Physica status solidi: Volume 26, Number 2 April 1 [Reprint 2021 ed.] 9783112496763, 9783112496756

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plxysica status solidi

VOLUME 26 . N U M B E R 2 • 1968

Classification Scherno 1. Structure of Solids 1.1 Alloys. Metallurgy 1.2 Solid-State Phase Transformations 1.3 Surfaces 1.4 Films 2. Non-Crystalline State 3. Crystallography 3.1 Crystal Growth 3.2 Interatomic Forces 4. Microstructure of Solids 5. Perfectly Periodic Structures 6. Lattice Mechanics. Phonons 6.1 Mossbauer Investigations 7. Acoustic Properties of Solids 8. Thermal Properties of Solids 9. Diffusion in Solids 10. Defect Properties of Solids (Irradiation Defects see 11) 10.1 Defect Properties of Metals 10.2 Photochemical Reactions. Colour Centres 11. Irradiation Effects in Solids 12. Mechanical Properties of Solids (Plastic Deformations see 10)see 10.1) 12.1 Mechanical Properties of Metals (Plastic Deformations 13. Electron States in Solids 13.1 Band Structure. Fermi Surfaces 13.2 Excitons 13.3 Surface States 13.4 Impurity and Defect States 14. Electrical Properties of Solids. Transport Phenomena 14.1 Metals. Conductors 14.2 Superconductivity. Superconducting Materials and Devices 14.3 Semiconductors 14.3.1 Semiconducting Films 14.3.2 Semiconducting Devices. Junctions (Contact Problems see 14.4.1) 14.4 Dielectrics 14.4.1 High Field Phenomena, Space Charge Effects, Inhomogeneities, Injected Carriers (Electroluminescence see 20.3; Junctions see 14.3.2) 14.4.2 Ferroelectric Materials and Phenomena 15. Thermoelectric and Thermomagnetic Properties of Solids 16. Photoconductivity. Photovoltaic Effects 17. Emission of Electrons and Ions from Solids 18. Magnetic Properties of Solids 18.1 Paramagnetic Properties 18.2 Ferromagnetic Properties 18.3 Ferrimagnetic Properties. Ferrites 18.4 Antiferromagnetic Properties (Continued on cover three)

physica status solidi B o a r d of E d i t o r s P. A I G R A I N , Paris, S. A M E L I N C K X , Mol-Donk, V. L. B O N C H - B R U E V I C H , Moskva, W. D E K E Y S E R , Gent, W. F R A N Z , Münster, P. GÖRLICH, Jena, E. G R I L L O T , Paris, R. K A I S C H E W , Sofia, P.T. L A N D S B E R G , Cardiff, L. N E E L , Grenoble, A. P I E K A R A , Warszawa, A. S E E G E R , Stuttgart, F. SEITZ, Urbana, 0 . S T A S I W , Berlin, M. S T E E N B E C K , Jena, F. STÖCKMANN, Karlsruhe, G. SZIGETI, Budapest, J . TAUC, Praha Editor-in-Chief P. GÖRLICH Advisory Board M. B A L K A N S K I , Paris, P. C. B A N B U R Y , Reading, M. B E R N A R D , Paris, W. B R A U E R , Berlin, W. COCHRAN, Edinburgh, R. COELHO, Fontenay-aux-Roses, H.-D. DIETZE, Saarbrücken, J.D. E S H E L B Y , Cambridge,P.P. F E 0 F I L 0 V,Leningrad, J . H O P F I E L D , Princeton, G. J A C 0 B S, Gent, J . J A U M A N N , Köln, E. K L I E R , Praha, E. K R O E N E R , Clausthal-Zellerfeld, R. KUBO, Tokyo, M. M A T Y A S , Praha, H. D. MEGAW, Cambridge, T. S. MOSS, Camberley, E. NAGY, Budapest, E. A. N I E K I S C H , Jülich, L. P A L , Budapest, M. RODOT, Bellevue/Seine, B. V. R O L L I N , Oxford, H . M . R O S E N B E R G , Oxford, R. Y A U T I E R , Bellevue/Seine

Volume 26 • Number 2 • Pages 383 to 770, K89 to K166, and A5 to A8 April 1, 1968

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Contents

Page

Original Papers R . F . EGOKOV, Β . I . R E S E R , a n d V . P . SHIROKOVSKII

Consistent Treatment of Symmetry in the Tight Binding Approximation

H . WAGENBLAST a n d S . A R A J S

Electrical Resistivity Studies of Iron-Nitrogen Solid Solutions . .

391 409

P . M . KARAGEORGY-ALKALAEV a n d A . Y i r . LEIDERMAN

Statistics of Inter-Impurity Recombination of Electrons and Holes in Semiconductors

419

A . SZYTULA, A . BUREWICZ, 2 . DIMITRIJEVIÖ, S . KRASKICKI, H . RZANY, J . TODOROVI)

1

P

(34)

into — (R + tv)p>

(0; (ft +

.

(35)

Using the obtained relations for expressing energy integrals as functions of the minimum number of independent parameters, repeating this procedure for every Qlp and every representation r} and F-y (where j j'), and using (28), we finally obtain the desired expression for the Hamiltonian's matrix components between two Bloch sums which include only independent parameters (energy integrals or certain of their combinations). 2.10 Calculation

procedure

a) Determine the point subgroup of the space group of the considered crystal. Find its irreducible representations (single-or double-valued) and set t h e m into the definite order. For the fixed set of initial functions it is sufficient to take those representations according to which these functions transform. Define the transformation of arbitrary vector under the operation of a from G0. b) After having classified atoms according to the coordinative spheres, choose the set of fixed vectors Qlp for every coordinative sphere (see Section 2.8). c) For every Qlp find the transformations a which do not change it and form t h e group Glp. Then find the generators for this group. d) For every Rn + t„ =j= Qp from a given coordinative sphere choose one of the transformations a which transform Rn + r„ into QlP (see (26)). e) Write down (29) for transformations defined in c). Solving the equations express all the energy integrals as functions of the minimum number of independent energy integrals. Use (17) and (19) for independent integrals. f) Compute the inner sums in (28): 2V"fc^Z>r>)I)f,,,(«), a

(36)

where a runs over the values determined by d). g) Substitute the results from e) and f) into (28) and obtain the final expressions for the matrix components. h) Determine other matrix components with the help of formulas of Section 2.6.

Notes: I t is efficient to choose the irreducible representations of the group G0 in such a way t h a t for appropriate a they would be reduced to irreducible representations for a maximum number of groups of the vectors Qlp. I t is considerably easier to point out the independent parameters (see Section 2.9) in this case. Actually, in such a case the functions cpl which are classified according to t h e irreducible representations of the group G0 simultaneously realize the irreducible representations of the groups Glp . Then q>l = (pi, where v is the row number of t h e i-th irreducible representation of Glp. According to (9) the energy integrals Jll'(Qp) are matrix components of the operator H T(Qlp) for the functions (pi and ïï = 2 / 3 / ? ; - 3 + ( T ) sin S cos f (sin 1

V

+ i cos i

rfy .

Similarly we obtain for case b) ^ = 1 and fi' = 2 : for v= I and v' = 1 : 2 ¿ « " D u t « * ) Dîa(«) =

0,

for v = 2 and v' = 2 : Ve"ir£>2Î( = 1 and r' = 2 : Z

ex

ë«™ D^i«)

DU*)

=

e« e

for v = 2 and r' = 1 : Z e , a f e T Dfiioi) £>U«)

—e

3

.2

= ~

e-«

tsri 3

' -

(cos |

3

e '

3

/ 3 sin I ) ] ,

_ 1 ^ (cos | + |/3 sin f ) J .

Then taking into consideration the relations between the energy integrals given above we have for the matrix component : < r 4 | ^ | r 6 > î l = 2 [i J Î i ( T ) ] s i n f

+

(cos y i ? - c o s f cos

sin £ cos -i- rj j +

i (sin y rj + cos £ sin y rj + (/3 sin f sin y rj j

.

The calculations for other cases are performed similarly. The results are listed in Tables 2 to 5. For brevity only a part of the matrix components is written. Other components may be obtained from them with the help of the relations given at the end of every table. Sign means that the corresponding matrix components have the same form and may be obtained from one another by changing the representation indices in the energy integrals (the function indices are conserved).

404

R. F . Egokov, B . I . R e s b r , and V. P. Shirokovskii

To simplify the notations we carry out the following: We omit the Hamiltonian H in the expression for the matrix components because our tables are suitable also for determining the orthogonality matrix, when we write 1 instead of H\ the indices of the sublattices are omitted (we know them from the table titles); we omit the representation indices in the energy integrals because we may easily recognize them from the knowledge of the matrix components; for one-dimensional representations we may even omit the function indices in the energy integrals. Finally, we separate real parameters everywhere. When the energy integrals are not real we establish the real combinations of them. They are written in square brackets, e.g. [i I 1 2 ( T ) ] , [ I n ( R ) + I h ( R ) ] , [i ( I U ( R ) - I h ( R ) ) ] , etc.

Table 2 Matrix components of the spin-independent Hamiltonian (« = 1 , 8 ' = 1) ( r t i r t )

1(0)

+

2

I(R)

(cos 2 s +

2 c o s | c o s tj)

( r t / r + )

2 i I(R)

(sin 2 £ — 2 sin £ cos tj)

( r t / r t l n

2

(cos 2 f -

cos | cos

( r t i r t ) u

2 i I12(R)

(sin 2 ( +

s i n f c o s tj)

(It/r$) ( r t i r f )

7(0) + 2 x l

( r t / r + )

n

I

n

(R)

7(B)

tj) —

2 ]/3 i 7 12 (it) cos | sin

+ 2 ^ 3

I

n

( R ) s i n £ s i n tj

I

n

( R ) s i n £ s i n r/

(cos 2 £ + 2 cos £ cos tj)'

2 i 7 n ( H ) (sin 2 £ +

s i n £ c o s tj) -

2

2

c o s £ c o s tj) +

2 ]/§ i IU(R)

I12{R)

tj

(cos 2 | —

( r t i r t ) n

I [7 U (0) + 7 22 (0)] +

( r f / r t ) i 2

2 i I12(R)

1U(R)

cos £ sin

n

(2 cos 2 £ + cos £ cos n ) + 3 7 2 2 (B) cos £ cos

(sin 2 | — 2 sin £ cos

tj) -

]/3 [7 n (K) - 7 2 2 (B)] sin £ sin

V V

(r3+//^)22 -f- [ J u ( 0 ) + 7 22 (0)] + 7 2 2 (R) (2 cos 2 | + cos £ cos tj) + 3 7 U ( R ) cos £ cos rj ( r j / T j i ) ¡¡p' ( r f i r j , ) ^ ' =

(r^jr^,)^!/ ( . r j i r f , w =

o

Consistent Treatment of Symmetry in the Tight Binding Approximation

405

Table 3 Matrix components of the spin-independent Hamiltonian (s = 1, = 2) (Ft/Ft)

2

(Ft I ID

-2

(ri/r^N

2

I(T)

cos £ [(cos 2/3 »7 + 2 cos | cos 1/3 Tj) + i (sin 2/3 rj - 2 cos f sin 1/3 »?)]

7(T)

sin £ [(sin 2/3

»7

-

7 n ( T ) cos £ [(cos 2/3 »7 -

2 cos f sin 1/3 rj) - i (cos 2/3

- 2 ]/3 7 U ( T ) cos £ sin | (sin 1/3 »7 + i cos 1/3 rj)

(Ft/rï)n

- 2 y ' 3 7 1 2 (T) sin £ sin £ (cos 1/3 rj -

(Ft/r^u

-2

(r+/rt)n

2

7 1 2 (T) cos £ sin I (sin 1/3 »? + i cos 1/3 rj)

(r£lF£)lt

2

/ J 2 ( T ) cos £ [(cos 2/3

(r+/rr)n

-2

(7^/r3-)12

cos £ sin 1/3 rj) -

i (cos 2/3 »7 -

cos £ cos 1/3 »7)]

cos f cos 1/3 rj) + i (sin 2/3 ij +

cos f sin 1/3 r/)]

cos | sin 1/3 rj) -

cos f cos 1/3 rj)]

2 )/3 7 n ( T ) sin £ sin £ (cos 1/3 rj -

12

i sin 1/3 rj)

(r+//V)u

( I f / r ^

i sin 1/3 rj)

cos £ cos 1/3 rj) + t (2 sin 2/3 »? -

-

7 1 2 (T) sin £ [(2 sin 2/3 »? 3

t sin 1/3 ,7)

cos f sin 1/3 rj) -

i (2 cos 2/3 »7 +

cos f cos 1/3 rj)]

-

cos | sin 1/3 r;) - i (2 cos 2/3 »7 +

cos | cos 1/3 »))]

-

7 2 1 (T) sin £ cos S (sin 1/3 »7 + i cos 1/3 rj)

-

7 2 1 (T) sin £ [(2 sin 2/3 »7 -

— 3

7 1 2 (T) sin £ cos f (sin 1/3 »7 + i cos 1/3 rj)

(rt/Fr)

=

(A+/A+) = (TV/TV) = ( r + / 7 Y ) = o

(7Y/7V)

~

(rt/rt)

(rr/r+)

~

(r+/7T)

(rrirthv

~

(r+/7Vh„

(JT/JY)~

( W h i

(A+/A-) 2 2

cos £ sin 1/3 »))] +

7 U ( T ) cos £ cos f (cos 1/3 »7 — i sin 1/3 rj)

V3 [7 1 2 (T) + 7 2 1 (T)] sin £ sin f (cos 1/3 »7 -

cos S sin 1/3 »7)] +

7 2 2 (T)] COS £ sin £ (sin 1/3 »7 + i cos 1/3 »7)

