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Ion-implanted, shallow-energy, donor centres in diamond: the effect of negative electron affinity
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Nuclear Instruments and Methods in Physics Research A 514 (2003) 69–78
Ion-implanted, shallow-energy, donor centres in diamond: the effect of negative electron affinity Johan F. Prins* Department of Physics, University of Pretoria, Pretoria, Gauteng 0002, South Africa
Abstract A summarised overview of selected results is presented, and their modelling, which have been generated by the research program pursued by myself in South Africa. It is shown that shallow donor states can be inserted into diamond by ion implantation of either oxygen- or nitrogen ions followed by low temperature (o600 C) annealing. It is argued that such low-temperature annealing is required to ‘‘quench’’ these donor flaws into the diamond lattice, because their energy levels are higher than the vacuum level. An analysis of the interface between such an n-type diamond and the vacuum, based on the accepted principles of band theory, leads to the additional conclusion that a dipole layer has to form at the surface. As in the case of a Schottky diode, this dipole generates a barrier to electron migration, from the n-type diamond. However, in contrast to a Schottky diode, a forward potential does not generate a forward current, but rather increases the barrier to electron egression. In order to ensure that electrons can overcome this barrier, the surface of the semiconductor has to be doped to a high density, such that electrons can tunnel out of the diamond’s surface. Electron-current flow into the anode initiates when these extracted electrons fill the whole gap between the diamond surface and the anode. All experiments, to date, show consistently that, after current flow has initiated, the electrons between the diamond surface and the anode are able to form a stable, highly conducting phase. By applying accepted, and proven, concepts from thermodynamics and quantum mechanics, it is concluded that the formation of such a phase is inevitable, and that it has to be superconducting. r 2003 Elsevier B.V. All rights reserved. Keywords: Diamond; n-Type doping; Ion implantation; Superconduction PACS: 74.70; 85.45.D; 68.55.L; 81.05.U
1. Introduction Owing to its extreme physical properties, diamond may find electronic applications, which are not possible when using silicon or any other semiconductor material. However, to generate *Corresponding author. Tel.: +27-11-477-8005; fax: +2711-477-3709. E-mail address: [email protected] (J.F. Prins).
useful electronic devices, controlled doping, with both acceptors and donors, is required. Substitutional boron atoms form acceptor centres in diamond situated at 0.37 eV above the valence band maximum [1]. These atoms can be incorporated into the diamond lattice during CVD growth [2] as well as using ion implantation [3]. Substitutional nitrogen atoms form donor levels in diamond, but they are situated at E1.7 eV below the conduction band minimum [4]. This is too deep
0168-9002/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2003.08.085
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to provide an adequate supply of electrons within the conduction band, at room temperature. At present, n-type diamond, ascribable to the incorporation of phosphorus, can be generated, either during CVD growth [5] or by ion implantation [6]. The ionisation energy, of this donor level, is 0.62 eV [7]. Shallower, donor levels are required if one wishes to make commercially viable electronic devices. If shallow donor levels could be generated to a high enough density, diamond substrates may find application as cold cathodes. Himpsel et al. [8] have found that electrons, which are photo-excited into the conduction band of boron-doped, p-type diamond, can exit the surface of the diamond without encountering any energy barrier, provided the surface is terminated with hydrogen atoms. The diamond substrate has negative electron affinity (NEA), in the sense that the electrons, which, in this case, have been photo-excited into the conduction band, find themselves at a higher energy than the vacuum level. This result has led to the conclusion that if one could find an n-type semiconductor with NEA, one should be able to extract conduction electrons into the vacuum, at room temperature, without needing a high voltage to generate field emission over a barrier. However, to date this type of behaviour has not been realised in diamond or, for that matter, in any other possible material. A consensus seems to be emerging that diamond may not be an inherent NEA material as anticipated, and that the NEA observed and reported for p-type, hydrogenterminated diamond, is an artefact generated by the dipole character of the C–H bonds at the surface [9]. However, my studies indicate that diamond is indeed an inherent NEA material, and that it could be this property that has made it difficult to generate shallow donors. In an NEA material, shallow donor levels should be situated at a higher energy than the vacuum level, and, because of this, it might only be possible to obtain them by generating thermodynamically, metastable donorlike flaws in the lattice. In this short overview, it will be shown how the latter assumption has led to the generation of shallow donor states by means of oxygen- or nitrogen-ion implantation. Next, a
band-structure analysis will be discussed, from which it has been concluded that electron extraction from such n-type diamond can only occur by tunnelling through a depletion layer at the surface, and not in the manner generally anticipated in the literature. In turn, it will be shown that the actual behaviour of electrons, extracted into the gap between the diamond surface and an anode, is also different from the anticipated behaviour.
