TETRAHEDRALLY BONDED CRYSTALS

This family of open structured crystals with low coordination number (z

=

⁴⁾

includes many important minerals and semiconductors and has attracted much exper-imental and theoretical attention. Their vibrational and associated thermal properties show interesting features, including a low frequency transverse acoustic branch which is important at low temperatures and leads to serious departures from Debye-like be-havior, and in many cases to negative values of the expansion coefficient. Such behavior is likely in open structure crystals (Section 2.6.3), being due to low fre-quency transverse modes (likened to the vibrations of a guitar-string) which are preferentially excited at low temperatures [BlaS7, BarS7a]. This pattern, already seen to a lesser extent in the alkali halides, extends also to other systems of open structure, including silicate crystals and silica glasses.

Non-Metals

3

2

_, /

, /

, / /

/ / /

, , ,

\

, ,

\

\

\

\

\

\

..

... - _ E'

bF2

----...

.. ·;:;:'·SaF₂

/

/i .·r

, .//

./

SrF2 ...•.•.. /

^{... /}

... /

,/ /

. /

---163

Fig. 5.6. ,,(T) for alkaline earth fluorides and PbF2. Hatched areas denote limiting values of

"b

^{h for the}

alkaline earth fluorides, and arrows show ,," [Whi80j.

164 ChapterS

5.5.1. Diamond and Zincblende Structure

As in the fluorite structure, the lowest acoustic velocity in these cubic crystals is for c' transverse modes propagated along the [110] direction with a [1 10

1

polarization;

but now all the ions are tetrahedrally coordinated. If we look at a three-dimensional model of this rather open structure, it is easy to see how such an acoustic wave (having an open space normal to the propagation direction) propagates along 'chains' without altering distances between nearest neighbors. Unless the bonds have appreciable covalency and hence strong angular rigidity, such modes will not only be of relatively low frequency but will also soften under pressure like a relaxed guitar string (Section 2.6.3). Ultrasonic measurements confirm that the velocity of this wave decreases under pressure for many members of the family; dc' / dP is negative, and increasingly so for the more ionic (less covalent) members (Table 5.4). A measure of the ionicity is a factor,

Ii,

defined and tabulated by Phillips [Phi73] (see Table 5.4 below). Crystals for which/; exceeds 0.78 are not likely to be stable in this cubic structure.

Neutron spectroscopy confirms a high degree of dispersion for these transverse acoustic modes. At temperatures in the region of E> /25 they may be fully excited, and contribute strongly to the sharp drop in E>D(T) (see Fig. 5.7) and the minimum in 'Y(T) (see Fig. 5.S). The decrease in 'Y from '}t) to 'Ymin results from two factors: dispersion, which weights these TA modes more heavily, and more importantly a decrease in the mode gammas (to become negative - or more negative) with increase in wave number, revealed both by model [Do166] and ab initio [Xu91] calculations, and by neutron diffraction under stress [Pay64]. Measurements of the Raman spectra under pressure for Zn chalcogenides also confirm that 'YT A at the zone boundaries may be much lower than at the low-frequency zone center (e.g., [Wei77]).

Table 5.4 lists diamond or zincblende structure crystals for which there are exten-sive low temperature data [BarSO]. Excepting diamond itself they all appear to have low-temperature ranges over which a is negative; thus 'Ymin is negative, although

'}t) is sometimes positive (though small). Barron et al. [Bar77a] show a correlation between '}t), c' and the ionicity,

Ii,

for a number of these crystals. Values of 'Yb^h marked with a query (?) in Table 5.4 are less reliable than elastic values, 'Y81, due to uncertainty in measuring the very small expansion below 10K.

Most of these crystals have 'Y values at higher temperatures which approach 0.7 or O.S. An exception is HgSe, for which a

<

2 x 10-⁶ K-¹ up to 500 K [Zhd66], giving 'Yoo

==

0.2. Pressure derivatives of both the shear moduli (c' and C44) are negative for HgSe, and the minimum value of a is comparable with that for CuCI (Fig. 5.9), namely about -8 x 10-⁶ K-¹ near 30 K.