J 2 2 ( T ) COS £ [(2 cos 2/3 »7 + + 3

(rttrtki

/ 2 2 ( T ) COS C COS £ (cos 1/3 rj -

/ 3 [7n(T) -

(r+/rf)n

(rï/r+hr

i (cos 2/3 rj -

7 U ( T ) cos £ [(2 cos 2/3 r, + cos f cos 1/3 rj) + i (2 sin 2/3 »7 + 3

cos f sin 1/3 t])]

i sin 1/3 rj)

/ n ( T ) sin C [(sin 2/3 »7 +

(r+/rf)n

(rt/rt)

-

+ 2 cos | cos 1/3 r/)]

cos | cos 1/3 IJ) + i (sin 2/3 t] +

(r+/r+)12

7 1 2 (T) sin £ [(sin 2/3 rj +

»7

=-(rt/rT)

~ (rr/rr) ~

u

~

(r+/r3-)i^

ot/7T)m~

(r+/r3+>!„

( r j i r j w -

( r f / n w

(r+/r+)

406

R . F . EGOROV,

B . I. R E S E R , and V . P . SHIBOKOVSKII

Table 4 M a t r i x c o m p o n e n t s of t h e s p i n - d e p e n d e n t H a m i l t o n i a n (s = (A/A)n,

(r4/r4)22

i

I

[Jn(0) +

t/n(0) +

A'i(0)] +

[/U(K) +

+

[»-(/„(H) lIn(R)

AVO)] +

- 0' (A/A) u

-

[

+

[i (IU(R)

(A/A)22

[ +

( r j r * ) 11

Ait«) + -

-M«) +

R

)

(cos 2 f +

2 cos | cos rj)

/ « ( « ) ) ] (sin 2 f -

2 sin £ cos r/)

/*i(it)]

2 cos | cos

-

J

n(

R

) +

[* (Iu(R)

~

(cos 2 | +

/ i ' i ( K ) ) ] (sin

sin | cos

A \ ( R ) ) ] • []/3 sin f sin rj -

i (cos 2 f -

cos | cos JJ)]

A ' i ( R ) ] • [}/3 cos | sin jj +

i (sin 2 £ +

sin f cos ij)]

n

+

-

sin f cos »?)] +

+

i (cos 2 | — cos | cos ?j)]

A ' i W ] • [^3 cos | sin rj +

» (sin 2 f +

sin f cos »?)]

Iii(R))]

i (cos 2 f -

cos f cos j?)]

' [^3 sin £ sin rj +

+

77)

i (sin 2 f +

R

) ] ' t / 3 cos S sin

2| — 2

V)

-

r

n (

( h i (

+

1)

I'^R)]

[» ( ^ n ( R ) — -in(JR))] • [ y 3 sin Saint] [

-

1 , tf =

-

(r4/r6)22 - [ JU(K) + JTi(-K) ] • [/3 cos f sin r) - i (sin 2 f + sin f cos t?)] (A/A)u

[ +

(rjrt)

[i (In(R)

( / y / » ^ ^

In(R)

[t ( I [

2 i

— / i i ( « ) ) ] • []/3 sin f sin rj — i (cos 2 f — cos f cos ?})]

n

{

IU(R)

+ R

) +

I'i(H))] Ih(R)

] * [(cos 2 f -

cos £ cos r,) +

]/3 i sin £ sin rj]

• [(sin 2 f +

sin £ cos ry) +

]/3 i cos f sin 17]

] • [(cos 2 F — cos f cos rj) -

[* ( I n ( R ) — A \ ( « ) ) ] • [(sin 2 f + 0

for

n *

/i'

(W)#.#.' ~ ( A / A W ' ~ ( A / A W -

sin g cos »/) -

]/§ i sin | sin JJ] |/3~i cos f sin 17]

+

-

Consistent Treatment of Symmetry in the Tight Binding Approximation Table 5 Matrix components of the spin-dependent Hamiltonian (s = 1, S ' = 2) (A/A)n

2

I n ( T ) cos C [(cos 2/3 »? + 2 cos ! cos 1/3 »?) + + i (sin 2/3 îj — 2 cos £ sin 1/3 »?)]

(A/A)I2 - 2

J12(X) sin C [(sin 2/3 »? + cos £ sin 1/3 »? - ]/3 sin | sin 1/3 r,) — i (cos 2/3 »? — cos £ cos 1/3 »? + y3 sin £ cos 1/3 /?)]

(A/A)II

2 [i / U ( T ) ] cos £ [(sin 2/3 »? + cos £ sin 1/3 »? - |/3 sin £ sin 1/3 »?) — i (cos 2/3 »? — cos £ cos 1/3 )? + ]/3 sin £ cos 1/3 »?)!

(A/A)I2

2 [i 7 12 (T)] sin C [(cos 2/3 »? + 2 cos £ cos 1/3 »?) + + i (sin 2/3 »? - 2 cos £ sin 1/3 »?)]

(A/Aka - 2

[i 7 n ( T ) ] cos f [(sin 2/3 »? + cos £ sin 1/3 »? + j/3 sin £ sin 1/3 »?) — i (cos 2/3 »? — cos | cos 1/3 »? — j/3 sin £ cos 1/3 »?)]

(A/A)I2

2 [i I 1 2 (T)] sin C [(cos 2/3 »? - cos £ cos 1/3 »? - ]/3 sin £ cos 1/3 »?) + + i (sin 2/3 »? + cos £ sin 1/3 rj +

(A/A)2I

2 [i /J 2 (T)] sin £ [(cos 2/3 r? - cos £ cos 1/3 »? +

sin £ sin 1/3 »?)]

sin £ cos 1/3 »?) +

+ i (sin 2/3 »? + cos £ sin 1/3 1 7 - / 3 sin £ sin 1/3 »?)] (A/A)I2 - 2

J 12 (T) sin C [(sin 2/3 »? + cos £ sin 1/3 »? + ^3 sin £ sin 1/3 1?) — i (cos 2/3 »? — cos £ cos 1/3 »? — ]/3 sin £ cos 1/3 »?)]

( A / A ) 11

2

/ „ ( T ) cosC [(cos 2/3 »? - cos £cos 1/3 »? + ]/3 sin £ cos 1/3 » ? ) + + i (sin 2/3 r? + cos £ sin 1/3 »? — /3 sin £ sin 1/3 »?)]

(A/A>22

2

7 U ( T ) cos C [(cos 2/3 » ? - c o s £ cos 1/3 » ? - ] / § sin £ cos 1/3 »?) + + i (sin 2/3 »? + cos £ sin 1/3 rj + f 3 sin £ sin 1/3 »?)]

(A/A)12 ~

sin £ [(sin 2/3 rj - 2 cos £ sin 1/3 rj) -

2

(/YR,)22 =

(A/A)N ~

(A/A)N~ -

(A/A);2

(A/A)22 ~ -

(A/A)N

(A/A)2I =

(A/A)I2

(A/A)I2 ~

(A/A)ia

(A/A)2I ~ -

(A/A)I2

i (cos 2/3 »? + 2 cos £ cos 1/3 »?)] (A/A)„

407

408

R. P. EGOROV et al. : Consistent Treatment of Symmetry

4. Summary

Up to now, the expressions for the matrix components of the Hamiltonian in the tight binding approximation were obtained with the help of the concrete form of atomic wave functions. To determine relations between the matrix components of the Hamiltonian different symmetry transformations were used for different pairs of functions and different lattice sites. We could never know whether all possible relations were taken into account or only a part of them. For complicated crystal structures with more than two atoms in the unit cell the pictorial geometrical approach becomes practically unapplicable. In addition, such approach fails in principle when the functions depend on spin variables. The procedure suggested in our paper is free from such difficulties. The complete and consistent use of symmetry of the problem allows to reduce all calculations to simple algebraic operations. The number of independent matrix components may be pointed out a priori; so, there is a reliable control during the computations. The choice of irreducible representations of the point group of the crystal given in this paper provides the most convenient from of the matrix components. To find the final expression it is necessary to know only the representations of the point symmetry group (i.e. the transformation law for the functions but not the functions themselves). Hence, considering the matrix components for different representations we thereby determine completely the form of the energy matrix and may take into account any number of atomic states (s-, p-, . . ., etc), in our computations. Thus, the results are applicable for any material with a given crystal structure. The calculations, performed for the h.c.p. structure, are not only illustrative. We hope to use the formulas obtained for investigations of the electron energy bands in rare-earth metals. References [ 1 ] J . C. SLATEB a n d G . F . KOSTER, P h y s . R e v . 9 4 , 1 4 9 8 ( 1 9 5 4 ) . [ 2 ] F . SEITZ, A n n . M a t h . 3 7 , 1 7 ( 1 9 3 6 ) .

[3] E. P. WIGNEB, Group Theory, Academic Press, New York/London 1959. [ 4 ] M . MIASEK, P h y s . R e v . 1 0 7 , 9 2 ( 1 9 5 7 ) .

(Received

October 23, 1967)

H , WAGENBLAST

and S.

ARAJS:

Resistivity Studies of Solid Solutions

409

phys. stat. sol. 26, 409 (1968) Subject classification: 14.1; 21.1.1 Edgar O. Bain Laboratory for Fundamental Research, United States Steel Corporation Research Center, Monroeville, Pennsylvania

Electrical Resistivity Studies of Iron-Nitrogen Solid Solutions By H . WAGENBLAST a n d S. ARAJS1)

The electrical resistivity of iron-nitrogen solid solutions prepared from high purity iron has been studied as a function of applied longitudinal magnetic fields u p to 60 kOe a t 4.2 °K. I t is demonstrated t h a t the presence of interstitial nitrogen increases the resistivity of iron and alters the shape of resistivity vs. magnetic field curves, particularly at low fields. By extrapolating the data to B = 0, where B is the internal magnetic induction, i t has been possible to determine t h a t the residual resistivity contribution per 1 a t % nitrogen in solid solution in iron is about 6.1 (xflcm. At 78 °K the value is approximately 7.0 u i i c m . Such a large positive deviation from the Matthiessen rule is characteristic for iron alloys with another transition metal as a solute. Magnetoresistivity data of iron-nitrogen solid solutions (but not pure iron) very approximately obey the Kohler rule. Die elektrische Leitfähigkeit von aus hochreinem Bisen präparierten Eisen-StickstoffFestkörperlösungen wurde in Abhängigkeit von einem longitudinalen Magnetfeld bis 60 kOe bei 4,2 °K untersucht. Es wird gezeigt, daß Stickstoff auf Zwischengitterplatz den Widerstand von Eisen erhöht und die Form der Widerstands-Magnetfeldkurve besonders bei niedrigen Feldern, ändert. Durch Extrapolation der Werte zu B = 0, wobei B die innere magnetische Induktion bedeutet, war es möglich, den Restwiderstandsbeitrag pro 1 a t % Stickstoff in Festkörperlösung in Eisen zu etwa 6,1 (iilcm zu bestimmen. Bei 78 °K beträgt dieser Wert näherungsweise 7,0 [xflcm. Derartig hohe positive Abweichungen von der Matthiesschen Regel sind f ü r Eisenlegierungen mit einem anderen Übergangsmetall als gelösten Stoff charakteristisch. Die Werte der Magnetowiderstandsänderung von EisenStickstoff-Festkörperlösungen (nicht jedoch für reines Eisen) folgen nur sehr näherungsweise der Kohlerschen Regel.

1. Introduction Recently we have initiated extensive studies of the electrical resistivity of different binary iron-base alloys [1], These studies are being done pimarily at 4.2 °K, because at this temperature the electron-phonon scattering is almost negligible, and thus it is possible to explore the electron scattering from the different types of impurity atoms. Such information is not only of importance for correlating the role of impurities on different physical properties of iron, but also has value for providing a better scale for evaluating the over-all purity of iron. Although some scattered data on the electron-impurity atom electrical resistivity exist for alloys in which the impurity atoms are located substitutionally, such information concerning the effect of interstitial impurities is extremely limited [2]. Thus, we are not aware of any studies on the electrical resistivity of iron-nitrogen solid solutions at 4.2 °K. For this reason we decided to study this system experimentally, and the results of such an investigation are reported in this paper. l

) Now at Clarkson College of Technology, Potsdam, New York.