2. Ion implantation and formation of shallow donor levels A short review of ion implantation into diamond for electronic applications has recently been published [10], and will thus not be repeated here. Suffice to point out that the major stumbling block, when attempting to dope diamond by means of ion implantation, is the thermodynamic metastability of this material. If the kinetic conditions are suitable, it will revert to a graphitic-type structure. To circumvent the problems caused by this metastability, two routines have been developed to dope diamond. They are known by the acronyms CIRA (Cold-Implantation-Rapid-Annealing) and LODDI (Low-Damage-Drive-In). In the CIRA routine, implantation is done at a low enough temperature, preferably liquid nitrogen or lower, to ensure that the point defects, generated by the impinging ions, do not diffuse during the implantation process. This is followed by rapid thermal heating to a suitable temperature at which the interstitial atoms can diffuse. When using the LODDI routine, a region, or layer, is generated by irradiation, such that it contains a suitable (low) density of vacancies. The dopant atoms are then implanted in an adjacent region, or layer, to allow some of the interstitial atoms (self and dopant), generated within this layer, to diffuse into the initial layer where they can meet up with the, previously generated, vacancies. In a comparative experiment using O+, B+ and + C ions for CIRA implantations with an annealing temperature of only 500 C, it was found that the oxygen-treated diamond layers conducted n-type with an activation energy of E0.32 eV
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[11]. As already mentioned in the introduction, the low annealing temperature was used in an attempt to ‘‘quench’’ in high-energy metastable donor states with concomitant high-energy electronic levels, which lie near to the conduction band. In a follow-up study [12], it was found that even shallower donor levels (at E0.28 eV) could be generated by similar nitrogen-ion implantation. These centres cannot relate to substitutional nitrogen, because, as mentioned above, it is known that the latter donor levels are situated deeper (E1.7 eV) below the conduction band. It has now been proposed that such ‘‘quenched-in’’ donor centres could consist of dopant atoms trapped in interstitial sites adjacent to vacancies. Further studies are in progress to study the metastable nature of these centres [13].
3. Donor levels above the vacuum level If donor levels are situated above the vacuum level, those electrons, thermally activated from them into the conduction band, find themselves also above the vacuum level. The electron-energy band diagram, in Fig. 1, shows the situation, which has been anticipated in the literature under such circumstances. It is believed that an anode would easily extract electrons (injected into the conduction band via an ohmic ‘‘back’’ contact) out of the front surface without encountering any energy barrier. However, the important point to note is that, according to the second law of Thermodynamics, this diagram is unrealistic, because it represents a transient, non-equilibrium situation. If Fig. 1 were applicable, it would mean that, owing to the offset w between the conduction band and the vacuum level at the surface, conduction electrons, arriving at the surface, would encounter a field accelerating them out of the substrate, even if there were no extraction field being applied. Such electrons thus make transitions from the conduction band to the vacuum level. By doing this, they leave positive charges behind which reside on ionised donor levels. These charges will attract the electrons back to the surface. However, as long as the vacuum level lies below the donor
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Surface
-
conduction band Fermi level
NEA = χ
vacuum level
n-type NEA semiconductor
vacuum
valence band
0
z
Fig. 1. The electron-energy band diagram at the surface of an n-type semiconductor with NEA, which is generally anticipated in the literature. Owing to the second law of Thermodynamics, this situation is not possible, because the energy levels are not in thermodynamic equilibrium (see text).
levels, these electrons will not be able to re-enter the diamond surface because all the available energy levels, which they can occupy, are at too high an energy. The electrons, accumulating outside, in the vacuum, will experience the positively charged surface as a two-dimensional ‘‘atomic nucleus’’ that causes them to become bound within quantum states [14]. These states will be situated within an ‘‘electron-charge layer’’ of width we adjacent to the surface. In turn, the ionised donors, left behind by the emitted electrons, have to accumulate within a depletion layer at the surface. A dipole, thus, develops which, after reaching the required strength, screens the field caused by the initial potential offset w; to, in this way, achieve thermodynamic equilibrium at the surface-to-vacuum interface. The latter dipole is similar to the one generated at a p–n-junction, except that the negative layer is now not a depletion layer within which the negative charge resides on ionised acceptors. An analysis of the equilibrium situation at room
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temperature renders the energy band diagram shown in Fig. 2 [14]. The electrons, furthest from the surface, are distributed around a Fermi level, which lines up with the Fermi level within the substrate. Those electrons, with the highest energy above the Fermi level, are at the vacuum level, or else they will not screen the electric field within the dipole. However, they are not free to drift away, because they are bound by the self-consistent potential well, which retains all the electrons at the surface within the field of the positive depletion layer. Nearer to the surface of the n-type substrate, the electrons are even more tightly bound within lower energy quantum states. In turn, within the positive depletion layer, band bending occurs. This forms a barrier to further net electron flow from the conduction band to the vacuum. Owing to the similarities between this dipole layer and a p–n-junction, one expects that when a forward potential appears over the dipole, the barrier to electron migration out of the n-type Surface
Initial energy position of the conduction band conduction band
++++++ Fermi level
n-type NEA semiconductor
----
NEA = χ vacuum level layer of electrons bound at surface self-consistent potential well
vacuum
valence band
electron-charge layer
depletion layer
-wn
we 0
z
Fig. 2. An electron-energy band diagram at the surface of an ntype semiconductor with NEA, which is in thermodynamic equilibrium, as required by the second law of Thermodynamics. A dipole has formed at the surface consisting of a positive depletion layer, just under the surface, and an ‘‘electroncharge’’ layer just outside the surface. The latter electrons are bound within quantum mechanical states. This dipole screens the field that would have been present, owing to the energy offset w; between the conduction band and the vacuum level.