There are limited thermal and elastic data for some other III-V compounds such as the phosphides. Useful sources of available data on the Group N and III-V compounds are the Landolt-Bomstein volumes IIII17a [Mad82]. Slack and Bartram [Sla75] have reviewed thermal expansion data for a number of diamond-like crystals, with emphasis on high temperature data and the importance of matching expansion values for technical purposes; the crystals include AIN, cubic BN, BP, GaP, cubic and polycrystalline SiC, diamond, Ge, Si and polycrystalline BeD (a wurtzite structure).

Non-Metals 165 Thble 5.4. Data for crystals of zincblende structure (and wurtzite in

brackets). Most are from review [BarSO]. Asterisk denotes room temperature data, el denotes use of elastic moduli andJi is Phillips

ionicity factor. For (?) see text

Fig. 5.7. Reduced plot of

a

^C against T for some diamond-type crystals [Gop66].

166

+1·0

>--

^-1-0

-2,0

CdTe

-

^...

_'\

\ \

_CU.0, /

" \

/

^/

'-

_{- , . /} ^/

0·01

/

/ / I / I / I I / /

/ /

/ / /

/ / /

1·0

ChapterS

Fig. 5.8. l' for some diamond-type crystals as function of reduced temperature [BarBO, Fig. 5.10].

T2 (K2) o 20 I 10 20 30 40 50 15

/

104

V·/·

T3 /

t /. ~

10 5

7". /~ ...-.~/' t ~ __ - 7/ ~ t} " ~.A"~10

",. ,/ ' ..,... T' " .... ","

... _

.... CdT.

-2 I -4¥

....

01 "11 '111 5 10 SO T (K)

ts

-15'" o

-

-8 Fig. 5.9. Values of C/TJ (J·mol-I·K-4) and -a/T3 (K-4) versus "[2 for CuC!. Inset shows a(T) for CuCI compared with CdTe. From [Bar80, Fig. 5.9].

f ~ ...

~

168 ChapterS

Silicon. This has particular significance as a reference material for thermal ex-pansion (see Section 3.3): it is cubic, readily available in a state of ultra-high purity (and therefore very reproducible), and the expansion has been carefully measured over a wide temperature range. Levels of uncertainty in a are :::; 10-⁸ K-ⁱ be-low room temperature. Values recommended by CODATA (Committee on Data for Science and Technology) [Whi97] are given in Table C.3 of Appendix C.

5.5.2. Wurtzite Structure Including Ice

Closely related to the zincblende structure is the wurtzite structure. This has hexagonal symmetry (ABAB ... packing) rather than cubic (ABCA. .. ), and the ther-mal expansion can therefore be anisotropic. A few of the III-V, II-VI, and I-VII compounds may exist in either zincblende or wurtzite form. Others such as ZnO, AIN, AgI, CdS, BeO are always wurtzite. The low-lying TA modes correspond to the

C44 and C66 elastic constants, which because of the different crystal axes correspond roughly to c' in zincblende.

There is not a great body of thermal data at low temperatures on this group. The Landolt-Bomstein volume on semiconductors [Mad82] gives information on band structure, optical properties etc. for many of the III-V compounds, but few thermal data at low temperatures.

CdS. Cadmium Sulphide has attracted some attention for its electroacoustic prop-erties, but values for a at temperatures below about 30 K are confined to polycrys-talline compacts. Curves of CXav(T) and yeT) for CdS are similar to those of CdTe (zincblende structure), after allowing for differences in E> [Bar80, p. 669].

ZnO. Values of the expansion coefficient of zinc oxide along the principal axes are shown in Fig. 5.10, a being negative below about 100 K.

Ice. The oxygen atoms in the hexagonal (normal) form of ice also form a wurtzite structure, being linked by hydrogen bonds. The light mass of the hydrogen or deuterium atoms and consequent low moment of inertia of the water molecules causes the frequencies of the translational vibrations of the molecules to be well-separated from the higher rotational frequencies; frequencies of the intramolecular vibrations are much higher again, and their contributions to Cv can be neglected over most of the cryogenic range.