410

H . WAGENBLAST a n d S . ABAJS

2. Experimental Considerations Alloys involving an interstitial solute are usually more difficult to produce than those involving substitutional solutes, partially because of the generally lower solubility and higher diffusivity of the interstitial. Considerable care must be taken to assure homogeneous distribution of the interstitial solute and to prevent the formation of any precipitated second phases. The purity of the solvent material must also be considered, since frequently the substitutional impurity content is comparable to the maximum interstitial solubility. For this reason, we fabricated specimens from starting material of two different impurity levels: a very high purity iron (hereafter referred to as Fel) which had been used in a previous study [3], and a vacuum melted iron (Fell) whose substitutional impurity content was more comparable to the subsequent interstitial nitrogen contents. The stock of iron F e l is identical to t h a t of the F e l described in [3]. This iron, produced by Oliver and Troy [4], contains a maximum of 4 atomic ppm substitutional impurities and 11 atomic ppm oxygen whereas F e l l contains approximately 1000 atomic ppm substitutional impurities. The preparation of dilute interstitial alloys is frequently accomplished by transferring the interstitial element from a gaseous compound to the pure metal at elevated temperatures, in contrast to the usual method of melt additions in the preparation of substitutional alloys. With this method, there is more danger of producing an interstitial gradient within the specimen if, for example, the gas treating time is insufficient. Such gradients would result in decreasing average interstitial contents in similarly treated specimens of increasing diameter. As a check for such an effect we cold-swaged and drew the F e l material into wire specimens with diameters of 0.38, 0.48, 0.61, and 0.68 mm. F e l l was similarly fabricated into specimens of 0.61 mm diameter. Portions of the latter material were also cold-rolled into sheet 0.056 mm thick for the chemical and weight gain analyses described below. The wires were then straightened by inserting them into quartz tubes of 1 mm i.d. and annealing in vacuum at 720 °C for « 1 5 min. Analysis for the small interstitial contents place a limitation on acuracy in the study of alloys involving interstitial solutes. In this investigation we used three methods: weight gain; chemical analysis; and equilibrium calculations using measured gas analysis, temperature, and barometric pressure. Weight gain measurements were made by encircling the specimens during each gas treatment with a 10 cm long, 1.25 cm diameter iron tube of 0.056 mm wall thickness. The tube was first weighed in the denitrogenized, annealed state and again after each gas treatment, the difference yielding the absorbed nitrogen. The weight of the tube was « 20 g and the sensitivity of the balance « 0.00005 g so that the sensitivity of the method, in principle, was +0.0003 w t % (+0.001 at%) nitrogen. In practice, we found the method to be less accurate, probably because of slight contamination of the tube while handling, and possible weight loss during the high temperature treatments. Chemical analyses were obtained by a modified Kjeldahl method from « 4 g portions of 0.056 mm thick iron strips, which were also included in each gas treatment. This method and its accuracy has been discussed in detail by Wriedt and Gonzalez [5]. Finally, gas specimens were collected at the gas output of the furnace and analysed volumetrically. Using these results and knowing the temperature of the furnace and barometric pressure, the nitrogen content of the specimens was calculated

Electrical Resistivity Studies of Iron-Nitrogen Solid Solutions

411

using equilibrium data of Wriedt [6]. In summary, the nitrogen content deduced from weight gain and gas analysis was used as a check on the results of the chemical analysis which are reported in this investigation. In our case, the nitrogenization was accomplished in the following manner: the four Fel specimens of different diameter, with »1/2 cm potential leads of iron spot-welded » 4.3 cm apart, were mounted inside of the weighing tube along with chemical analysis strips and five F e l l specimens without potential leads. This assembly was suspended in a vertical tube furnace at room temperature and the furnace flushed with dry hydrogen to remove any residual air. The furnace temperature was then raised to «¿750 °C, and the atmosphere changed to wet hydrogen satuiated at room temperature (pressure ratio: PnJP]i t o ^ 30) for « 2 hours. The purpose of this treatment was to remove any residual carbon and nitrogen, and also to produce a larger grain size («¿0.4 mm diameter with some preferred texture of [110] parallel to the wire axis) than could be accomplished at the lower nitrogenizing temperature. This step was included only in the first run. The gas was then changed back to dry hydrogen and the temperature lowered to 470 °C. Nitrogenization was accomplished by introducing a predetermined flow of ammonia into the hydrogen stream for at least 40 hours. At no time was the ammonia content of the gas sufficient to produce a nitride precipitate by exceeding the nitrogen solubility at 470 °C (0.18 at%). The specimens were then quenched to retain the nitrogen in solid solution by a free-fall of » 7 0 cm into a 0 °C brine solution and immediately stored in liquid nitrogen until measurement. It has been established that this method is effective in preventing precipitation during the quench. Four specimens of different diameters were mounted in a sample holder used previously for studies of the longitudinal electrical magnetoresistivity of chromium alloys [7], This holder can be placed into the cylindrical section (5 cm inside diameter) of a superconducting solenoid capable of providing magnetic fields up to 60 kOe. All electrical resistivity measurements as a function of magnetic field, reported in this paper, were made at 4.2 °K. Since the sample holder containing the specimens was always dipped into liquid nitrogen before being placed in the solenoid cryogenic system, electrical resistivity data were also collected at this temperature. The resistivity of each sample was determined using the standard four-probe method. The iron potential leads, spot-welded to the specimens at the beginning of the investigation, were attached to copper potential wires in the following manner: sections of 3 mm diameter rosin core solder (Dutch Boy, Grade 688) about 1.5 mm long were washed in ethyl alcohol, dissolving the rosin and leaving a hole of » 1.5 mm diameter. The iron potential leads were clamped to the copper wires by placing the ends in the 1.5 mm hole and squeezing the whole section so that a good electrical connection was made. Because of the softness of the solder it was very easy to remove the sample potential leads from the copper potential wires after completion of the particular resistivity run. This technique resulted in less deposition of solder on the potential leads than conventional soldering, which could possibly have caused difficulties when the specimens were nitrogenized as described above. The current contacts were made by attaching the looped ends of the specimen to a brass block (to which current leads were soldered) using a small set screw. The electrical accessories necessary for the electrical resistivity measurements have been described elsewhere [7].

412

H . WAGENBLAST a n d S . ARAJS

Fig. 1. Electrical resistivity of iron a n d a n i r o n - n i t r o gen solid solution containing 0.012* a t % nitrogen a t 4.2 as a f u n c t i o n of applied longitudinal magnetic field. Source of i r o n : F e l

Fig. 2. Electrical resistivity of i r o n - n i t r o g e n solid solutions containing 0.057 2 a n d 0.10 4 a t % nitrogen a t 4.2 °K as a f u n c t i o n of applied longitudinal magnetic field. Source of i r o n : F e l

Fig. 3. Electrical resistivity of i r o n - n i t r o g e n solid solutions containing 0.13« a n d 0.15 8 a t % nitrogen a t 4.2 °K as a f u n c t i o n of applied longitudinal magnetic field. Source of i r o n : F e l

Electrica] Resistivity Studies of Iron-Nitrogen Solid Solutions

413

3. Results and Discussion The main objective of the present investigation was to determine the contri-

iron: one is for annealed (at 1025 °K) and furnace-cooled Fel, the other for quenched (from 998 °K) Fel. The resistivity values for the annealed iron are lower than those for the quenched iron, probably because the quenched material contains some residual carbon in solution. Fig. 1 shows t h a t the electrical resistivity of pure iron at 4.2 °K rapidly decreases with increasing applied magnetic field, reaches a minimum value at about 1 kOe, and then gradually increases with larger field values. Although the behavior of the magnetoresistivity below 1 kOe is definitely associated with changes in magnetic domain configurations, the detailed electron scattering mechanism is not fully understood. I t has been suggested that the decrease occurs because of the decrease in the domain wall concentration [8]. Another possibility has been proposed by Berger and de Vroomen [9], which is based on the observation t h a t the field experienced by the conduction electrons when Ha = 0 is the magnetic induction Β (22 kG for iron and iron-nitrogen alloys), and the assumption t h a t the transverse magnetoresistivity is larger than the longitudinal effect. I t may be seen from Fig. 1 to 3 that a low field minimum also exists in iron-nitrogen alloys containing 0.0123 and 0.0572 a t % nitrogen, but is absent in samples with higher nitrogen contents. I n these latter cases, the effect of an applied magnetic field is opposite to t h a t described above. The electrical resistivity has a minimum value at H& = 0, rapidly increases with increasing field, and above 1 kOe exhibits a more gradual increase with H&. These results indicate that the low field magnetization process in polycrystalline iron is strongly influenced by small amounts of dissolved nitrogen. I t would be of interest to study the magnetic behavior of single crystals of iron-nitrogen alloys and to observe the domain arrangements as a function of the applied magnetic field. As mentioned previously, the determination of the residual electrical resistivity of iron alloys is complicated by the domain and grain boundary effects. I t is difficult to eliminate magnetoresistive effects in a polycrystalline iron alloy sample since even the magnetic fields arising from the electrical currrent needed for the resistivity determination can influence the magnetic domain arrangement and hence the electrical resistivity of the sample. For this reason, the points associated with H& = 0 in Fig. 1 to 3 are to some extent uncertain. This is completely consistent with other results on pure iron described elsewhere [3], Because of these effects, it appears that a better residual resistivity determination, in a relative sense, can be made by applying a longitudinal magnetic field sufficient to saturate the sample and then determining its electrical resistivity. Since conduction electrons in a ferromagnetic substance are experienc-

414

H. W a g e n b l a s t and S. Abajs Table 1 Electrical resistivities of iron-nitrogen solid solutions at 4.2 °K (source of iron: F e l ) N

q ((xilcm)

concentration

(at%)

0 (annealed) 0 (quenched)

B

=

0

0.003 4 0.005 0 0.087 5 0.349, 0.636 5 0.825 8 1.05,

H

&

=

0

0.0072, 0.0090 6 0.1070 6 0.3614 5 0.(54270 0.8288 o 1.056 5

= 1 kOe 0.0039 0 0.0056 6 0.0991 6 0.3599j 0.6446 5 0.8326» 1.0628

ing the B field [10, 11], given by

B = H + 4

ti

M,

where H is the internal magnetic field and M the volume magnetization, in principle the best technique for the residual resistivity measurement is to extrapolate the resistivity data from the high field region to B = 0. This is not an unambiguous extrapolation since one can see from Fig. 1 to 3 that additions of nitrogen to iron cause considerable changes in the slopes of the q vs. H a curves where q is the measured electrical resistivity at 4.2 °K. Nevertheless, we have attempted to determine the electrical resistivity of iron and iron-nitrogen solid solutions at B = 0 by extrapolating from the high field region as shown in Fig. 1 to 3. These extrapolated values are denoted by g0 and are tabulated along with the q data at H& = 0 and H& = 1 kOe in Table 1. The residual electrical resistivity values, due to dissolved nitrogen in iron are plotted as a function of nitrogen content in Fig. 4. The points associated with 4.2 ° K were determined using the aforementioned extrapolated values, from which the value of the nitrogen-free, quenched iron (since the alloy samples were also quenched) was subtracted. From the curve of vs. N concentration at 4.2 ° K one can estimate that 1 a t % nitrogen causes about a 6.1 u.Qcm contribution to the electrical resistivity of iron at 4.2 °K. The upper solid curve in Fig. 4 represents the quantity pN at 78 °K as a function of N concentration using = 0 as the reference point. The slope of this curve is about 7.0 ¡xilcm per 1 a t % nitrogen, clearly indicating that the Mat-

0

0.02

M

0.06

008

0.10 QV Q14 0.16 0.« N concentration (at %)

F i g . 4. I n c r e a s e in the electrical resistivity of iron a t 4.2 ° K , B = 0 ; 78 ° K , fl* = 0 ; and 2 9 8 ° K , i f » = 0 due to dissolved nitrogen

Electrical Resistivity Studies of Iron-Nitrogen Solid Solutions

415

thiessen rule is not obeyed by iron-nitrogen solid solutions. The deviation, defined by AeN (78 °K) =

(78 °K) -

eN

(4.2 °K) ,

is positive and is about 0.9 [xQcm per 1 at% nitrogen at 78 °K. A deviation of this order of magnitude generally occurs in binary iron base alloys with transition metals as solutes [12]. Large positive deviations strongly dependent upon temperature at lower temperatures are expected according to Campbell et al. [12] when É > 0 + / É ? O - =)= 1 where is the residual electrical resistivity for spin-up electrons and ¡30_ that for spin-down electrons. This ratio is related to the detailed nature of the impurity shielding by electrons. In fact, when g0+/g0_ =)= 1 the shielding is predominantly by the 3d-electrons [13, 14], and one can expect different shielding for the two directions of spin. For alloys such as those with aluminum and silicon, where the shielding is primarily due to the s-electrons, the d-bands remain undisturbed [1'5], and g0+/(?o- í=s 1 implying much smaller deviations from the Matthiessen rule. The results on the electrical resistivity behavior of iron-nitrogen solid solutions described in this paper seem to imply that nitrogen dissolved in iron acts in such a manner as expected for a solute for which Qo+IQo- =1= 1- It would be of interest to study in detail the deviation ApN(T) as a function of temperature. I t may be remarked that the resistivity measurements on iron-nitrogen alloys containing up to 0.06 at% nitrogen by Dijkstra [16] at 298 °K give a line with a steeper slope («* 9.1 [lOcm per 1 at%) which lies above our 78 °K curve, in support of the above mentioned picture. This is shown as the dotted line in Fig. 4 which was estimated from Fig. 8 of [16] by assuming the electrical resistivity of the iron used by Dijkstra to be 10.2 [j.i2cm. The electrical resistivity curves as a function of Ha for the F e l l alloys (not shown) are similar to those of the Fel nitrogen alloys with 0.0123 and 0.0572 a t % nitrogen. However, plots of £>N vs. N concentration for this material at 4.2 and 78 °K have slopes of approximately 6.6 and 7.0 ¡j.Qcm/at% nitrogen, respectively, indicating that Matthiessen's rule is more nearly obeyed for ironnitrogen alloys if the substitutional impurity content is comparable to the interstitial content. We have also plotted vs. N concentration curves for iron-nitrogen alloys at 4.2 °K using the resistivity values at H a = 0 and = 1 kOe. For brevity no details are presented in this paper. There is considerably more scatter of points, particularly for the Ha = 0 data because of the difficulty in achieving a completely demagnetized state in the resistivity samples. Nevertheless, the conclusion is that essentially the same value of for 1 at% nitrogen as that for the B = 0 case is found. There is a significant difference between the resistivity curves as a function of Ha for nitrogen-free Fel and Fell. The F e l curves for both annealed and quenched samples are concave upward as can be seen from Fig. 1 while those for F e l l are concave downward. Such differences have been observed before [3] and may result, we believe, from slight differences in the textures of the polycrystalline wires of the two different purities. As surprising effect which is not fully understood at the present time is the ability of small amounts of nitrogen to change the curvature of the resistivity vs. H& curve of high purity iron (Fel). Since there are no texture changes, the observed behavior implies a considerable change in the electron relaxation time due to the presence of nitrogen as a solute 27

physica 26/2

416

H. Wagenblast and S. Arajs

in iron. It should be mentioned that the effect of nitrogen on the curvature is less pronounced in the case of F e l l . When a magnetic field is applied to a ferromagnetic metal, its magnetoresistivity is determined by two separate processes. First, there is the ordinary effect resulting from the B field action on the kinetic behavior of the conduction electrons. Second, because of the existence of ferromagnetic domains, there is also a specific contribution from this source at low magnetic field values. These ordinary effects can generally be analysed by testing the data with respect to the Kohler rule which states that if a) the electron collision can be described by means of a relaxation time, b) the rigid band approximation is valid and the volume of occupied electron states does not change appreciably in size over the range of temperature and purity considered, c) changes in the temperature and purity simply alter the relaxation time by the same factor, and d) no magnetic breakdown occurs, then eo

\e I

This equation is known as the Kohler rule for a ferromagnetic metal where the electrons are sensing the B field. The quantity Aq is simply where q is the electrical resistivity at, say, 4.2 ° K for different values of B, and q0 is the resistivity when B = 0.