substrate should immediately decrease, such that a current will flow. However, this does not happen in the present case, as one can see when the two situations are compared: (a) The dipole of a p–n-junction is sandwiched between two conductors within which the fields, caused by an applied potential, may be neglected compared to the field that appears over the insulating dipole. The latter field lowers the Fermi level in the p-type material, relative to the Fermi level in the n-type material, making it possible for electrons to flow from the power supply, through the ntype semiconductor to the dipole interface. At the interface, they are pulled through the dipole, by the applied field within it, into the p-type semiconductor, through which they can flow back to the power supply. (b) In the case of electron-extraction from an NEA, n-type substrate, the dipole is sandwiched between the n-type semiconductor and the vacuum. The latter has no free carriers and is thus an insulator. Thus, the applied field, generated by an anode potential F; appears over both the dipole and the region between the anode and the electron charge layer. This effectively lowers the vacuum level outside the electron-charge layer. It should now be recalled that the reason why the dipole formed, in the first place, was to screen the field that had been caused by the potential energy offset w relative to the vacuum level. Thus, any additional, applied field will have a similar effect. In order to reach a state of thermodynamic equilibrium, both the depletion layer and the electron-charge layer will increase further in width from wn to wnf ; and we to wef ; respectively, until the applied field is also screened. The energy band diagram, after this has been achieved, is shown in Fig. 3. It should be noted that an extraction field only appears after the applied potential has exceeded a critical value FW ¼ W DEF ; where W is the work function of the anode metal, and DEF is the difference in energy between the conduction band edge and the Fermi level of the n-type substrate [15]. The potential FW
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Initial energy position of the conduction band
Surface
χ χeff conduction band
++ + + ++ ++ Fermi level
n-type NEA semiconductor
e(Φ -ΦW)
-vacuum level -with field -vacuum -Fermi level
a n o eΦW d e
of anode electron-charge layer
valence band depletion layer
-wnf
0
z
wef
Fig. 3. The electron-energy band diagram, when applying an electric field to an NEA, n-type semiconductor, in an attempt to extract electrons into the vacuum. Owing to the workfunction of the anode, an extraction field only appears after the applied potential F exceeds a critical magnitude FW : The applied field then lowers the vacuum level outside of the semiconductor surface, and the dipole at the surface expands to screen this additional offset eFVAC ¼ eðF FW Þ in the potential energy. In other words, the applied potential effectively increases the original NEA w to have a value of weff ¼ w þ eFVAC :
is needed to initially align the vacuum levels of the substrate and the anode. Accordingly, the field, which needs to be screened, is generated by the potential FVAC ¼ F FW ; as shown in Fig. 3.
4. Electron tunnelling into the vacuum Because the Fermi level of an NEA, n-type semiconductor will always be at a higher energy than the Fermi level of any metal, the only way to establish an ohmic contact to a metal would be to generate a tunnelling junction between the two materials. One method is to dope the surface and near-surface region with a very high density of shallow donors before depositing the metal. Although an ‘‘ohmic contact’’ to an insulator does not make any sense, my research has led to the postulate that the only way to ensure that electrons keep on flowing easily from the conduction band of an NEA, n-type semiconductor into
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the vacuum, would be to, similarly, generate an ‘‘ohmic’’-tunnelling region or ‘‘contact’’ [15]. For example, doping the surface region to a high enough density ensures that the depletion layer, which has to develop at the surface of such a semiconductor, must be very narrow, such that when applying an extraction field, by means of an anode, the electrons are able to easily tunnel through it into the electron-charge layer. By increasing the applied voltage to the anode, more electrons should keep on adding to the electron-charge layer, in this way causing it to expand towards the anode. At some critical applied potential FC ; the electron-charge layer should just fill the whole distance dgap between the semiconductor surface and the anode, as shown in Fig. 4. A catastrophic, metastable equilibrium point is then reached, and this will be disturbed when the potential F is increased to be larger than FC : The self-consistent potential well will then expand into the anode. The anode is a conductor, and the self-consistent potential well cannot be maintained within it. Thus, the electrons, which enter the anode, become free to move. An electrical contact is established between the anode and the electron-charge layer, which must result in a field being established to attract electrons into the anode in order to equalize the Fermi level outside its surface to the one within the anode. This field will add to the field, which the dipole expansion is trying to neutralize. A direct field should, thus, be established owing to the offset between the Fermi levels at the surface of the diamond and the Fermi level in the anode. In other words, the potential, FVAC ¼ ðFC FW Þ; now adds to the offset FW between the Fermi levels, in order to increase this offset to FC : This is also shown in Fig. 4. At this point it should be pointed out that for a large enough gap distance dgap ; the applied potential might become so large, that the electron-charge layer breaks down before it can fill the whole gap [14]. However, if this happens, the electrons will still experience the field caused by the offset between the two Fermi levels. Electric current flow will initiate. At position z; as measured from the substrate surface, the potential increase FðzÞ is now equal to zEgap ; where Egap is the field between the diamond
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74 Initial energy position of the conduction band
JðzÞ may be written as
Surface
2eFðzÞ JðzÞ ¼ enðzÞvðzÞ ¼ enðzÞ m
original position of vacuum level
n-type NEA semiconductor gap filled with electrons
conduction band
e(ΦC - ΦW )
a n o d e
Fermi level
eΦW eΦC valence band
a n o d e
0
z
Fig. 4. The electron-energy band diagram, at the critical applied potential FC ; where current flow, between the semiconductor surface and the anode, initiates. The selfconsistent potential well, within which the electrons were bound within quantum states (see Figs. 2 and 3), has now expanded to fill the whole gap. A further slight increase in potential allows the electrons at the Fermi level of the electroncharge layer, to enter the metal anode. An electrical contact is established between the semiconductor surface and the anode. The whole applied potential FC now manifests itself as an offset between the Fermi levels of the semiconductor and the anode, which generates a field that makes current flow possible. The electrons are now not bound anymore and thus act as charge carriers. The depletion layer is shown with a very narrow width, because it is assumed that the region, just below the surface, has been doped to a high density in order to facilitate tunnelling of the electrons through this layer (see text).
