 1 1 3 (1 969) [ Soviet Phys. -

Solid State 11_, 81 (1969)].

41. A. L. Mears, D. Phil Thesis, Oxford University (1969).

42. R. A. Wood and R. S. Stradling, Conf. on High Magnetic Fields

and their Applications, Nottingham (The Inst, of Phys. and the

Phys. Soc. , London, 1969), p. 47.

43. E. Yamada and T. Kurosawa, Proc. 9th Int. Conf. on Phys. of

Semicond. , Moscow (Nauka, Moscow, 1968) p. 805.

43

FORM NBS-114A (1-71) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. Technical Note 614

2. Gov t Accession No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE Longitudinal Magnetoresistance in Polar Semiconductors

5. Publication Date

6. Performing Organization Code

7. AUTHOR(S) 8. Performing Organization Robert L. Peterson

10. Project/Task/Work Unit No. 2710412

9. PERFORMING ORGANIZATION NAME AND ADDRESS NATIONAL BUREAU OF STANDARDS, Boulder Labs

DEPARTMENT OF COMMERCE Boulder, Colorado 80302

11. Contract/Grant No.

12. Sponsoring Organization Name and Address

Same as Item 9.

13. Type of Report & Period Covered

Sept. '70 - Sept. '71

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.)

The magnetophonon effect due to resonant interactions of charge

carriers with optical phonons in nondegenerate polar semiconductors is studied when several elastic scattering mechanisms are simultaneously

active also. The simple model of the band structure, and the displaced

Maxwellian distribution function are used. Analytical expressions for

drift mobility are given for B = and the quantum limit, and are compared

with the results of other techniques where possible. The numerical

calculations show that whenever magnetophonon oscillations are seen in the ohmic regime, maxima occur near the resonance magnetic fields for

all lattice temperature and strengths of scattering mechanisms which

compete with the optical phonon scattering. This is in contrast with

typical experimental findings, in which minima usually (but not always)

appear near the resonance fields for ohmic, longitudinal, conditions.

The explanation of the minima must therefore be sought in the use of a distribution function in which carrier-carrier scattering is less important.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Magnetophonon effect; magnetoresistance; semiconductors; transport theory.

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22. Price

$. 50

SCOMM-DC 66244-P71 GPO 84 1 - 47 1

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JOURNAL OF RESEARCH reports National Bureau of Standards research and development in physics, mathematics, chemistry, and engineering.

Comprehensive scientific papers give complete details of the work, including laboratory data, experimental

procedures, and theoretical and mathematical analyses. Illustrated with photographs, drawings, and

charts. Published in three sections, available separately: • Physics and Chemistry

Papers of interest primarily to scientists working in these fields. This section covers a broad range of physical and chemical research, with major emphasis on standards of physical measurement, fundamental

constants, and properties of matter. Issued six times

a vear. Annual subscription: Domestic, $9.50; for- eign, $11.75*. • Mathematical Sciences

Studies and compilations designed mainly for the mathematician and theoretical physicist. Topics in mathematical statistics, theory of experiment design, numerical analysis, theoretical physics and chemistry, logical design and programming of computers and computer systems. Short numerical tables. Issued quarterly. Annual subscription: Domestic,

$5.00; foreign, $6.25*. • Engineering and Instrumentation

Reporting results of interest chiefly to the engineer and the applied scientist. This section includes many

of the new developments in instrumentation resulting from the Bureau's work in physical measurement,

data processing, and development of test methods.

It will also cover some of the work in acoustics, applied mechanics, building research, and cryogenic

engineering. Issued quarterly. Annual subscription:

Domestic, $5.00; foreign, $6.25*.

TECHNICAL NEWS BULLETIN

The best single source of information concerning the Bureau's research, developmental, cooperative and

publication activities, this monthly publication is designed for the industry-oriented individual whose

daily work involves intimate contact with science and

technology

for engineers, chemists, physicists, re- search managers, product-development managers, and company executives. Annual subscription: Domestic,

$3.00; foreign, $4.00*.

• Difference in price is due to extra cost of foreign mailing.

Applied Mathematics Series. Mathematical tables, manuals, and studies. Building Science Series. Research results, test methods, and performance criteria of building materials, components, systems, and structures. Handbooks. Recommended codes of engineering and industrial practice (including safety codes) de- veloped in cooperation with interested industries, professional organizations, and regulatory bodies. Special Publications. Proceedings of NBS conferences, bibliographies, annual reports, wall charts, pamphlets, etc. Monographs. Major contributions to the technical

literature on various subjects related to the Bureau's

scientific and technical activities. National Standard Reference Data Series. NSRDS provides quantitative data on the physical and chemical properties of materials, compiled from

the world's literature and critically evaluated. Product Standards. Provide requirements for sizes, types, quality and methods for testing various industrial products. These standards are developed cooperatively with interested Government and industry groups and provide the basis for common understanding of product characteristics for both buyers and sellers. Their use is voluntary.

Technical Notes. This series consists of communications and reports (covering both other agency and

NBS-sponsored work) of limited or transitory interest. Federal Information Processing Standards Publications. This series is the official publication within

the Federal Government for information on standards adopted and promulgated under the Public Law

89-306, and Bureau of the Budget Circular A-86

entitled, Standardization of Data Elements and Codes

in Data Systems.

Consumer Information Series. Practical information, based on NBS research and experience, covering

areas of interest to the consumer. Easily understandable language and illustrations provide useful back- ground knowledge for shopping in today's technological marketplace.

NBS Special Publication 305, Supplement 1, Publications of the NBS, 1968-1969. When ordering, include Catalog No. C13. 10:305. Price $4.50;

foreign, $5.75.

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