02-

S

Q1-

0.050.03-

A

ao2-

0.01-

o Fell • aon3

•

/

0.005 20

0.0572

A 0.10.\ > V 0.13s • 0.15s

i

30

i i i ti i 50

200

atVoN.Fel

I l i l t 300 500 B/g0(k6jiQ- 1cm->}-

Fig. 5. The Kohler plot for iron-nitrogen alloys at 4.2 °K

Electrical Resistivity Studies of Iron-Nitrogen Solid Solutions

417

Fig. 6. The Kohler plot for high purity iron at 4.2 °K

Ehrlich et al. [17] measured the magnetoresistivity of polycrystalline nickel and some nickel alloys at low temperatures and found t h a t the d a t a for the alloys at 4.15 °K approximately follow the Kohler plot, b u t t h a t the data for pure nickel do not. As far as we know, the magnetoresistivity of iron base alloys has not been explored from this point of view. For this reason, we present the Kohler plots for some of our iron-nitrogen solid solutions in Fig. 5. In these plots we have used data above H & = 1 kOe, because B/gJkB/ia'cm1)only these values are due to the ordinary effect mentioned above. From this figure we can observe t h a t the iron-nitrogen alloys do not obey the Kohler rule exactly b u t t h a t the deviations are not very large. I t is interesting to note t h a t nitrogenfree pure iron (Fel) shows extremely large departures f r o m t h e Kohler plots of the iron-nitrogen alloys. This can be judged f r o m Fig. 6. Contrarily, t h e nitrogen-free less pure iron (Fell) fits reasonably well on t h e curves associated with iron-nitrogen solid solutions. I n this respect our results are similar t o those of Ehrlich et al. for nickel and nickel binary alloys. I t m a y be remarked t h a t the large deviations of the Kohler plot for F e l from those of the iron-nitrogen alloys is not due to the difficulty of determining o 0 for high purity iron samples. I n fact, it is not possible to find a reasonable value of Q0 such t h a t t h e pure iron data will satisfy the Kohler plot of the alloys. The detailed mechanism of this behavior is not understood at the present time. Acknowledgements

The authors are grateful to J . W . Conroy for his technical assistance with the experimental aspects of this study and to R . M. Fisher for stimulating discussions and review of this paper. I n addition, we wish to acknowledge t h e gas analysis by H. A. Hughes, the chemical analysis by C. Sharp, and texture determination by R . S. Cline. References [1] S. ARAJS, Electrical Resistivity of Binary Iron-Base Alloys at 4.2 °K, unpublished investigations. [2] F. PAWLEK and K. REICHEL, Metallwissenschaft und Technik 12, 1 (1958). [3] S. ARAJS, B. F. OLIVER, and J. T. MICHALAK, J. appl. Phys. 38, 1676 (1967). [4] B. F. OLIVER and E. W. TKOY, Oxidation Zone Refining of Iron, to be published in Rev. Metall.. [5] H. A. WRIEDT and O. D. GONZALEZ, Trans. AIME 221, 532 (1961). [6] H . A. WRIEDT, unpublished data. [7] S. ARAJS and WM. E. KATZENMEYER, J. Phys. Chem. Solids 28, 1459 (1967). [8] A. I. SUDOVTSOV and E. E. SEMENENKO, Soviet Phys. — J. exp. theor. Phys. 8, 211 (1959). 27«

418

H. WAGENBLAST and S. ARAJS : Resistivity Studies of Solid Solutions

[9] L. BERGER and A. R. DE VROOMEN, J . appl. Phys. 36, 2877 (1965). [10] J . SMIT, P h y s i c a 1 7 , 6 1 2 ( 1 9 5 1 ) .

[11] J . R. ANDERSON and A. V. GOLD, Phys. Rev. Letters 10, 227 (1963). [12] I . A . CAMPBELL, A . PERT, a n d A . R . POMEROY, P h i l . M a g . 1 5 , 977 ( 1 9 6 7 ) .

[13] J . FRIED EL, J . Phys. Radium 23, 693 (1962). [14] A. GOMES, J . Phys. Chem. Solids 27, 451 (1966). [15] N . P . MOTT, A d v . P h y s . 1 3 , 3 2 5 (1964).

[16] L. J . DIJKSTRA, Philips Res. Rep. 2, 357 (1947). [17] A . C. EHRLICH, R . HTTGTTENIST, a n d D . RIVIER, J . P h y s . C h e m . S o l i d s 2 8 , 2 5 3 ( 1 9 6 7 ) . (Received

November

27,

1967)

P . M. KABAGEORGY-ALKALAEV a n d A . Y U . LEIDERMAN : S t a t i s t i c s of R e c o m b i n a t i o n

419

phys. stat. sol. 26, 419 (1968) Subject classification: 13.4; 20.3; 22 Institute

of Technical

Physics,

Academy

of Sciences

of Uzbek SSR,

Tashkent

Statistics of Inter-Impurity Recombination of Electrons and Holes in Semiconductors By P . M . KABAGEOKGY-ALKALAEV a n d A . Y u .

LEIDEBMAN

Recombination statistics for semiconductors are derived taking into account transitions between impurities forming spatially localized recombination pairs. Expressions for the recombination rate and lifetime are given. The cases of weak and strong deviations from the thermodynamic equilibrium are studied.

IIocTopoeHa cTaracTHKa peKOMGHnauiin, ywrtiBaiomaH nepexonti MesKay npnMeCHMH, 06pa3yK>mHMH npOCTpaHCTBeHHO JI0KajIH30BaHHyj0 peKOMSHHaUHOHHylO napy. ITojiyqeHbi BbipaJKemiH HJIH CKOPOCTH peKOMCHHauHH H BpeMeH >KH3HH. PaccMOTpeH cJiyiaii Majiux HapymeHHtt TepMOflHHaMHHecKoro paBHOBecHH. Cjiynaii Sojibiimx ypoBHeft HHJKeKitim paccMOTpeH 6e3 yieTa H3M6H6HHH 3apH^a jioBymeK. 1. Introduction The modern theory of recombination [1] suggests the lack of immediate transitions between different and spatially isolated localized centres. The exchange of electrons between such centres is only possible if the carriers become free through intermediate transitions. Until recently this imagination principally agreed with the observed effects. Moreover, theoretical investigations of carrier recombination on various types of single- and multiplecharged traps in most cases were confirmed by the results of the experimental work. Investigations of photoconductivity in comparatively high-doped Ge at helium temperatures [2] succeeded in discovering the effect of inter-impurity recombination. This phenomen, however, which is related to the recombinative charge exchange between acceptor and donor impurities forming the so-called acceptor-donor pairs, is not compatible with the interdiction of immediate transitions of carriers between localized levels. The emissive character of this recombination enables one to include observations of luminescence for its investigation. At present, the radiative inter-impurity recombination, apart from Ge [2], has been found in Si [3, 4], GaAs [5], GaP [6], ZnS-phosphore [8] and SiC [9]. Aukerman and Millea [10] investigated the statistics of recombination through donor-acceptor pairs. This work suggests t h a t an acceptor-donor pair forms a single localized complex with a behavior similar to t h a t of a triple-charged trap of the Sah-Shockley type [11]. Therefore, it is not sure whether the model under examination has a direct connection, in the full sense of the word, with the inter-impurity recombination. However, it may be interesting to examine the statistics of recombination including immediate transitions between the imparities.

420

P . M . KARAGEORGY-ALKALAEV a n d A . YTJ. LEIDEKMAN

2. Formulation of the Problem The rate of recombination

under steady-state

conditions

Let us assume that inter-impurity recombination transitions take place between two localized levels E1 and E2 forming a spatially localized recombination pair (Fig. 1). The localized levels E1 and E2 are due to the introduction of two different impurities into a semiconductor. There is no need to concretize the type of these impurities. I n a particular case the recombination pair may be of the acceptor-donor type, however, it is also possible that they may be either of acceptor-acceptor or of donor-donor type. I t is assumed that the "activation energies" of the impurities forming the pairs, E(, — E1 and E2 — Ev, differ from those of the same impurities in an isolated state. Such non-coupled centres may act as independent traps (trapping or recombination levels). Their effect on the processes connected with impurity recombination is given in Section 5. Transition of carriers from level E1 to level E2, as can be seen from above, is in most cases of an emissive nature. The energy of photons emitted may be written as follows: hv = E%-

[(Ec -

Ex) + (E2 -

Ev)]

- E ^

+

A,

where Eg energy gap, Eph emitted photon energy, A = u)] . Comparing the factors A, B and C with one another it becomes obvious that the following condition is fulfilled at all times: B* > 4 A C .

422

P. M. K a r a g e o r g y - A l k a l a e v and A. Ytr. Leidekman

It is possible to derive an approximate expression for the rate of recombination 1 ) by augmenting this inequality, ancl we get JJ

J 2 N\cm C p 2 (p n wf) (p + p12) + c12 Nt [Cni (n + % 2 ) + c p 2 (p + 2>u)] C

=

Cni Cp2 (n +

from which we obtain the dependence of filling of impurity levels upon the density of free carriers and the parameters of centres, ^ cm c P 2 n (n + nn) (p + p12) + c12 Nt {[c n i (n + nn) + c p 2 p „ ] n + c p 2 n?} 11

12

~

(» + W11) {Cni c p 2 (» + m u ) (i> + i>i2) + c12 Nt [c n i {n + nn) + cp2 (P+Pu)]}

^

cm c p 2 p12 (n + n12) (p + pl2) + c,2 Nt {c n i n (p + y 12 ) + c p 2 p,2 (p + ff12)}

~

(i> + i>i2) {Cn 1 CP 2 (1 + »Ii) (p + Pn) + c12 iv t [c n l (w + n12) + c p 2 (i> + ;p u )]}

(H)

'

(12)

When the density of recombination pairs Nt is comparatively small, i.e. Cni c P 2 (n + nn) (p + p12) > Nt c12 c ni « ( l + ^

+ Cp.to +

fc)

(13)

the filling rate of levels E1 and E2 by electrons and the rate of recombination are determined by the simpler expressions ft2 « p12l(p + p12) ,

ftl & n\(n + nu) , U = c12 N? (pn-

«?)/(» + r^) (p + p12) .

(14) (15)

It is interesting to examine the dependence of the recombination rate U on the filling rate of the traps: a) If the bottom level is filled with holes (p12 p) and the top one by electrons (m12 n) then the recombination rate V = cl2 remains independent of the free carrier densities. This means that the recombination radiation intensity does not depend upon the current. b) If both levels are considerably filled with carriers of one type then U is linear dependent on the density of free carriers of the other type; if p12 p, n, then U «

c12

njnn

,

and when p12 p, n nn, then U ¡=s; c12 p/p12. In these cases the intensity of recombination radiation is proportional to j or \/j. c) The law of recombination becomes bimolecular u ^ ^ N t l ^ L ,

(16)

"•11 Pl2

if the bottom level E2 is filled with electrons (p12 p) and the top one E1 with holes (nn '¡¡¡> n). In this case various dependences radiation intensities on the current exist: L ~ j, L j3l2, L ~ j2. In the case of a radiative transition between the levels E1 and E2 it is possible to determine the relative radiation intensity L(j) in dependence on the recombination rate U. If a current is injected into the semiconductor, the dependence n(j) and p(j), as a rule, is of a linear type. However, it is quite possible that at high injection levels n, p j/j (e.g. [12]).

Statistics of Inter-Impurity Recombination of Electrons and Holes

423

3. Small Disturbances of Thermodynamic Equilibrium Let us deal with the limiting case of small disturbances of thermodynamic equilibrium: |Aw| = |n — n0\ < w 0 , |Ap| = \p — p0\ < ^ p 0 a n d |A/t| = |/t — The neutrality condition of a semiconductor derived from Poisson's equation may be expressed as follows: A/tl -

Ap-An-Nt-

=

Nt • A/„

(17)

0 .

If in (11) and (12) A/ t l and Af t 2 are interpreted as functions / ( U , n) and / ( U , p) respectively and using the neutrality condition (17) we can derive from (14) expressions for the lifetime of non-equilibrium electrons and holes, r n = A n j U and'rp —

ApjU, Tp 0 K +

+ *ad

P11

+

+ Tn 0 (Po + P12) j 1 + -^t .(Po+Pn)2

P12 (P0+P12V

(n„ + nn) (p0+p12) ; Pa \ Nt. I 1 + ^t (P0+P12)7 n0+Po +

Ntp„

(P0+Ä2)2.