and the anode caused by the offset between their respective Fermi levels. The gain in kinetic energy Tkin ðzÞ for an electron, with mass m; which enters the gap, may be written as 1 pðzÞ2 Tkin ðzÞ ¼ mvðzÞ2 ¼ ¼ eFðzÞ ¼ ezEgap : 2 2m
ð1Þ
Because the electrons enter the gap from the minimum energy of the conduction band, their initial speeds are equal, or very near, to zero. Thus vðzÞ and pðzÞ can be taken as the actual speed and momentum of the electrons at position z: If the density of the electrons is nðzÞ; the current density
1=2 :
ð2Þ
This equation proves that, even with no atoms in the gap, a potential is still needed to drive the current through the gap, i.e. the gap with electrons has an effective resistivity rðn; FÞ: Thus, if after reaching the critical juncture, demonstrated in Fig. 4, the electron density nðzÞ increases, the resistivity within the gap will decrease. The applied potential will, accordingly, redistribute over the diamond and the gap, to drive an increased current around the circuit.
5. Electron extraction from n-type diamond with an ‘‘ohmic’’-tunnelling contact to the vacuum (experimental results) The following results are from the first, pioneering experiment using low-energy oxygen-ion implantation [15]. Since then, similar results have been repeatedly obtained when doping diamond with either oxygen- or nitrogen-ion implantation to generate shallow donors. This substrate was a high-purity (natural type IIa) diamond with an area of 3.6 3.6 mm2. It was decided to start off with ion energies well below 1 keV in order to just dope the very-near surface region. The ions were extracted through a hole, from a direct-current (DC) oxygen-plasma, into the diamond surface, using an extraction potential of 150 V, and the current density measured from the diamond, to the ground potential, E0.3 mA cm2. The diamond was then annealed up to E400 C, while monitoring its resistance with a potential of 10 V across it. After cooling, the sheet resistance was E5 106 O: After another anneal for 3 h at 400 C, in a high-purity argon atmosphere, without applying a potential, the sheet resistance was found to have increased to E1 1011 O, and the diamond conducted like an oxygen-doped diamond, which had been doped by keV ion implantation. This result indicates that the injected oxygen ions did not just form donor states very near to the surface, but also deeper lying donors owing to LODDI-type oxygen-interstitial
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in-diffusion, and that, during annealing, without an applied potential, the near surface states are less stable than these deeper-lying ones. The diamond was again treated with low-energy oxygen ions, and annealed by heating it in vacuum with an applied potential of 10 V. Afterwards, the sheet resistance was E1.75 106 O. From the results above, the deduction was made that the implanted surface then, most probably, consisted of a deeper lying, lighter-doped, n-type layer and an overdoped surface layer that may be able to form an ‘‘ohmic’’-tunnelling contact to the vacuum, provided the diamond is an NEA material. A typical result obtained when applying a field, with a gold-plated anode sphere of radius 0.5 mm, is shown in Fig. 5. The measurements were made while maintaining the distance between the diamond surface and the anode at 10 mm. At first, while increasing the positive potential on the 0.6 0.4
Current (mA)
0.2
Gap width: dgap = 10 µm Behaviour after current flow has stabilised
0.0 Critical potential at which current starts flowing
-0.2 -0.4 -0.6 -1000
-500
0
500
1000
Probe ("anode") potential (V) Fig. 5. Experimentally measured electrical current when a potential is applied to an anode, in order to extract electrons out of the surface of an n-type, diamond doped, to a very high donor density near its surface, by means of oxygen-ion implantation. The distance between the diamond surface and the spherical anode (with radius 0.5 mm) was 10 mm. Initially, no current flow occurred with increasing positive potential, until a critical potential of E420 V had been exceeded. After the current had then been allowed to stabilize, its behaviour was the same whether the potential was positive or negative. It is argued that the latter behaviour is only possible if the electrons, between the diamond and the anode, have formed a superconducting phase.
