VOLUME 55, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 25 NOVEMBER 1985

Magnetic Excitations in Amorphous Germanium Studied by High-Field Calorimetry

R. van den Berg and H. v. Lohneysen //. Physikalisches Institut der Rheinisch-Westfalischen Technischen Hochschule Aachen, D-5100 Aachen, West Germany

(Received 19 June 1985) The specific heat of amorphous Ge films evaporated at room temperature shows a strong mag­

netic field dependence at very low temperatures (0.07 K < T < 1.5 K). This effect is quantitative­ly attributed to magnetic excitations arising from localized unpaired electronic states (dangling bonds). It is shown that other kinds of excitations, e.g., tunneling states known from other amor­phous solids, do not contribute measurably to the specific heat.

PACS numbers: 65.40.-f, 71.25.Mg, 71.50. + t, 72.80.Ng

The low-temperature properties of tetrahedrally bonded amorphous semiconductors have been the subject of much research recently. In this Letter, the nature of low-lying elementary excitations in amor­phous Ge (a-Ge) is investigated through a specific-heat study in high magnetic fields. In principle, three different kinds of excitations have to be considered: (1) structure-induced low-energy excitations common to amorphous solids, (2) excitations in the "electron glass" of localized electrons, and (3) magnetic excita­tions due to the localized electrons carrying spin \ .

An intriguing problem has been the existence or not of structure-induced low-energy excitations in tetra-hedral semiconductors^"^ which are found in virtually all other amorphous solids, e.g., random-network glasses such as vitreous silica, polymers, and metallic glasses.^ They give rise, for instance, to a linear specific heat at very low temperatures. It is generally believed that these excitations arise from tunneling of atoms or groups of atoms between the two lowest lev­els of a double-well potential, and hence they are gen­erally termed two-level tunneling systems (TLS). It has been suggested^ that TLS should be less numerous in tetrahedral semiconductors because of their rather ' 'closed" structure compared with other amorphous solids. While early measurements of phonon scatter­ing by low-energy excitations in a-Ge and a-Si '"^ gave conflicting results, a specific-heat study below 1 K re­vealed a linear specific-heat contribution in a-Ge,^ which, however, was attributed to magnetic excitations arising from localized unpaired electrons (dangling bonds), and not to TLS. Since a microscopic model for the structure-induced TLS is still lacking, an un­

equivocal assignment of the low-lying excitations in tetrahedral semiconductors would be very important. If indeed there is a contribution by dangling bonds, the specific heat should depend on an applied magnetic field.

This Letter reports on the results of a calorimetric study of a-Ge below 1 K in magnetic fields up to 6 T. In brief, we observe a strong field dependence of the specific heat which can be quantitatively accounted for in terms of magnetic excitations, with negligible con­tributions from either TLS or electronic excitations. Thus, evaporated <2-Ge (with a high concentration of dangling bonds) provides an interesting system with which to study spin-y amorphous antiferromagnetism. There have been various theoretical attempts^^"^^ at explaining the absence of magnetic ordering^^ in disor­dered spin-y systems such as (crystalline) Si:P where the exchange interaction between donors is of compar­ably longer range than between dangling bonds in a-Ge. In addition to settling the above-mentioned prob­lem of TLS, our results shed some light on the mag­netic excitations of a disordered dilute spin-y magnet with well-localized moments.

Amorphous Ge films were prepared by electron-beam evaporation of 99.999% Ge onto a single-crystalline Si substrate (2x3 cm^, 0.02 cm thick) at room temperature. In order to check for the possible influence of preparation conditions, three different evaporation setups were employed, with varying base pressures p and evaporation rates R (see Table I). The mass density as determined for sample 3 was 5.25 g/cm^, i.e., close to that for crystalline Ge. Here we discuss only the measurements performed on sample

TABLE I. Properties of investigated a-Ge samples. Symbols are explained in

Sample

1 2 3

P (mbar)

2x10"^ 2x10"^ 5x10-^

R (A/s)

100 40 20

A^ESR

(cm-3)

5x10^^ 5.4xl0^« 6.0x10^^

^ S c h

(cm-^)

1.0x10^9 8.5x10^^ 8.5x10^^

(J/g K^)

3.0x10"^ 4.5x10-^ 5.8x10"^

n

0.9 1.3 1.6

the text.

(cm-^)

9.8x10^^ 9.3 xlO^^ 8.0x10^^

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V O L U M E 55, N U M B E R 22 PHYSICAL REVIEW LETTERS 25 NOVEMBER 1985

2. The results for the other samples were qualitatively similar and are summarized in Table I. The specific heat was measured with the standard heat-pulse tech­nique, as described in detail elsewhere.^'^"^ Fields up to 6 T were provided by a superconducting magnet. The only change in the sample-holder arrangement was the use of a small piece of a Matsushita resistor as a thermometer because of its low magnetoresistivity. It was calibrated in various magnetic fields against a Ge resistor located in a compensated low-field region of the magnet.