[ ( A + P11)2

Tn0 (Po + Pn) + t p 0 (»0 + »12) j l

. K + %1)2 »11 j l + Nt K + %i)2

(»0 + nu) (P0+P12)

+ Po - Art»o

(»0

(18)

Pi 2

Pi 1

: +

(m0 + w12):1 (19)

K + JI12) . 2

where T n 0 = [c n i -ZVt]-1. t po = [c p2 ATt]~\ r a d = [c12 iVJ" 1 . In general r n 4= r p . However, if the density of recombination centres Nt is comparatively small (Nt smaller than the least value of n0, p0, n12, pn) the charge variation on them need not be taken into account, and this automatically leads to T n — Tp — T0 — Tpo

n0 + n n n0 + Po

„

TnO

Po + Pu ; »0 + 2>o

, „

r ^ad

(»0 + %i) (Po + Pia) N t (»0 + Po)

(20)

The sum of two terms in (20) is similar with the expression derived by Shockley and Read [1] for the lifetime at low injection levels. However, the values w12 and pn are related to different impurity levels. In the following limiting case when density of recombination centres is great (N t -> 00) we derive from (18) and (19) r n = T n 0 (1 + Pu/Po) > Tp = Tpo (1 +

nnlno)

(21)

which likewise agrees with result [1], Analysis of the dependence of r 0 , Tn and r p upon the position of the Fermi level F0 in the band gap necessitates the clarification of some of the properties of impurity levels that form a pair. Let us assume that the level El is located in the upper whereas the level E2 in the lower half of the band gap and, furthermore, the activation energy of the first level to be less than that of the second one, Ec — E1 < E3 — Ev

.

424

P . M . KABAGEORGY-ALKALAEV a n d A . Ytr. LEIDERMAN

Therefore it has to be accepted that the characteristic densities and energies go conform with the inequalities n, > n12 > pn , E1 > 2 E, - E2 > E, > E2 > 2 E, - E1, (22) where Et = 1/2 [E c + + fcT In (iVv/iV"c)]. Now it is possible to investigate the lifetime variation in accordance with the estimations given in (22) in dependence on F0 (i.e. n0) variations: 1) In the n-type case, when n0 > nn and p0 < pu, n

n

>

p12

>

Pn

TpO + Tno

ad

T

rpo n0 + Tno Pn ( l + \

Pj3 N t

2(l + *ad-°|iN

Pul

\

(23)

'

t

Pn

\

+

(24)

Pi*}

n

n /

Tn

(25)

It is obvious, when r n 0 and r po p o represent values of the same order, Xp

n

aPii

~ Tp 0 -f- Ta d "

(26)

J J

2) In a weak w-type specimen sufficiently far from intrinsic conductivity with M] > t i j ^ i i w e obtain _

,

Po

,

rpo + T n o - +

r

r

n n a d

TpO n0 + ^nOPo

(27)

Pn

^-,

Ei Po0 \Pl2 n9

+

N t

PoU { * L \Pl2

+

N

t

\

Pi« I

(28)

Nt

\

rhlj

(29)

? * ) Po I

N.

VnOPo + Tp0 »0

\%j

n0lj

\»u no I 3) In the case of weak p-type conductivity, when nx < p0 < p12, T

p0

Po

h T n0 + T ad N

TpO n0 + Tn0 Po

(30)

NtPo' t

(Pn

Po

\ Po

, Po

\Po

Tn0?>0 + Tp0

1 +

»o Wo

Po +

Ni

N

PiJi

Pi

\ no

M

+

n

\

Nt

\

Pn!

(31)

J

%i/| l*h2

t

nl

nn/

(32)

425

Statistics of Inter-Impurity Recombination of Electrons and Holes

4) For a strong p-type semiconductor with p0 > T0 «

Tpo

Po

b Tn0 +

pi2 (33)

Tad T T > -"t

(34)

Tnofo +

TpO » 1 2 ( l

\

+

»12 /

+

Tad

A

t

(l

\

+

»11 /

(35)

Thus, it can be seen that the lifetime of minority carriers for a welldefined type of conductivity (cases 1) and 4)) does not depend on the charge of the centres irrespective of their density. In order to reduce the effect of centre charge variations on the lifetime in a material with weakn- or p-type conductivity (cases 2) and 3)) it is necessary to impose additional restrictions on the parameters of the recombination pair. These specific conditions shall not be considered here. A graphical representation of the dependence r0(F0) (and of its item components) is given in Fig. 2 which has been obtained analysing (20). The items composing r 0 correspond to the successive steps of the recombination transitions of electrons. I t is to remark that the lifetime defining step is that which is the slowest under the given conditions, whilst fast steps only yield a poor contribution to the lifetime r. When the density of recombination pairs Nt is great, the inter-impurity transition represents the "fast"-type stage and the third item in (20) corresponding to it has a negligible effect on r 0 . A detailed analysis in the case of strong n- or p-type conductivity shows the first and the second items in (20) to possess negligible maxima. The third item in (20), i.e. the time defined by inter-impurity transitions possesses a welldefined maximum at intrinsic conductivity. I t agrees with the maximum of the total lifetime To max

^

1

^Tno +

Tpo

+

Fig. 2. Dependence of the lifetime upon the Fermi level position (formula (20)). The continuous curve corresponds to r 0 , the dotted curves describe its component items. Expressions in brackets describe separate sections of the curves

»11 Pl2

Ev SErE} Ej

(36)

E, 2ErEt, Ef

F0

Ec —

426

P . M . KARAGEORGY-ALKALAEV a n d A . YTJ. L E I D E R M A N

When the activation energy of the levels complies with a inequality inverse to (22), for the characteristic densities we get Pía >

u > «i > Pn > %2 •

(37)

n

The positions of the impurity levels E1 and E2 will only alter when n^ > n0 > > p12 and nn > p0 > p12. Emphasis should be made that inequalities (22) and (37) do not exhaust all possible positions of E1 and E2 which, apparently, also could be located on the same side above or below the level E{. As is evident from formulas (23), (27), (30), and (33) the inter-impurity lifetime components increases with increasing temperature, because nu and p12, grow exponentially with temperature. This is due to thermal quenching of the interimpurity radiative recombination observed in [13]. I n a well-defined n-type semiconductor the activation energy of the thermal quenching effect coincides with the activation energy of the level E2 whereas in a well-defined p-type semiconductor with the activation energy of the level Ex. When the type of the conductivity is weaker accentuated the activation energy of the quenching effect is defined by the sum of the activation energies of the pair-forming impurity levels. Thermal quenching in an intrinsic semiconductor, according to (36), is only possible when E1 — E2 < 1/2 (Ec — Ev). Otherwise a thermal stimulation of the inter-impurity recombination transition will take place. 4. Strong Disturbances of Thermodynamic Equilibrium Let us consider the case (ignoring charge variation of recombination pairs, i.e. An = Ap) of density variations of the free carriers comparable with the densities of equilibrium carriers or exceeding them. Then, the dependence of the lifetime upon the injection level is easily obtained from (10), T = T

®

1 + a An + b(Anf

'

(38)

where a = c n i c p 2 (n0 + p0 + % + p12) d,

6 = c„icp2d,

c = {n0 + Po)" 1 ,

d = { c n l c p 2 (n0 + nn) (p0 + p12) + c12 Nt [ c n l (n0 + n^) + c p 2 (p0 + Pu)]}"1

.

At low injection levels, when a Aw < 1, c An < 1 , 6 An < a, r = r 0 is given by formula (20). The following cases appear with increasing injection level: 1) c An > 1, a An < 1 , 6 An < a, then r = r 0 (n0 + p0)jAn, the lifetime decreases with increasing injection level. 2) c An > 1, a An > 1, b An < a. In this case x m r0 ajc = const and saturation of r = T(An) occurs. 3) c An > 1, a An > 1 , 6 An > a. This is the range of very large injection levels, if r r 0 (6/c) An, the lifetime increases with increasing injection level. I t is to note that Shockley-Read statistics [1] discloses such a growth of lifetime to be only possible at low injection levels. The recombination rate attains a saturation value t/ l i m = r a ( j Nt. Thus, Uy,m is defined by the transition probability between the impurities and linearly grows with increasing density of pairs.

Statistica of Inter-Impurity Recombination of Electrons and Holes

427

When r An, and the inter-impurity recombination is regarded as the major process for a current flow, then in a field approximation (for example refer to [14]) the electrical field intensity in a semiconductor specimen remains independent from current "j". Thus, the growth of lifetime with the injection level defines the "break-down" character of the dependence of the current upon the voltage. 5. Inter-Impurity Recombination in Presence of Independent Traps In such a case recombination takes place along two parallel channels: a) Inter-impurity recombination at the rate £/ ad , b) Common recombination across traps (described by Shockley and Read [1]) at the rate UT. The total recombination rate is m U=Uai+UT^ (n p (39) r-e}, M + B N, where M = (n + n12) (p + p12) , R = (Cni Cpa)"1 [c„i (n + n12) + c p 2 (p + 2>i2)] , Q = [Cpi (n + Wir) + C"1 (p + Plr)]" 1 • Let us consider the problem of the dependence of the inter-impurity luminescence intensity upon the density of one of the pair-forming impurities and assume that the excitation intensity is constant (U — const). The charge of the impurities may be negligible. Specially take the concentration N t of the levels with activation energy El equal to const, whereas the concentration N2 of the levels with energy E2 can be changed. If N2 = 0, £7ad = 0, U = Ut, the whole recombination traffic passes through the traps Now introducing impurities of the second type pairs are formed. Their number equals to N2 as long as N2 < A7j. Regular Shockley-Read recombination Ut only takes place through (Nt — N2) traps contributing partly to the whole recombination rate U. The remaining part refers to the inter-impurity recombination m *7ad= U M + R N + c

Q(NiV

2

m M + B N,

(40)

The number of pairs grows with rising N2, UT is decreasing whereas f / a d increases until N2 equals to Nt. Then Vr = 0, U = J/ a d , the recombination process goes over the inter-impurity channel. Further growth of N2 again initiates appearance of non-coupled impurities with concentration N2 — jVj and makes the portion of inter-impurity recombination to V decrease. Now U&d

=

N\ M + RN\

U N\ M + BN f

(41) Q(N2-

jyj

I t should be emphasized that the recombination radiation intensity connected with the inter-impurity recombination L ~ E/a d grows with the increasing N2, attains its maximum at N2 = N1 and decreases with further growth N2.

4 2 8 P . M . KABAGEOBGY-ALKALAEV

and

A . Y U . LEIDERMAN

: Statistics of Recombination

Acknowledgements

The authors are much indebted to A. E. Yunovich and I. V. Ryzhikov for their most useful discussions. References and W . T . R E A D , J B . , Phys. Rev. 8 7 , 8 3 5 ( 1 9 5 2 ) . and S. M. R Y V K I N , Fiz. tyerd. Tela 6 , 1203 (1964). S H . M . K O G A N , T . M . L I E S H I T Z , and V . I . SIDOBOY, Physics of Semiconductors, Proc. Seventh Internat. Conf., Paris 1964, Academic Press 1964. A. Horn and R. E N C K , Congr. internat. physique des semiconducteurs, Jouve, Paris

[ 1 ] W . SHOCKLEY

[2] V. P. [3] [4]

DOBREGO

1964.

[5] [6] [7] [8] [9] [10] [11] [12]

M. F. M I L L E A and L. W. A U K E R M A N , J . appl. Phys. 37, 1788 (1966). D. G . T H O M A S , M. G E R S H E N Z O N , and J . J . H O P F I E L D , Phys. Rev. 131, 2397 (1963). D. G . T H O M A S , M. G E R S H E N Z O N , and F. A. T B U M B O R E , Phys. Rev. 133, A 269 (1964). F. E. W I L L I A M S , J . Phys. Chem. Solids 12, 265 (1960). I. V. R Y Z H I K O V , I I . Conf. Electroluminescence, Dnepropetrovsk 1967. L. W. A U K E R M A N and M . F. M I L L E A , Phys. Rev. 148, 759 (1966). C H . T. S A H and W. S H O C K L E Y , Phys. Rev. 109, 1103 (1958). A . YTR. L E I D E B M A N and P . M . K A B A G E O R G Y - A L K A L A E V , Radiotekhnika i Electronika 10, 720 (1965).

[13] K . MAEDA,

J.

[ 1 4 ] M . A . LAMPERT

Phys. Chem. Solids 2 6 , 5 9 5 ( 1 9 6 5 ) . and A . R O S E , Phys. Rev. 1 2 1 , 2 6 ( 1 9 6 1 ) . (Received January 2, 1968)

A.

SZYTULA

429

et al. : Neutron Diffraction Studies of a-FeOOH phys. stat. sol. 26, 429 (1968) Subject classification: 4; 18.3; 18.4; 22.8

Institute of Nuclear Physics, Cracow (a), Laboratory of Structure Research of the Jagiellonian University, Cracow (b), Institute of Nuclear Sciences "Boris Kidrii", Vinia (c), and Department of Inorganic Chemistry of the Poznan University (d)

Neutron Diffraction Studies of «-FeOOH By A.

SZYTUIA

( b ) , A . BTJREWICZ ( d ) ,

S. KRASNICKI

(a), H .

A. WANIC

RZANY

(a), a n d

DIMITRIJEVK!!