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anode from zero to higher values, no current flow occurred, until at E420 V, current flow suddenly initiated. From the discussion above, one may conclude that, at this point, the self-consistent potential well, and, thus, the electron-charge layer, broke down. Once current flow had initiated and stabilised at higher voltages, the potential was changed between +1000 V (the maximum of the power supply) in steps down to 1000 V, while monitoring the current. Current could now be measured for potentials lower than 420 V, as well as negative potentials (see Fig. 5). For the same magnitude of the potential, the same current flow occurred in both directions when the polarity was changed from positive to negative. It was also found that one could now switch off the potential, even for a week and longer, without the conducting material, within the gap, disappearing. When subsequently switching on, at any potential between 1000 and 1000 V, the current started flowing. Furthermore, when pushing the probe against the surface, exactly the same curve was measured; i.e. the current was, within experimental error, the same, whether there was a gap or not. Even when breaking the vacuum, and removing the system from the vacuum chamber, the stable conducting material could persist. One could then actually observe the gap between the diamond and anode probe. No extraneous material could be seen within this gap. Furthermore, subsequent, careful microscopic examination of the diamond and probe surfaces showed no evidence, whatsoever, of any contamination, like, for example, a filamentary material, which could account for the observed conduction. The experiment has been repeated on a large number of alternative substrates. Not in one of the latter cases could the same, or even similar, results be obtained. Only when diamond substrates were employed, where their surfaces had been highly doped with shallow donors, could commensurate results be measured. Furthermore, in all the latter cases the current flow was not a function of the gap size between the diamond substrate and the anode, but related directly to the resistivity of the diamond. This means that the voltage drop over the gap was negligibly small: within experimental error, zero.
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After exhaustive, multiple experiments, the only acceptable conclusion that could be reached has been that the stable, highly conducting phase, which formed within the gaps between the n-type diamonds and their respective anodes, has to consist entirely of electrons.
6. The requirement for superconduction to establish a steady-state current As already discussed above, the situation depicted in Fig. 1 is not physically possible. The dipole structures, shown in Figs. 2 and 3, have to form in order to establish equilibrium, as required by the laws of Thermodynamics. The reason, why these dipoles form, is the presence of unscreened fields at the surface. Thus, when a field appears, for any reason, at the surface, the depletion layer will again start to inject more electrons, into the vacuum, in an attempt to also eliminate this field. Current flow initiates after the re-appearance of a field at the surface, as shown in Fig. 4. The depletion layer at the surface will, thus, react to eliminate this field, and the electron density within the gap will, in concert, start to increase further. According to Eq. (2), an increase in the density of the electrons causes either the current to increase or the potential over the gap to decrease. In fact both happens [15]. Thus, as long as the depletion layer adds electrons, steady-state current flow, as thermodynamically mandated, cannot be achieved. This leads to the unequivocal conclusion that a steady-state condition can only be generated when the field, within the gap, becomes zero while, at the same time, a current still flows between the diamond and anode. The only material that can fulfil both conditions at the same time is a superconductor. Accordingly, the second law of Thermodynamics has to fail, or Mother Nature has to find a way to form a superconducting phase, which has to consist entirely of electrons. In this respect, it is relevant to consider the BCS theory for superconduction [16] within metals at low temperatures. The electrons are believed to interact via phonons to form so-called Cooper pairs. Because the individual electrons are fermions with half-integral
spin, the Cooper pairs have integral (zero) spins. They can thus act like bosons, which are able to form a collective state represented by a single, coherent wave function: akin to Bose–Einstein Condensation of atoms during cooling to temperatures near absolute zero [17]. It is this state that is superconducting.
7. Electron pair formation without phonon interaction In the present situation, the electrons cannot form pairs by the Cooper mechanism, because there are no vibrating atoms within the gap between the diamond cathode and the anode, and, thus, also no phonons. One may consider the possibility of magnetic coupling between two electrons owing to their spins. However, it is well known that this would be a very weak interaction [18]. Except if one wants to re-consider the concept of ‘‘hidden variables’’, the needed interaction, which could lead to the required pair formation, should rather be sought as a logical consequence of the existing interpretation of Quantum Mechanics. The Pauli Exclusion Principle manifests itself as a much stronger, apparent force, which causes two electrons to pair (spin up and spin down) when they are occupying the same quantum orbital. In fact, the strongest chemical bond, namely the covalent bond, is formed by this interaction. For an average electron density ngap ; between the n-type diamond and the anode, the average volume per electron becomes n1 gap : Thus, the average distance between electrons in the z1=3 direction follows as Dz ¼ ngap : Because, according to the Pauli Exclusion Principle, electron pairs, with spin-up and spin-down, can occupy the same limited space, the uncertainty /DzS of the position of an electron, may be written as 2 1=3 /DzS ¼ : ð3Þ ngap From Eq. (1) the quadratic uncertainty in the momentum of such an electron may be written as /Dp2z S ¼ 2me/DFðzÞS ¼ 2meEgap /DzS:
ð4Þ
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Thus, the Heisenberg Uncertainty Relationship may be written as [19] /Dp2z S/Dz2 S ¼ 4meEgap n1 gap X
h2 16p2
ð5Þ
where h is Planck’s constant. But, as concluded above, the average density ngap must keep on increasing as long as Egap a0: A point must, thus, be reached at which ngap becomes so large, for the prevailing value of Egap ; that the left-hand side becomes equal to the right-hand side in Eq. (5): i.e. a further increase in ngap has to violate Eq. (5). At this critical point, each electron pair is forced to stay within the coordinate constraints imposed by the latter equation: i.e. each pair occupies a new type of spatially restricted ‘‘orbital’’, and the latter ‘‘orbitals’’ fill the space between the diamond surface and anode completely. If one of the electrons attempts to move as an independent entity, it implies that it has to exit the volume it shares with its partner and enter an adjoining volume. However, the latter volume will now already be occupied by an electron pair. The Pauli Exclusion Principle will thus prevent a third electron from entering. Thus, any current within the gap has to, henceforth, involve the combined motion of electron pairs; i.e. each pair, or ‘‘orbital’’ containing two electrons, has to move as a single charge carrier with a charge of 2e; and these charge carriers have integral (zero) spin; they are boson-like entities.