Figure 1 shows the specific heat C of an a-Ge film measured without an applied magnetic field and in a field of 5 = 6 T. For comparison, C for one a-Ge film measured earlier^ (sample 3 of Ref. 5) in zero field is indicated. Data for sample 1 of Ref. 5 coincide with the present data. Hence the zero-field data agree quite well, showing a roughly linear specific heat at our lowest temperatures which levels off near 0,4 K and rises again more steeply above 1 K. More accurately, C =^ aT" with AZ ~ 1; see Table I. The specific heat is strongly affected upon application of a large magnetic field. As is evident from Fig. 1, the linear term at low temperatures disappears, and the specific heat be­comes very small below 1 K. In fact, it follows a T^ temperature dependence and the magnitude of this Debye-type T^ specific heat agrees reasonably well with an extrapolation of earlier data^ obtained above L5 K; cf. the dash-dotted line in Fig. L

10"

o> 10

10

I I I I I 111

B=OT B=6T

^:

1°

I i ml J I I I M III J L 0.1

T(K)

FIG. 1. Specific heat C vs temperature T of an evaporated amorphous Ge film in zero magnetic field B and in ^ = 6 T. The dashed line represents earlier zero-field data for other a-GQ films (Ref. 5), The dash-dotted line represents an ex­trapolation of the zero-field T^ contribution as measured above 1 K (Ref. 1 ).

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The strong magnetic field dependence of C below 1 K convincingly shows that the linear term observed for 5 = 0 must be of magnetic origin, as has been conjec­tured earlier.^ Because the tunneUng-states model as it currently stands predicts an atomic tunneling motion, the strong magnetic field dependence definitely rules out an interpretation of the linear term in our a-Ge samples in terms of tunneling states. From our data in 6 T, we can establish an upper limit for the coefficient of a magnetic-field-independent linear term which might be attributable to TLS, ^XLS < 1 X 10""^ J/g K l This is more than an order of magnitude smaller than a XLS ii vitreous silica, indicating that an overcon-strained random network such as a-Ge does not sup­port TLS.^^ In this respect it is interesting to mention recent thermal-conductivity measurements^ on a-Ge from which some evidence for TLS was inferred. In particular, it was found that the TLS density in U-GQ depends on the sample's mass density, with the TLS density decreasing when the mass density increases to­ward that of crystalline Ge. Our measurements lend strong support to the conjecture^ that in a dense tetrahedral random network the TLS density is indeed largely suppressed. In addition, they show that high-field calorimetry is capable of separating a possible^ TLS contribution to the specific heat in less dense a-Ge samples from that of magnetic excitations.

The vanishing of the linear specific heat in high magnetic fields also rules out any sizable contribution from excitations of localized electrons across the Fer­mi level i?F- This is supported by the fact that the electronic density of states at ^p as estimated from the T" /"^ law for the hopping conductivity would yield^^ a linear specific heat about 3 orders of magnitude lower than measured in i? = 0. The possible opening of a Coulomb gap ^ at Ej: for T—^ 0 in a-Ge would de­crease the specific heat even further.

Figure 2 shows the specific heat of a-Ge in inter­mediate fields. The zero-field linear term at low tem­peratures is largely diminished already in 2 T, and in­stead a hump in C builds up around 1 K, suggestive of a Schottky anomaly. This hump is shifted to higher temperature for 5 = 4 T. On closer inspection of Fig. 1, a small hump can also be seen for 6 T above 1.5 K, although this is almost within the scatter of the data because of the rapidly rising T^ specific-heat contribu­tion.

The excess specific heat AC =-C — (37^, where /8= 1.5x10"^ J/g K" , indeed closely resembles a Schottky anomaly with energy splitting AE ^gfi^B. As an example. Fig. 3 shows AC for B = 2 T. Note that our Schottky fit for two energy levels with equal degeneracy (solid line) has only one adjustable param­eter, i.e., the height of the maximum, since the posi­tion of the maximum is determined by the applied magnetic field. Hence AC arises from the Zeeman splitting of dangling-bond states. This is corroborated

VOLUME 55, NUMBER 22 P H Y S I C A L R E V I E W L E T T E R S 25 NOVEMBER 1985

10*

M i i i { r—I M M i i | r j p ~

LUJl _i i_ 0.1

T(K)

FIG. 2. Specific heat of a-Ge in magnetic fields of 0, 2, and 4 T. The solid lines are intended as guides to the eye.

by the agreement between the number of Zeeman-split states contributing to the specific heat as calculated from the Schottky maximum (e.g., A^sch= 8.5x 10^^ cm~^ for sample 2) and the spin density as determined from ESR measurements (A^ESR = 5 .4X 10^^ cm~^). Since the ESR determination of spin densities is known not to be very accurate, the agreement is quite satisfactory. A similarly satisfactory agreement has also been found for the other films; see Table I.