The thermodynamic properties have been discussed in a classic analysis by Lead-better [Lea65], who exploited differences between H20 and D20 ice to separate the contributions of translational and librational modes to the heat capacity and thermal expansion. He thus obtained moments

<

wⁿ

>

and approximate Griineisen parame-ters for the different parts of the spectrum. These correlated well with spectroscopic and Debye-Waller data. Small additional effects in Cp due to incipient orientational ordering of the molecules [Hai72] are shown in Fig. 1.12.

Non-Metals 169

8

-f

^InO

;::-...

~ 5

'f' ~

~

,

2

;::-...

t

'f' ~

~iUr---~~.---~---.r-;---~

~

Fig. 5.10. Linear coefficients of expansion of Zno [Iba69]. The calculated curves were obtained by fitting to a frequency distribution of two Einstein peaks (8E = 107 K and 590 K).

170 ChapterS The thermal expansion is not markedly anisotropic: the principal coefficients of linear expansion differ by less than 2% near 273 K, with values of about 50 x 1O-6K-^{I .} Crystals of both H20 and D20 have been measured down to 20 K [Dan62b], showing similar a values for the two isotopes which become negative below 63 K. At 20 K, 'Y(T) is about -0.9, rising to about 0.6 above 150 K [Lea65].

5.5.3. Phenacites, Cuprite

The mineral phenacite, Be2Si04, is another open structure, with tetrahedrally coordinated cations (Be++) and 3-coordinated anions. Slack and Huseby [Sla82] have reviewed the thermal behavior of a number of phenacite-type compounds because of their potential interest as low expansion materials. They are also reviewed by D. C.

Palmer in the chapter on 'stuffed derivatives of the silica polymorphs' in Silica, see [Hea94, Ch. 3].

Many show a marked departure from Debye behavior in heat capacity with 8

c

falling to 8~in '" 0.680 at T '" 0.06 80. They are all non-cubic, and those with small average expansion coefficients at 300 K include:

For willemite (Zn2Si04) there are expansion data on single crystals extending down to 2 K [Whi88]; al. is negative below 290 K and all is negative below 150 K. There are no reliable data on Cp or on B below 50 K, but extrapolation indicates that 'Y falls from 0.6 at high temperatures to about 0.1 at room temperature and less than -1 at low temperatures (see Section 8.4.3).

Cuprite. Another material of potential technical interest because its low coor-dination leads to negative expansion is the cubic oxide of copper, cuprite (CU20), see [Bar80, p. 674]. Structurally it consists of two interpenetrating networks not connected to each other by any primary CuO bonds; in each network, the 0 atoms form a diamond structure, being linked by Cu atoms midway between them, so that each 0 is surrounded tetrahedrally by Cu and each Cu has two nearest neighbor 0 atoms linearly arranged. Both the c' and C44 stiffness constants have negative pressure derivatives, leading to '}\) ~ -4 and negative values of expansion coefficient below 280 K. Other parameters are 80 ~ 185 K and Bo = 110 GPa.

Non-Metals 171 Thble 5.5. nata for some ceramic oxides used at low temperatures [BarBO, Tou77].

AI /1 = (1293 -14.2) /1293' Asterisk denotes room temperature data

eo

^{80 c~93 Cl293 I:J/l) ')t) 'Y293}

There are a number of ceramic oxides which are used over a wide temperature range for structural supports, insulators, reference standards, film substrates etc. They have little in common crystallographically but are tough and stable materials. In Table 5.5 are given technical data on Ah03 (alumina or sapphire), MgO (magnesia), Ti02 (rutile), a-Si02 (quartz), Th~ (thoria), and yttria stabilized zirconia (Zr02 + 9 mol%

y 203). Some further data on length changes at intermediate temperatures are given in Table C.2 of Appendix C. Data sources are compendia of CINDAS [Tou77], American Institute of Physics Handbook [IGr72], and [Bar80, p. 674]. For the yttria stabilized zirconia, data are from Collins et al. [CoI85a]. The data for this zirconia should be fairly representative also of calcia- and magnesia-stabilized zirconia.