8. Formation of a Bose–Einstein-like Condensate After extending the statistics developed by Bose, for photons, to particles with mass, and before it was known that there are two types of particles with half-integral and integral spins, Einstein [17] postulated the formation of (what is today known as) a Bose–Einstein Condensate, as follows: ‘‘When a given number of particles approach each other sufficiently closely, and move sufficiently slowly, they will together convert to the lowest energy state possible’’. It is now known that these particles have to be, or act like bosons. It is generally accepted in the literature that the formation of such a condensate requires the
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constituent particles to be cooled to extremely low temperatures near absolute zero. One should now note that, in the present case, these exact conditions are approached for the charge carriers injected into the gap between the ntype diamond substrate and the anode. And this occurs without cooling! Furthermore, because a threshold condition can be reached where the electrons form pairs, a Bose–Einstein-like Condensate becomes possible at room temperature. This implies that the speeds of the electron pairs are now, within the limits of the Heisenberg Uncertainty Relationship (as applied to bosons for which the uncertainty in position is the same as the gap width), the same at the cathode as at the anode. No acceleration takes place anymore, which, in turn, implies that, although a current is flowing, there is no field within the gap. A superconducting state has formed, and steadystate current flow can proceed, as required by the laws of Thermodynamics.
9. Summary and conclusion Diamonds can be doped by ion implantation to generate metastable donor flaws with donorelectron energy levels situated near to the conduction band edge. Experiments on electron extraction from such n-type diamonds, where their surfaces had been doped to a high density with shallow donors, consistently show that a stable, highly conducting material, consisting entirely of electrons, form between the diamond’s surface and the anode. Furthermore, the resistance of this material is independent of the gap width between the anode and the diamond. In fact, it is the same as when the anode touches the diamond, thus indicating that it might be zero. A theoretical analysis based on the accepted principles, used in the band theory of solids, leads to the unequivocal conclusion that a steady-state current can only flow into the anode, if the field, within the gap between the diamond and the anode, becomes zero. Furthermore, the conditions required for the formation of such a superconducting material follow logically when the Pauli Exclusion Principle and Heisenberg Uncertainty Relationship are
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applied to the situation. The electrons are forced to pair and form boson-like entities, which then, in turn, form a Bose–Einstein-like Condensate at room temperature. The experimental results, as well as the theoretical analysis, independently and jointly, provide compelling evidence that a superconducting phase can form, and actually has formed, at room temperature.
References [1] A.T. Collins, A.W.S. Williams, J. Phys. C 4 (1971) 1. [2] N. Fujimori, H. Nakahata, T. Imai, Jpn. J. Appl. Phys. 29 (1990) 824. [3] J.F. Prins, Mater. Sci. Rep. 7 (1992) 271. [4] R.J. Farrer, Solid State Commun. 7 (1969) 685. [5] S. Koizumi, M. Kamo, Y. Sato, H. Ozaki, T. Inuzuka, Appl. Phys. Lett. 71 (1997) 1065.
[6] J.F. Prins, Diamond Relat. Mater. 8 (1999) 1635. [7] H. Sternschulte, K. Thonke, R. Sauer, S. Koizumi, Phys. Rev. B 59 (1999) 12924. [8] F.J. Himpsel, J.A. Knapp, J.A. van Vechten, D.E. Eastman, Phys. Rev. B 20 (1979) 624. [9] J.B. Cui, J. Ristein, L. Ley, Phys. Rev. Lett. 81 (1998) 429. [10] J.F. Prins, Semicond. Sci. Technol. 18 (2003) S27. [11] J.F. Prins, Phys. Rev. B 61 (2000) 7191. [12] J.F. Prins, Semicond. Sci. Technol. 16 (2001) L15. [13] J.F. Prins, Diamond Relat. Mater. 11 (2002) 612. [14] J.F. Prins, Semicond. Sci. Technol. 18 (2003) S125. [15] J.F. Prins, Semicond. Sci. Technol. 18 (2003) S131. [16] J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 108 (1957) 1175. [17] A. Einstein, Sitzungber. Preuss. Akad. Wiss. 1925 (1925) 3. [18] R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics. Vol. III: Quantum Mechanics Addison-Wesley, Reading MA., 1965, p. 4/14 [19] W. Greiner, Quantum Mechanics: An Introduction, Springer, New York, 1994, pp. 47–50.