(a), J . TODOROVI6

W . WOLSKI

(C),

(C),

(d)

Neutron spectrometric investigations are made of goethite (a-FeOOH) of mineralogical and synthetic origin (a-FeOOH of natural isotopic composition) and also of a-FeOOD using a monochromatic neutron beam of A = 1.12 A. Some powder diffraction patterns in the temperature range from —190 to + 1 2 0 °C are measured. The obtained coordinates for all atoms (including those for hydrogen) in the unit cell are given. The existence of antiferromagnetic spin ordering at temperatures below + 8 0 °C is observed. A model is proposed in which the spins are aligned parallel to the 6-axis. E M J I HccjiejioBaH reTMT (a-FeOOH) MimepaJiorimecKoro riponcxomHCHHH H C H H TeTHHecKHft (ecTecTBeHHoro iisoTonimecKoro cocTaBa) a TaKJKe AeitTepH30BaHHbiM reTHT a-FeOOD. liccjienoBaiiHH Bejiwct B BHHia (lOrocjiaBHfl) npH peaKTope P A . ¿JjIH H3MepeHHft HCn0JI30BaH0 KpaKOBCKHtt IICHTpOHHblii KpHCTajUIHHeCKHH CneKTpoMeTp H nynoK HeilTpoHOB c HJIHHOH BOJIHLI A = 1,12 A. TeMnepaTypa o6pa3iia H3MeHHJiaci> B npeaejrax OT —190 no +120 °C. yToiHeHo KOopminaTbi ATOMOB Htejie3a H KHCJiopona B 3JieMeHTapHoit HiettKe, a T A W W E B nepBbitt pa3 6 M J I O onpejjejreHO nojiOHteHiie aTOMOB Bonopona. B H J I O 06Hapy>KeH0 cymecTBOBaHHe aHTH$eppoMarHHTHoro ynopHHoneirHH cnHHOB H O H O B >«ejie3a B TeMnepaTypax HHJKC + 80 °G. IlpHBOHHTCH MOHejI enHHOBOft pemeTKH B KOTOpoii enHHH napajiejIbHBI OCH b.

1. Introduction a-FeOOH (goethite), long considered a paramagnetic substance, is one of the polymorphic modifications of the hydrated ferric oxide (FeOOH). The antiferromagnetism of goethite was established quite recently when Hrynkiewicz and Kulgawczuk [1, 2] discovered the Zeeman splitting of its Mossbauer spectrum. This stimulated the interest of various investigators and a number of papers appeared [3, 4, 5, 6, 7, 8]. They revealed the existence of two forms of goethite: the A-goethite possessing two internal magnetic fields, and B-goethite having one internal magnetic field, acting at the iron nuclei. The most intriguing property of A-goethite is the appearance of two antiferromagnetic critical points, and T 1 ^, manifesting themselves in the temperature variation of the internal fields. One of the fields vanishes at = 340 °K and the other at T = 370 °K. The rontgenographic investigations hitherto performed, neither give the positions of hydrogen atoms nor throw any light on the problem of magnetic structure. I t was hoped that part of the lacking information could be obtained by applying the neutronographic method.

430

A. SzYTUtA et al.

2. Samples and Auxiliary Measurements Unfortunately, suitable single crystals of goethite are unavailable and powder samples had to be used to the disadvantage of accuracy. At our disposal were: 1. 2. 3. 4.

synthetic A-goethite of natural isotopic composition, synthetic deuterized A-goethite (a-FeOOD), A-goethite of mineralogical origin, B-goethite of mineralogical origin.

Samples 1 and 2 were prepared by the method described in papers [21] and [22], X-ray diffraction patterns of the samples did not show any significant departures from the data tabulated in [9]. The measurements of the Mossbauer spectra at room temperature gave the following intensities of internal fields in the samples: No. No. No. No.

1 2 3 4

= 335 - H 1 = 355 - H 1 = 355 - H = 357

± ± + +

15 kOe, H 2 = 288 + 15 kOe; 15 kOe, # 2 = 301 ± 15 kOe; 15 kOe, H 2 = 302 + 15 kOe; 15 kOe.

These may be considered as a proof of the structural identity of samples 1, 2, and 3. Sample 1 contained about 3.5 mass% of water in excess of the stoichiometric composition. The temperature dependence of magnetic susceptibility was measured [10] in the range 77 to 600 °K for samples 1 and 3. The single ¿-point, at T = 330 °K, was found in sample 1. This corresponds roughly to i) = 340 °K, the lower Neel point obtained by the Mossbauer method [2], Samples 1 and 2 were in a state of very fine pulverization. The average cubic size of grains, measured by the BET method, was equal to 105 A [10]. Samples 3 and 4 were different parts of a single mineral lump. Sample 3 was prepared (pulverized in a mortar) from black-coloured fragments and sample 4 from brown-coloured ones. The diameter of grains was about 104 to 105 A. 3. Crystallochemical Structure According to the results of X-ray diffraction studies [11, 12, 13] goethite belongs to the D\% space group [14]. There are four "molecules" in the orthorhombic unit cell and the atoms are believed to be located at two planes, y = -j and y = normal to the 6-axis. The coordinates of Fe and 0 atoms have been found, but the data of different authors are not quite consistent. The general positions in the group D\% are: (x, z), (x, z), + x, -j, — z), (t -

i> i + *)• Table 1 Lattice constants of goethite Reference Goldsztaub [11] Böhm [18] Peacock [13] Strunz [14] This paper

«0 (A) 10.00 10.01 9.937 10.02 9.95

' MA)

e» (A)

3.03 3.04 3.015 3.04 3.01

4.64 4.60 4.587 4.64 4.62

431

Neutron Diffraction Studies of a-FeOOH F i g . 1. Neutron diffraction p a t t e r n s of a - F e O O D

In order to find the nuclear (atomic) coordinates neutronographically the neutron diffraction patterns at 393 °K were measured. The measurements were carried out at Vinca on the Cracow Neutron Spectrometer [15], using X = 1.124 A, in the angular range of 2 6 = 6 to 68°. In Table 1 the obtained unit cell dimensions are compared with X-ray data. For structure calculations only the "peaks" up to 2 d = 47° were used because of the excessive overlapping of lines at higher angles. The calculations were made by an "Odra 1004" electronic computer. The set of coordinates corresponding to the minimum value of the agreement factor R was found, where

Scattering angle 28

E\I0(hkl) hkl

T> ~

Ic{hkl)|

S hkl

Ie(h

—»-

k I)

'

I 0 (h 1c I) observed relative intensity, I J h k I) calculated relative intensity. These are compared (see Table 2) with the coordinates obtained rontgenographically by Hoppe [12] and the ones obtained neutronographically by Busing et al. [19] for the isostructural compound AlOOH. The ii-values given for samples 2 and 3 were calculated for the same set of coordinates. Table 3 gives the relative intensities of the "peaks" observed and calculated on the basis of the obtained coordinates. As can be seen in Fig. 1, the (101) peak could not be isolated from the background, probably because of the remnants of the magnetic scattering. The application of a higher temperature was not recommended because of the possible decomposition of the sample. The diffraction patterns of samples 3 and 4 showed no significant differences. Table 2 Atomic coordinates

X

Hoppe

y z

This paper

X

y z 28

physica 26/2

Fe

Oi

o2

+0.146 +0.250 -0.045

-0.200 +0.250 +0.310

-0.047 +0.250 -0.200

—

+0.145 +0.250 -0.045

-0.199 +0.250 +0.288

-0.053 +0.250 -0.198

-0.08 +0.25 -0.38

H

_ —

432

A. SzYTULA et al. Table 3 Observed and calculated neutron diffraction intensities a-PeOOH

a-FeOOD

hkl

I 0 for sample 3 I 4

100 200 001 101 201 300 010 110 301 210

011

400,111 002 211, 102, 202 311 410, 302 501, 112, 212, 402, 312 511, 601 103, 020, 412, 203, 220, 021,

B

1.000 401

411 600 502 610 121

0 1.280 0.004 0.166 0.325 0.775

I L

J J

j-

0.083 0.185 0.527 0.508

0 0.434 0 2.749 0.527 0 0 0 0.475 0.349 0 1.000 0.015 1.599 0.018 0.008 0.071 1.488 2.591 0.149 1.334 1.041

0 0 0 0 0 0 0 0 0.183

0 0.008 0 0.082 0.020 0 0 0 0.170

l.ooo 0 1.255 0 0.159 0.203 0.758 0.062 0.203 0.528

|

0.490

j.

1.485

4.8%

0 0.466 0 2.740 0.315 0 0 0 0.411 0.342 0 1.000 0 1.781 0 0 0 1.229 2.589 0.178 1.042 1.009 |

1.454 8%

0 0.612 0 2.653 0.408 0 0 0 0.476 0.350 0 1.000 0 1.837 0.016 0 0.072 1.408 2.449 0.144 1.122 0.939 |

1.753 9%

4. Crysiallomagnetic Structure Neutron diffraction patterns taken at 83 °K did not show any. superlattice lines but the intensities of certain peaks were markedly changed in respect to the R T pattern (see Fig. 1). This means that the magnitudes of the chemical and magnetic unit cells are the same. The temperature variation of the intensities, especially for the reflections (200) and (101) of a-FeOOD, proves the existence of antiferromagnetic spin ordering below the Neel point T N = = 362 + 5 °K. Turov [16] lists four possible spin ordering modes in a system containing four paramagnetic ions in a crystal of the D\\ group: one ferromagnetic and the other antiferromagnetic. For these three AFM modes the magnetic structure factors were calculated and compared (see Table 4) with the observed ones derived from the differences of the two neutronographic patterns taken at 83 and 393 °K respectively. The magnetic scattering form factor of trivalent iron ions [17] was used in the calculations, such a valency being confirmed by the quadrupole splitting of Mossbauer spectra of the samples. Only the A1-mode gives satisfactory agree-

433

Neutron Diffraction Studies of a-FeOOH Fig. 2. The spin arrangement [in the unit cell of goethite

S3» Ç^Oxygen

(^)Iron

°Hydrogen

Table 4 Observed and calculated magnetic structure factors of a-FeOOD

2 • t f l f (calculated) hkl

100

Pattern

+- + 0

Pattern A 2

Pattern A 3

(observed)

10.44

14.95

0

+

+

++

200

21.24

1.63

21.24

20.45

001

1.61

18.94

1.61

0.78

101

11.09

0.94

0.57

10.88

ment and the spins must be aligned parallel to the 6-axis because the (010) reflection did not appear in the patterns. In such a case spin canting or weak ferromagnetism are forbidden [16] by symmetry and goethite must be considered a collinear antiferromagnet (see Fig. 2). The Neel point, estimated from the temperature dependence of the (200) and (101) reflections of sample No. 2, coincides approximately with No anomalies connected with the existence of T 1 ^ were discovered. However, one should remember the severe reduction in the accuracy of neutron powder diffraction patterns of hydrogen containing structures. Moreover, the "peaks" of samples 1 and 2 were broadened. This broadening, when interpreted as the results of very fine pulverization of the sample, gives the following effective diameters of the grains: d(ioi) = 60 A ,

d(SOi) = 70 A ,

¿(200) =

¿(311) = 120 A .

80 A ,

These fit well with 105 A found by the B E T method and with the fact that goethite crystallizes in the form of needles [20], Acknowledgement

The authors are very much indebted to Prof. D. P. Grigoriev, Head of the Mineralogy Department of Leningrad Mining Institute, for delivering the mineralogical samples of goethite. 28«

434

A. SZYTTOA et al. : Neutron Diffraction Studies of a-FeOOH

References [1] A. Z. HRYNKIEWICZ and D. S. KTTLGAWCZUK, Acta phys. Polon. 24, 689 (1963). [2] A. Z. HRYNKIEWICZ,

D. S. KULGAWCZUK, a n d K . TOMALA, P h y s . L e t t e r s (Nether-

lands) 17, 93 (1965). [3] T . TAKADA, M . KIYANA, Y . BANDO, T . NAKAMURA, M . SHINGA, T . SHINJO, N . YAMAMOTO, Y . ENDCH, a n d H . TAKAKI, J . P h y s . S o c . J a p a n 1 9 , 1744 ( 1 9 6 4 ) . [4] M . J . ROSSITER a n d A . E . M. HODGSON, J . i n o r g . n u c l e a r C h e m . 2 7 , 6 3 (1965). [5] P . VAN DER WOUDE a n d A . J . DEKKER, p h y s . s t a t . sol. 1 3 , 181 (1966). [ 6 ] I . DEZSI a n d M . FODOR, p h y s . s t a t . sol. 1 5 , 2 4 7 ( 1 9 6 6 ) .

[7] G. W. OOSTERHOUT, Proc. Internat. Conf. Magnetism, Nottingham 1964 (p. 529). [ 8 ] J . STANKOWSKI,

W . WOLSKI,

and

B . KLIMASZEWSKI,

Ogölnopolska

Konferencja

Spektroskopii i Elektroniki Kwantowej, Poznan 1966 (p. 185). [9] V. I. MIKHEEV, Rentgenometricheskii opredelitel mineralov, Gosgeoltekhizdat, Moskva 1957. [10] A . SZYTUTA, A . BTJREWICZ, K . DYREK, A . HRYNKIEWICZ, D . KULGAWCZUK, OBUSZKO, H . RZANY, a n d A . WANIC, p h y s . s t a t . sol. 1 7 , K 1 9 5 ( 1 9 6 6 ) .

Z.

[11] S. GOLDSZTAUB, Bull. Soc. Fran?. Miner. 68, 6 (1935). [12] W. HOPPE, Z. Krist. 103, 73 (1940). [13] M. A. PEACOCK, T r a n s . R o y . Soc. Canada 36, 116 (1942).