Sapphire (a-alumina). a-A1203 is trigonal and so has anisotropic expansion; but it has been available in single crystal form as a Standard Reference Material for both heat capacity (SRM720) and thermal expansion (SRM732), being recommended as a reference material by both CODATA [Whi97] and IUPAC [Mar87a] (see also Ch. 3).

Magnesia. MgO has cubic rocksalt structure, is stable with a high Debye

e,

and is used as a substrate for thin films of high temperature superconductors.

Quartz. Si02 can exist in many forms: a-quartz, f3-quartz, cristobalite, tridymite, coesite, stishovite, as well as in the vitreous state (see [Hea94, Ch. 1]).

The stable form below 846 K is a-quartz, which has an open structure of trigonal symmetry composed of regular Si04 tetrahedra linked by shared oxygen atoms

172

o ...

COl COl

...

~

CO>

'"

... ~

CO>

...

~

'"

Chapter S

N ';

~

~

'"

N

""

₈

Co :I

i

_...

.8

'"

°5 §

~

=

.1&

..

"

:~ ^~ vi li: ail

Non-Metals 173 In a-quartz, however, helical arrangements of the tetrahedra occur parallel to the hexagonal c axis which permit the Si-O-Si angle joining the tetrahedra to change by cooperative rotation or 'tilting' of successive tetrahedra. This flexibility ac-counts for the relatively large compressibility (see low value of B in Table 5.5) of a-Si02 (and of a-Ge02), and the large positive thermal expansion at room temperature compared to other forms of silica [Wri76]; the tetrahedra themselves are fairly rigid, but the change in macroscopic dimensions when tilting is in-duced by pressure or temperature is relatively large (see for example the neutron diffraction studies under pressure [Car81]). Barron et al. [Bar82] discuss the ex-pansion, heat capacity and elastic constants over the range of stability and show how E)c and the principal 'Y's change with temperature (Fig. 5.11). Theoretical models indicate that although tetrahedral tilt is the largest effect in the change of geometry with temperature [Bar87], tetrahedral distortion makes a comparable effect on the macroscopic expansion at low temperatures.

Rutile. Many oxides have the structure of the mineral rutile (Ti~), including Ge02, Sn02, Mn02. The symmetry is tetragonal and the expansion anisotropic.

Another tetragonal compound with the rutile structure is paratellurite (Te02), for which E)~

=

²⁶⁵

±

10 K. At room temperature all

=

^{5.6 x 1O-6K-1 and a.L}

=

10 x 10-6K-l, giving 'Y = 0.9, while at liquid helium temperatures 'Y ~ -lowing to the influence of a soft c' transverse mode [Whi90a].

Thoria. Th02 is another stable high melting point ceramic, having the cubic fluorite structure. It is discussed briefly in Section 5.11.3, in association with its isomorph U02 which becomes antiferromagnetic below 30 K.

Zirconia. Zr02 also has the cubic fluorite structure at high temperatures. The structure can be stabilized at lower temperatures by addition of a few per cent of yttria, calcia or magnesia. It is then important as a high strength ceramic, an oxygen sensor and as a substrate for film deposition. The low temperature heat capacity has a linear (T -) term, indicative of 'disordered' tunnelling states, discussed in more detail under Section 5.7 on glasses and glass-ceramics.

Low temperature values of Cp

/T

³ and a/T3 for Zr02 + 9 mol% Y 203 are shown in Fig. 5.13 [CoI85a] with data below 1 K from [Ack84]. The measured heat capacity

174 ChapterS

includes a Schottky contribution from magnetic impwities, evident below 5 K and centered about 1.5 K. Subtracting this term leaves:

Cp (in p.J. g-l. K- 1) = 1.9T

+

^{Debye T3 term (eo = 540 K)}

Values for 'Y are about 4 near 10 K and 1.7 at 293 K (see Table 5.5).

5.7. GLASSES AND GLASS CERAMICS

In document at Low Temperatures (pagina 168-180)