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Ion-implanted, shallow-energy, donor centres in diamond: the effect of negative electron affinity Johan F. Prins* Department of Physics, University of Pretoria, Pretoria, Gauteng 0002, South Africa
Abstract A summarised overview of selected results is presented, and their modelling, which have been generated by the research program pursued by myself in South Africa. It is shown that shallow donor states can be inserted into diamond by ion implantation of either oxygen- or nitrogen ions followed by low temperature (o600 C) annealing. It is argued that such low-temperature annealing is required to ‘‘quench’’ these donor flaws into the diamond lattice, because their energy levels are higher than the vacuum level. An analysis of the interface between such an n-type diamond and the vacuum, based on the accepted principles of band theory, leads to the additional conclusion that a dipole layer has to form at the surface. As in the case of a Schottky diode, this dipole generates a barrier to electron migration, from the n-type diamond. However, in contrast to a Schottky diode, a forward potential does not generate a forward current, but rather increases the barrier to electron egression. In order to ensure that electrons can overcome this barrier, the surface of the semiconductor has to be doped to a high density, such that electrons can tunnel out of the diamond’s surface. Electron-current flow into the anode initiates when these extracted electrons fill the whole gap between the diamond surface and the anode. All experiments, to date, show consistently that, after current flow has initiated, the electrons between the diamond surface and the anode are able to form a stable, highly conducting phase. By applying accepted, and proven, concepts from thermodynamics and quantum mechanics, it is concluded that the formation of such a phase is inevitable, and that it has to be superconducting. r 2003 Elsevier B.V. All rights reserved. Keywords: Diamond; n-Type doping; Ion implantation; Superconduction PACS: 74.70; 85.45.D; 68.55.L; 81.05.U
1. Introduction Owing to its extreme physical properties, diamond may find electronic applications, which are not possible when using silicon or any other semiconductor material. However, to generate *Corresponding author. Tel.: +27-11-477-8005; fax: +2711-477-3709. E-mail address: [email protected] (J.F. Prins).
useful electronic devices, controlled doping, with both acceptors and donors, is required. Substitutional boron atoms form acceptor centres in diamond situated at 0.37 eV above the valence band maximum [1]. These atoms can be incorporated into the diamond lattice during CVD growth [2] as well as using ion implantation [3]. Substitutional nitrogen atoms form donor levels in diamond, but they are situated at E1.7 eV below the conduction band minimum [4]. This is too deep
0168-9002/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2003.08.085
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to provide an adequate supply of electrons within the conduction band, at room temperature. At present, n-type diamond, ascribable to the incorporation of phosphorus, can be generated, either during CVD growth [5] or by ion implantation [6]. The ionisation energy, of this donor level, is 0.62 eV [7]. Shallower, donor levels are required if one wishes to make commercially viable electronic devices. If shallow donor levels could be generated to a high enough density, diamond substrates may find application as cold cathodes. Himpsel et al. [8] have found that electrons, which are photo-excited into the conduction band of boron-doped, p-type diamond, can exit the surface of the diamond without encountering any energy barrier, provided the surface is terminated with hydrogen atoms. The diamond substrate has negative electron affinity (NEA), in the sense that the electrons, which, in this case, have been photo-excited into the conduction band, find themselves at a higher energy than the vacuum level. This result has led to the conclusion that if one could find an n-type semiconductor with NEA, one should be able to extract conduction electrons into the vacuum, at room temperature, without needing a high voltage to generate field emission over a barrier. However, to date this type of behaviour has not been realised in diamond or, for that matter, in any other possible material. A consensus seems to be emerging that diamond may not be an inherent NEA material as anticipated, and that the NEA observed and reported for p-type, hydrogenterminated diamond, is an artefact generated by the dipole character of the C–H bonds at the surface [9]. However, my studies indicate that diamond is indeed an inherent NEA material, and that it could be this property that has made it difficult to generate shallow donors. In an NEA material, shallow donor levels should be situated at a higher energy than the vacuum level, and, because of this, it might only be possible to obtain them by generating thermodynamically, metastable donorlike flaws in the lattice. In this short overview, it will be shown how the latter assumption has led to the generation of shallow donor states by means of oxygen- or nitrogen-ion implantation. Next, a
band-structure analysis will be discussed, from which it has been concluded that electron extraction from such n-type diamond can only occur by tunnelling through a depletion layer at the surface, and not in the manner generally anticipated in the literature. In turn, it will be shown that the actual behaviour of electrons, extracted into the gap between the diamond surface and an anode, is also different from the anticipated behaviour.