[14] H. STRUNZ, Mineralogische Tabellen, Akad. Verlagsgesellschaft Geest u. Portig K.G., Leipzig 1957. [15] S. RRA&NICKI, J . PAWELCZYK, a n d H . RAPACKI, N u k l e o n i k a ( W a r s z a w a ) 7, 2 2 3 ( 1 9 6 2 ) .

[16] E. A. TTIROV, Fizicheskie svoistva magnitouporyadochennykh kristallov, Akad. Nauk SSSR, Moskva 1963. [17] C. SHULL a n d Y . YAMADA, J . P h y s . S o c . J a p a n 1 7 , S u p p l . B - I I I , 1 ( 1 9 6 2 ) . [ 1 8 ] J . BÖHM, Z . K r i s t . 6 8 , 5 6 7 ( 1 9 2 8 ) . [ 1 9 ] W . R . BUSING a n d H . A . LEVY, A c t a c r y s t . 1 1 , 7 9 8 ( 1 9 5 8 ) .

[20] T. SHINJO, J . Phys. Soc. Japan 21, 917 (1966). [21] R. FRICKE and G. HÜTTIG, Hydroxyde und Oxydhydrate, Akad. Verlagsgesellschaft Geest u. Portig K.G., Leipzig 1937 (p. 316 to 344). [ 2 2 ] W . WOLSKI, A . BUREWICZ, a n d J . SKRZYPCZAK, Z . a n o r g . a l l g . C h e m . 3 5 1 , 6 3 ( 1 9 6 7 ) . (Received

December

4,

1967)

J . MACHACKOVA: Ferromagnetic Resonance Measurements of F e r r i t e Films

435

phys. s t a t . sol. 26, 4 3 5 (1968) Subject classification: 19; 18.3 Institute

of Solid State Physics,

Prague

Ferromagnetic Resonance Measurements of Magnesium-Manganese Ferrite Films By J . MACHÄÖKOVÄ

T h e aim of this work is t o p r e p a r e ferrite films w i t h a n a r r o w F M R a b s o r p t i o n line a n d t o determine some magnetic p a r a m e t e r s of polycrystalline a n d single crystal films a n d t h e i r changes w i t h t e m p e r a t u r e . T h e influence of t h e r m a l t r e a t m e n t on t h e a b s o r p t i o n line half width a n d on t h e values of t h e e x t e r n a l static resonance field H is i n v e s t i g a t e d . T h e excit a t i o n of m a g n e t o s t a t i c modes is observed for perpendicular orientation of H a n d t h e film. T h e changes of these modes a f t e r polishing t h e surfaces of t h e films a r e i n v e s t i g a t e d . F e r r i t s c h i c h t e n m i t einer schmalen FMR-Absorptionslinie w u r d e n hergestellt u n d einige magnetische P a r a m e t e r der polykristallinen u n d einkristallinen Schichten sowie ihre Temp e r a t u r a b h ä n g i g k e i t b e s t i m m t . Der E i n f l u ß einer t h e r m i s c h e n B e h a n d l u n g auf die H a l b wertsbreite der Absorptionslinie u n d auf die W e r t e des ä u ß e r e n , s t a t i s c h e n Resonanzfeldes H wurde u n t e r s u c h t . Anregung von m a g n e t o s t a t i s c h e n Moden f ü r senkrechte Orient i e r u n g v o n H u n d Schicht w u r d e n b e o a b c h t e t u n d die Ä n d e r u n g dieser Moden n a c h Oberflächenpolieren der Schichten u n t e r s u c h t .

1. Introduction The films of Ni, Mg, Mn, Fe, and Mg-Mn ferrites were prepared by chemical transport reaction according to a modified method of Ksendzov [1]. As transport gas we used HCl. The single crystal films grew epitaxially on the (100) plane of MgO and polycrystalline films were prepared on the Si substrates. The thickness of the films was varied over the range 0.5 to about 50 ¡j.m. The largest halfwidth of the FMR absorption line AH was observed for Fe 3 0 4 films (1500 Oe). The smallest AH was observed for Mg-Mn ferrite films which were studied therefore in more detail. 2. Structure and Composition Transmission electron diffraction of Mg-Mn ferrite films removed from the substrates of MgO showed that the films were single crystals with the lattice constant (8.4 + 0.08) A. The microscopical observations showed t h a t the films grew as a replica of the substrate so that all cracks, steps, and faults of the substrate surface were visible on the films too. We also observed a great number of overgrowths in the form of pyramids on the film surfaces (Fig. l a , b). For polycrystalline films of Mg-Mn ferrites the (100) or (111) planes of Si Were used as the substrate. X-ray analysis showed that these films were polycrystalline with the same lattice constant (8.4 + 0.01) A without regard to choice of the plane of substrate.

J. Macháóková

436

r M U s . 9 ft 0 . ,;, < • "«sF „ ' ' ... l' # I P ft. ? m. * »"si f *« i» ® «

'¿é an a ! "

|P

*

% « ^ 3

'

i

1

jSÉjÉIII11 * nil Hp IÜ ' '* • s SL

t ?

fafk A,„- k -

»¿

•

r

*4 » 1 M**: • Ml ** • B B j ,

•

',"-" fiáfr-7 "SÍ"

«4*

f'?

if

a

b

Fig. 1. Microphotographs of single crystal films of Mg-Mn ferrites. a) (860 x ) , b) (16000 x )

The surface of polycrystalline films was remarkably inhomogeneous (Fig. 2). The dimension of these triangular formations is comparable with the thickness of the films. These formations were randomly oriented but their triangular surfaces were parallel to the plane of the substrate. Chemical analysis was only carried out for one specimen. Owing to low weight of the specimen the Mg and Mn ions were simultaneously analysed and the estimated composition was (Mg + Mn) 155 Fe 1 4 5 0 4 . The composition of the starting bulk ferrites, from which the films were prepared, was Mgi.iMn0.i3Fe1.77O4+y. 3. Resonance Measurements

The resonance measurements were carried out on JES-3BQ spectrometer mostly in the K-band. The disc-shaped samples ( 0 = 2 mm) were placed in the cylindrical cavity with TE012 mode.

437

Ferromagnetic Resonance Measurements of Mg-Mn Ferrite Films

Fig. 2. Microphotographs of polycrystalline films of Mg-Mn ferrites (860 X )

3.1 Single crystal

films

The external resonance static magnetic field i f as a function of angle in the (100) plane displayed fourfold symmetry for Mg-Mn ferrite films corresponding to the cubic symmetry with negative magnetocrystalline anisotropy constant K x . For deducing the resonance conditions by the method of effective fields [2] we took into account only the first anisotropy constant K v We supposed the demagnetizing factors in the film plane to be zero and simplified the strain to an isotropic one in the film plane. The influence of the overgrowths (Fig. la, b) was not taken into account because the polishing of the surface of these films did not show remarkable influence on the value of H. For parallel and perpendicular orientation of the static magnetic field with respect to the film plane we obtained

(tJ=K+K

(1)

+4 t/),

(2)

and (3) V

M

s

Here M e is the saturation magnetization, ojij k the resonance frequency, Tx = 3Ao ., 1 is a factor which characterizes the influence of strain on the value of the Ms resonance static field, X is the magnetostriction constant, the strain, //¿ji the external static magnetic field parallel to the [i j k] direction, y = spectroscopic splitting factor, //B and h have usual meaning.

;

g js the

438

J . MACHACKOVA

Table 1 Single crifstal films 20 °C 4 n Mt + Ti (G) 9 2 K1

Polycrystalline films

- 1 9 0 °C

20 °C

10

3820 ± 150

2060 ± 130

3150 ± 410

1.98 ± 0.01

2.01 ± 0.01

2.00 ± 0.03

2.04 ± 0.06

2530 ±

-m;

-190 ±

20

AHno (Oe)

255 ±

85

425 ± 145

AH l m (Oe)

380 ±

20

580 ±

Aff„oi (Oe)

385 ± 185

-980 ±

30

|

-

- 1 9 0 °C

-

305 ±

55

660 ±

360 ±

60

920 ± 140

60

40

700 ± 190

The values of the ^-factor, effective field of magnetocrystalline anisotropy 2K and magnetization (whose value is not separable from the factor Tx) were calculated from equations (1) to (3) both at 20 and —190 °C. The mean values together with their dispersion from measurements on 8 samples the thickness of which was in the range from about 0.5 to 10 [i.m are given in Table 1. The mean values of halfwidth of the resonance absorption line measured in the direction [i j is denoted by AHij k . 3.2 Polycrystalline

films

When we measured the resonance static magnetic field for various orientation in the film plane of Mg-Mn ferrite films, we observed no anisotropy at 20 and - 190 °C. The resonance conditions for parallel and perpendicular orientation of the static magnetic field (H\\ and Hj_) are +

+

(4)

and ( y ) = H

±

- * n M

s

-

(5)

T2,

where T 2 characterizes the effect of strain. The influence of the magnetog

crystalline anisotropy of randomly oriented grains ~ [3] was not taken 8 into account. The mean values of the gr-factor, 4 jt J f s + T2, and the halfwidth of the resonance absorption line from the measurements on 12 samples the thickness of which was in the range from 5 to 50 ¡i.m are also given in Table 1. 3 . 3 Thermal

treatment

We used single crystal films which were chemically removed from the substrates (the vinegar acid or the ammonium salts). The thermal treatment of

Ferromagnetic Resonance Measurements of Mg-Mn Ferrite Films

439

Fig. 3. Variation of the external resonance static field with temperature for 1,1 2,2' 3,3' 4,4'

single crystal film, single crystal film after thermal treatment, polycrystalline film, polycrystalline film after thermal treatment.

Here the figures and the primed ones correspond to perpendicular and parallel orientation of H , respectively. For single crystal films the parallel orientation corresponds approximately to the [110] direction

these films (the area of which was about 1 to 2 mm 2 ) at 600 °C on the air for half an hour resulted in both decreasing the halfwidth of the resonance absorption line down to 120 Oe (for AH110) and the changes of the resonance field for perpendicular and parallel orientation. For the latter case the external field was approximately in the [110] direction (Fig. 3). These changes may be ascribed to the decrease of strain in the film (if we suppose a^ < 0), eventually to the decrease of magnetization by treatment at a temperature lower than that at which the films were prepared [4], The same treatment of polycrystalline films had no remarkable influence on the AH and the changes of the static resonance fields were smaller than in single crystal films. 3.4 Magnetostatic

modes

For polycrystalline films we observed the excitation of a number of modes on the resonance absorption line for perpendicular orientation of H (Fig. 4a, b). These modes were predominantly unevenly situated. As it was noted on the beginning, the polycrystalline films had remarkable inhomogeneities on the surface (Fig. 2). We observed a relation between the intensity of these modes and the dimension of the inhomogeneities. During the polishing of the films (abrassive powder 800 and floated whiting) we observed successive decreasing of the intensity of these modes up to complete disappearing. The values of the static resonance magnetic field did not change by polishing. We also measured very small crystals of Mg-Mn ferrites which were prepared by chemical transport reaction. The dimension of these crystals was about 100 ¡xm. These

440

J. Machácková

Ferromagnetic Resonance Measurements of Mg-Mn Ferrite Films

441

crystals both fixed on the polished surfaces of the Mg-Mn films and isolated gave a number of lines the values of static magnetic field of which correspond to the values of magnetic fields for modes under observation. These magnetostatic modes were apparently excited by changes of the static magnetic field as result of the inhomogeneities of the surface. 4. The Measurements of Magnetization by the Inductive Method The disadvantage in the determination of the magnetic parameters of films by FMR consists in the impossibility of separating the magnitude 4 n Ma from the influence of strain. At the present time measurements of the magnetic moment of films by the inductive method are performed and preliminary values of 4 n Ms at room temperature for two polycrystalline films were about 1900 G. For comparison the magnetization of starting bulk ferrites Mg1.1Mno.13Fej.77O4 in powder form was measured in the same way and the value of magnetization was 2300 G. 5. Discussion and Conclusions a) The films of Mg-Mn ferrites were prepared by chemical transport reaction. The lattice constant (8.4 + 0.01) A (we take into account only a lattice constant for polycrystalline films with respect to the accuracy of measurements) is in the range from 8.366 A (MgFe 2 0 4 ) to 8.457 A (MnFe 2 0 4 ). b) We can only say, according to one chemical analysis, t h a t the composition of films differs from t h a t of the starting bulk material. I n the future the investigation of both magnesium and manganese ferrite films will be suitable to account for the changes in composition between the films and the bulk ferrites as a result of the different transport velocities of different oxides. c) From the measurements of magnetization by the inductive method we preliminarily observed t h a t 4 71 Ma for films is smaller than for starting bulk ferrites. This may be caused by different chemical composition and by the influence of the preparation of films, especially by cooling. d) The resonance measurements showed t h a t the calculated values 4 n Ma + Tt are larger for single crystal films than for polycrystalline ones. This fact together with the observed influence of thermal treatment indicates t h a t the strain is larger in single crystal films. e) The value of the spectroscopic splitting factor g for films agrees with t h a t for starting bulk ferrites. f) No essential difference was found between the half width of the resonance absorption line for single crystal films and t h a t of polycrystalline ones at room temperature. For explaining the observed line widths we may take into account the influence of inhomogeneities such as strain, surface, and structure inhomogeneities, eventually the possibility of exciting various modes of spin wave resonance which are unseparable. The higher porosity of polycrystalline films may also influence the line widths. The larger temperature change of AH for polycrystalline films could be ascribed to the contributions to the line broadening due to crystal anisotropy IK I [5] as at low temperature the c o n d i t i o n - j - > M s holds.