From our measurements, a coherent description of the low-lying magnetic excitations in a-Ge is possible. For isolated dangling bonds in zero applied field, only the dipolar interaction between spins or the hyperfine interaction can lift the spin degeneracy. The corre­sponding energy splittings are a few millikelvins at most,^^ i.e., at least an order of magnitude below our lowest accessible temperature. Hence the linear specific heat in zero field must be attributed to exchange-coupled clusters of dangling bonds with a constant distribution P(^.E) of exchange splittings, as conjectured earlier.^ As is evident from Fig. 1, the linear specific-heat term has a cutoff near 0.4 K for room-temperature-evaporated films. This suggests an upper limit for the exchange splitting AiS' ax == 1 K. For the sake of simplicity, we will assume in the fol­lowing qualitative discussion that the predominant clusters will be pairs of dangling bonds. The general line of the argument should be unaltered by an exten­sion to larger clusters while quantitative calculations will become very difficult. In a magnetic field, the

FIG. 3. Excess specific heat AC (T^ contribution sub­tracted) for B^2 T. The solid line is a fit of a Schottky function with A^ = gfiBB to the data.

singlet ground state of a pair of dangling bonds will be unaffected while the triplet state (separated from the former by AE in zero field) will split into three Zee-man levels. For gfjusB » AE, the occupation proba­bility for the singlet state will be very small for k^T < AE, and hence there will be no contribution to C from this pair at these low temperatures. Thus for g/jL^B > A£'jnax5 the linear specific heat at low tem­peratures will disappear altogether as is indeed ob­served. Of course, as T approaches g^^B/k^, transi­tions between the magnetic-field-split Zeeman levels become possible, leading to a Schottky-type anomaly (broadened somewhat because of the distribution of exchange splittings) with a maximum just at the same position as the Schottky anomaly due to single dan­gling bonds discussed above. As can be seen from Fig. 3, the measured specific heat exceeds the Schottky curve somewhat at low tempertures. This could be due to the fact that the condition g^x^B » A^^^^x is not fulfilled for 2 T.

From the entropy 5', we can calculate the number of spins Ns==S/(k^ ln2) contributing to the linear specif­ic heat up to the cutoff. As can be seen from Table I, Ns is about 10% of the total number of spins A' sch-The rapid decay of the exchange interaction with dis­tance prevents most of the spins from forming clus­ters. The fact that P(AJE') = const must be attributed to a cancellation of the strong variation of J(R) by the distribution function P(R) of distances R between dangling bonds. The question of whether eventually a dipolar freezing of the more distant spins sets in at very low temperatures must remain open.

Finally, we note that the linear specific heat in our amorphous spin-y system bears strong resemblance to the well-known linear specific heat in spin glasses. Our system, however, differs from "typical" spin glasses where either long-range interactions exist (as in Ruderman-Kittel-Kasuya-Yosida systems) or—in the case of short-range interactions—the concentration

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V O L U M E 55, N U M B E R 22 P H Y S I C A L R E V I E W L E T T E R S 25 NOVEMBER 1985

of magnetic moments exceeds the percolation thresh­old (as in, e.g., Eu^^Sri-^j-S). In the present case, the linear specific heat simply stems from a constant distri­bution of singlet-triplet splittings due to the inherent spatial disorder. The freezing of relatively close spins into local singlet pairs due to the short-range exchange interaction prevents the system from going into a more cooperative spin-glass-like state. For more dis­tant pairs, the magnitude of the exchange interaction rapidly drops below the other interactions present such as dipolar and hyperfine interactions.

In conclusion, we have shown that the low-temperature specific heat of a tetrahedral amorphous semiconductor, a-Ge, can be quantitatively attributed to magnetic excitations arising from dangling bonds, with negligible contributions from electronic and struc­tural excitations (TLS). In particular, the absence of TLS in a-Ge provides a stringent test for future micro­scopic theories of these excitations.

We acknowledge helpful discussions with Professor W. Sander. We are grateful to Dr. W. Beyer and Dr. J. Kastner for their help with sample preparation, and Dr. R. Roll for the ESR measurements . A useful dis­cussion with Dr. L. Schweitzer is also acknowledged. This work was performed within the research program of Sonderforschungsbereich 125, supported by the Deutsche Forschungsgemeinschaft.

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