442

J . MACHACKOVA : Ferromagnetic Resonance Measurements of Ferrite Films

g) The resonance magnetic case they

observed magnetostatic modes for perpendicular orientation of the static magnetic field were probably due to the change of static field as a result of the inhomogeneities of the surfaces, and in this can be a source of information about the quality of surface. Acknowledgements

The electron diffraction and the X-ray and chemical analyses were kindly provided by M. Janatka, L. Cervinka, and A. Novak, respectively. The author wishes to express her thanks to Dr. S. Krupicka and Ing. J . Kanturek for their valuable discussions. The author is also grateful to Ing. V. Sik and K . Suk for their experimental assistance. References [1] [2] [3] [4] [5]

J . KSENDZOV, Rep. Conf. Magnetic Oxides, Liblice 1966. J . MACDONALD, Proc. Phys. Soc. A64, 968 (1951). P . CLABRICOATS, Microwave Ferrites, London 1961 (p. 57). R . PAUTHENET, Ann. Phys. (France) 12, 730 (1952). S. V. VONSOVSKII, Ferromagnitnyi rezonans, Moskva 1961 (p. 249). (Received January 3, 1968)

B. HEINKICH et al.: Paramagnetic Susceptibility of Ni-I Boracite

443

phys. stat. sol. 26, 443 (1968) Subject classification: 18.1; 18.4; 22 Institute

of Physics,

Czechoslovak

Academy

of Sciences,

Prague

The Temperature Dependence of the Paramagnetic Susceptibility of Ni-I Boracite By B . HEINRICH, J . ZiTKOVA, a n d J . KACZER

The decrease of the paramagnetic susceptibility in Ni-I boracite with decreasing temperature is explained taking into account the thermally excited paramagnetism of the sublattice of the Ni s+ ions. Der Abfall der paramagnetischen Suszeptibilitat in Ni-I-Borazit mit fallender Temperatur wird mit der Annahme eines thermisch angeregten Paramagnetismus des Untergitters der Ni2+-Ionen diskutiert. Recent measurements on the temperature dependence of the magnetic susceptibility of ferromagnetoelectric N i - I boracite by Ascher et al. [1, 2] showed the existence of two maxima, a sharp one at 60 °K and a flat one at 120 °K. The second maximum at 120 °K they attributed to a magnetic phase transition from the paramagnetic to an antiferromagnetic state. Since the shape of the second maximum does not seem to correspond to a phase transition we propose explaining it on the basis of the temperature-excited paramagnetism of the Ni 2 + sublattices [3] and their antiferromagnetic exchange interactions. According to Ascher et al. the magnetic structure of low N i - I boracite is of the m'm2' type. The metal ions can therefore be divided into three magnetic sublattices which correspond to three non-equivalent positions of the low boracite Ni 2 + ions [4]. From the asymptotic behaviour of the magnetic susceptibility it follows that the interactions in each of these magnetic sublattices are of antiferromagnetic character. The nearest neighbourhood of the metal ions of the high boracite N i - I (see Fig. 1) has a slightly deformed tetragonal symmetry, the exact symmetry is an orthorhombic one of the D 2 d type. The mechanism of the temperature-excited paramagnetism involves a singlet (S = 0) ground state and a magnetic (S = 1) excited state. The stabilization of the (S = 0) ground state for the electron configuration (d)8 of the Ni2+ ion in the crystal field of axial symmetry can occur in two different ways: Firstly, as Ballhausen and Liehr [3] pointed out, this stabilization may be due to the strong crystal field in the planar complexes. Secondly, the latter crystal field may be overcome by a strong effective field produced by the two anions in the axial axis. In what follows, we shall show that in our case of N i - I boracite the second possibility is more plausible because of the following reasons. The experimental measurements (Ascher et al.) on the temperature dependence of the magnetic susceptibility of the Ni-Cl and Ni-Br boracites did not show any flat maxima. It follows that the magnetic behaviour of the Ni 2 + ions is effectively influenced by the presence of the halogen ions, and that the contribution of the oxygen ions to the crystal field potential cannot predominate the contribution of the halogen ions.

444

B . HEINRICH,

J . ZÎTKOVÂ, a n d J . K A C Z É B

a-ifdzz)

Fig. 1. Slightly deformed tetragonal surrounding of the N i " ion in N i - I boracite. N i - O : 2.04 A, N i - I : 3.02 A, 0 - 0 : 2.9 A, I - O : 3.44 A

Fig.2. Energy spectrum of the configurations (d) 1 (a) and (d) a (b) of the Ni 8 + ion in N i - I boracite

It is well known that halogen ions have strong covalent bonding [5]. Hence the contribution of these ions to the effective crystal field overcomes the contribution of the oxygen ions. The covalent bonding is the strongest in the case of I-ions. It follows that in our case of Ni-I boracite the second possibility with the strong effective axial crystal field will take place. The energy levels for the single (d)1 electron and the (d)8 electron configuration of the Ni 2+ ions taking into account this second possibility are plotted in Fig. 2. From the simple calculations we get that the strong effect of covalency of I~ ions allows stabilization of the non-magnetic level before the magnetic one 3 E (this case corresponds to the situation in Fig. 2a(y) if the condition A > 3 F2 + 90 Ft (1) is fulfilled. Here the Condon-Shortley parameters for Ni 2+ ions are F t « 100 cm - 1 and F2 m 1000 cm - 1 [3]. The expression (1) is derived neglecting the Coulomb interaction of the orbital levels. Taking into account this interaction, the value of the energy on the right-hand side of the condition (1) becomes a little smaller. The justification of the explanation of the flat maximum of the Ni-I boracite susceptibility by means of temperature-excited paramagnetism of the Ni 2+ ion sublattices is made by comparing the course of the experimentally measured and the theoretically calculated curve of the paramagnetic susceptibility. The influence of intrinsic exchange interactions of Ni-I boracite is included in our calculation of the susceptibility by means of the standard phenomenological method of the molecular field [6]. An estimate of the molecular field coefficients is made from the asymptotic behaviour of the measured paramagnetic susceptibility. In the paramagnetic state the magnetic susceptibilities for each of the sublattices are given by the relation

v ^

Xi

2*°f 1-lw'

(2)

445

Paramagnetic Susceptibility of Ni-I Boracite

where y_et is the susceptibility of the given lattice not considering the molecular field, and X is a corresponding coefficient of the effective molecular field; i = 1, 2, 3. For the Ni 2 + metal ion lattices with the energy spectrum of Fig. 2 b we have Zef

,N

1

*

AIT

i +

¥

(3)

exp

[TtJ

where N is the number of atoms in each sublattice, A is thé Avogadro number, A E is given by Fig. 2 b . The asymptotic behaviour of the susceptibility is then given by X = 3 where

20

T + 6+XC

T

6C +TC

(4)

N 3 2 AE : A 14 Tie Comparing expression (4) with the asymptotic behaviour of the experimentally measured susceptibility, we take Tc = 320 ° K

« 2.2

The theoretically calculated susceptibility and its comparison with the experimental course is plotted in Fig. 3. Hence we obtain for the energy separation [3] of the non-magnetic and magnetic levels AE œ A -

3 F2 -

90 F,=

1.9 JcTma,x = 160 c m " 1 ,

(5)

from which we obtain an approximate value A « 12000 c m . As mentioned above, the experimental results of the magnetic susceptibility measurements by Ascher et al. justify the great influence of the covalency effect of the halogen ions on the magnetic behaviour of boracites. Since the covalency effect of the CI - and B r " ions is not so strong as that of the I - ion, the energy spectra of the Ni 2 + ions of the Ni-Cl and B i - B r boracites are not expected to fulfill the energy separation conditions for the stabilization of the non-magnetic levels. The single (d) 1 electron energy spectrum would correspond either to the case of weak axial distortion in Fig. 2 a (a) or to the case in Fig. 2 a (P). I n the energy spectrum of Fig. 2 a (a) the condition for stabilization of the non-magnetic level before the magnetic one ^ j 1 ) becomes -1

A' > 1 2 F2 + 45 F£

Fig. 3. Temperature dependence of the experimentally measured and theoretically calculated paramagnetic susceptibility oi N i - I boracite

16500 cm"

(6)

TCK)-

) This case is entirely analogous to that of the weak tetragonal distortion of the planar complexes discussed by Ballhausen and Liehr [3], 1

446

B. HEINRICH et al. : Paramagnetic Susceptibility of Ni-I Boracite

The temperature-excited paramagnetism for this case is hardly probable because the energy value A' of the stabilization condition (6) is too large. Even if the energy spectra of the single (d) 1 electrons are as in Fig. 2 a (¡3), the energy separation condition (1) for the Ni-Cl and N i - B r boracites will not be fulfilled because of the weaker covalency of the Cl~ and B r - ions compared to the I - ion case. In conclusion it can be said that the observed flat maximum of the N i - I boracite magnetic susceptibility can be explained by the temperature-excited paramagnetism of the Ni 2 + ion sublattices. At the same time when comparing experiment and theory we obtain the value of the ¡/-factors and the coefficients of the internal molecular fields, which for an antiferromagnetic state lead to the value of the molecular field of the order 10 6 Oe. 2 ) Acknowledgements

The authors would like to thank Dr. E . Simanek for his helpful comments and Dr. Z. Frait and Dr. K . 2dansky for their interest. References [ 1 ] E . ASCHER, M . RIEDER, M . SCHMID, a n d M. STOSSEL, J . appi. P h y s . 3 7 , 1 4 0 4 ( 1 9 6 6 ) .

[2] E. ASCHER, A propos de la ferromagnetoélectricité, Conf. held at the Seminar of Phys. of the Univ. of Neuchatel, June 20, 1966. [ 3 ] C. J . BALLHAUSEIX a n d A . D . LIEHR, J . appi. Chem. 8 1 , 5 3 8 ( 1 9 5 9 ) . [4] T . ITO, N . MORIMOTO, a n d R . SADANAGA, A c t a c r y s t . 4 , 3 1 0 ( 1 9 5 1 ) .

[5] Following e.g., the Nephelauxetic series; see e.g. J . S. GRIFFITH, The Theory of Transition Metal Ions, Cambrigde University Press 1961. [6] CH. KITTEL, Introduction to Solid State Physics, Wiley, New York 1956. (Received December 15, 1967)

2 ) From relation (1) it is obvious that in our case the non-magnetic level JAX approaches to the limit of its stabilization. The stabilization of the Ni 2 + ion magnetic level after a transition into the ordered magnetic state could be explained by a change of the crystal parameters due to the lattice deformation, since during the phase transition the sense of the deformation of the metal ion lattice leads just to a decrease of the energy separation of the one-electron levels (%) and (e).

A. B. KUNZ: Notes on a Valence Electron Model of Atoms

447

phys. stat. sol. 26, 447 (1968) Subject classification: 13.4; 22.5.2 Department of Physics, Lehigh University, Bethlehem,

Notes on a Valence Electron Model of Atoms1) By A . B . KUNZ

The exact solutions to the "valence-electron model Hamiltonian" are considered. This model was suggested in recent articles by Wang. It is shown that for a certain range of C in the potential term — C r~2, solutions to the problem exist which are formally identical to the usual solutions for the hydrogen atom problem. It is also seen that for a wide range of the parameter C, there exist no solutions for which the Hamiltonian is Hermitian. The implications of this are discussed, and uses of this model considered. Es werden die exakten Lösungen des „Hamiltonoperators des Valenzelektronen-Modells" untersucht. Dieses Modell wurde kürzlich von Wang vorgeschlagen. Es wird gezeigt, daß für einen bestimmten Bereich von C im Potentialterm —C r~2 Lösungen des Problems existieren, die formal mit den gewöhnlichen Lösungen des Wasserstoffproblems identisch sind. Es wird auch gezeigt, daß für einen großen Bereich des Parameters C keine Lösungen, für die der Hamiltonoperator hermitesch ist, existieren. Die Folgerungen daraus werden diskutiert und die Benutzung des Modells untersucht. 1. Introduction In a recent work discussing the absorption spectrum of an alkali-earth cation impurity in an alkali-halide host crystal, Wang has introduced a "valence electron model of atoms" [1], He proposes to use this simplified model to provide atomic or ionic wave functions which would be useful in solid state calculations. The model employed by Wang assumes that the outhermost electrons can be described by a single-particle wave function which sees a potential given as =

I e2

C e2

(D

The parameter C is chosen with the aid of experimental data and is real. I is the ionicity of the atom plus one (i.e., for neutral K I = 1, and for K + 1 = 2). This form for the potential is assumed to be valid for all values of r in the range 0 5S r 5S oo. The effective Hamiltonian is given as R = In order to solve the equation

n2

2m

Je2 C P2 — — —2 . r r

Hy> = Erp

(2) '

v

(3)

Wang employs a variational procedure and hydrogenic trial wave functions. The use of this procedure is discussed in the Appendix. I t is the intent of this note to discuss exact solutions to equation (3) using the Hamiltonian given by equation (2). Work supported in part by U.S. Air Force Office of Scientific Research, Contract No. 1276-67. 29

physiea 26/2

A. B. Kunz

448

2. Mathematical Development Atomic Hamiltonians of the type (2) are not new. In fact, this idea is suggested by Fock and Petrashen [2] who used it to obtain asymptotic solutions to the Hartree-Fock equation. Fock and Petrashen develop this form by expanding the Hartree-Fock potential seen by an electron in a power series of r - 1 . This series expansion is only valid for the range 1 < r iS oo. However, by truncating this series at the term a2r~2, one might hope to obtain some meaningful solutions on the range 0 5S r oo [3]. Potentials of this type also arise in discussions of the "quantum defect model" for atoms or solids [4]. One wishes to solve le2 Ce2 ÏL y) — E y) . (4) r 2m To proceed, one assumes y(r) = Rni(r) Yf{d,