2. Ion implantation and formation of shallow donor levels A short review of ion implantation into diamond for electronic applications has recently been published [10], and will thus not be repeated here. Suffice to point out that the major stumbling block, when attempting to dope diamond by means of ion implantation, is the thermodynamic metastability of this material. If the kinetic conditions are suitable, it will revert to a graphitic-type structure. To circumvent the problems caused by this metastability, two routines have been developed to dope diamond. They are known by the acronyms CIRA (Cold-Implantation-Rapid-Annealing) and LODDI (Low-Damage-Drive-In). In the CIRA routine, implantation is done at a low enough temperature, preferably liquid nitrogen or lower, to ensure that the point defects, generated by the impinging ions, do not diffuse during the implantation process. This is followed by rapid thermal heating to a suitable temperature at which the interstitial atoms can diffuse. When using the LODDI routine, a region, or layer, is generated by irradiation, such that it contains a suitable (low) density of vacancies. The dopant atoms are then implanted in an adjacent region, or layer, to allow some of the interstitial atoms (self and dopant), generated within this layer, to diffuse into the initial layer where they can meet up with the, previously generated, vacancies. In a comparative experiment using O+, B+ and + C ions for CIRA implantations with an annealing temperature of only 500 C, it was found that the oxygen-treated diamond layers conducted n-type with an activation energy of E0.32 eV
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[11]. As already mentioned in the introduction, the low annealing temperature was used in an attempt to ‘‘quench’’ in high-energy metastable donor states with concomitant high-energy electronic levels, which lie near to the conduction band. In a follow-up study [12], it was found that even shallower donor levels (at E0.28 eV) could be generated by similar nitrogen-ion implantation. These centres cannot relate to substitutional nitrogen, because, as mentioned above, it is known that the latter donor levels are situated deeper (E1.7 eV) below the conduction band. It has now been proposed that such ‘‘quenched-in’’ donor centres could consist of dopant atoms trapped in interstitial sites adjacent to vacancies. Further studies are in progress to study the metastable nature of these centres [13].
3. Donor levels above the vacuum level If donor levels are situated above the vacuum level, those electrons, thermally activated from them into the conduction band, find themselves also above the vacuum level. The electron-energy band diagram, in Fig. 1, shows the situation, which has been anticipated in the literature under such circumstances. It is believed that an anode would easily extract electrons (injected into the conduction band via an ohmic ‘‘back’’ contact) out of the front surface without encountering any energy barrier. However, the important point to note is that, according to the second law of Thermodynamics, this diagram is unrealistic, because it represents a transient, non-equilibrium situation. If Fig. 1 were applicable, it would mean that, owing to the offset w between the conduction band and the vacuum level at the surface, conduction electrons, arriving at the surface, would encounter a field accelerating them out of the substrate, even if there were no extraction field being applied. Such electrons thus make transitions from the conduction band to the vacuum level. By doing this, they leave positive charges behind which reside on ionised donor levels. These charges will attract the electrons back to the surface. However, as long as the vacuum level lies below the donor
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Surface
-
conduction band Fermi level
NEA = χ
vacuum level
n-type NEA semiconductor
vacuum
valence band
0
z
Fig. 1. The electron-energy band diagram at the surface of an n-type semiconductor with NEA, which is generally anticipated in the literature. Owing to the second law of Thermodynamics, this situation is not possible, because the energy levels are not in thermodynamic equilibrium (see text).
levels, these electrons will not be able to re-enter the diamond surface because all the available energy levels, which they can occupy, are at too high an energy. The electrons, accumulating outside, in the vacuum, will experience the positively charged surface as a two-dimensional ‘‘atomic nucleus’’ that causes them to become bound within quantum states [14]. These states will be situated within an ‘‘electron-charge layer’’ of width we adjacent to the surface. In turn, the ionised donors, left behind by the emitted electrons, have to accumulate within a depletion layer at the surface. A dipole, thus, develops which, after reaching the required strength, screens the field caused by the initial potential offset w; to, in this way, achieve thermodynamic equilibrium at the surface-to-vacuum interface. The latter dipole is similar to the one generated at a p–n-junction, except that the negative layer is now not a depletion layer within which the negative charge resides on ionised acceptors. An analysis of the equilibrium situation at room
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temperature renders the energy band diagram shown in Fig. 2 [14]. The electrons, furthest from the surface, are distributed around a Fermi level, which lines up with the Fermi level within the substrate. Those electrons, with the highest energy above the Fermi level, are at the vacuum level, or else they will not screen the electric field within the dipole. However, they are not free to drift away, because they are bound by the self-consistent potential well, which retains all the electrons at the surface within the field of the positive depletion layer. Nearer to the surface of the n-type substrate, the electrons are even more tightly bound within lower energy quantum states. In turn, within the positive depletion layer, band bending occurs. This forms a barrier to further net electron flow from the conduction band to the vacuum. Owing to the similarities between this dipole layer and a p–n-junction, one expects that when a forward potential appears over the dipole, the barrier to electron migration out of the n-type Surface
Initial energy position of the conduction band conduction band
++++++ Fermi level
n-type NEA semiconductor
----
NEA = χ vacuum level layer of electrons bound at surface self-consistent potential well
vacuum
valence band
electron-charge layer
depletion layer
-wn
we 0
z
Fig. 2. An electron-energy band diagram at the surface of an ntype semiconductor with NEA, which is in thermodynamic equilibrium, as required by the second law of Thermodynamics. A dipole has formed at the surface consisting of a positive depletion layer, just under the surface, and an ‘‘electroncharge’’ layer just outside the surface. The latter electrons are bound within quantum mechanical states. This dipole screens the field that would have been present, owing to the energy offset w; between the conduction band and the vacuum level.
substrate should immediately decrease, such that a current will flow. However, this does not happen in the present case, as one can see when the two situations are compared: (a) The dipole of a p–n-junction is sandwiched between two conductors within which the fields, caused by an applied potential, may be neglected compared to the field that appears over the insulating dipole. The latter field lowers the Fermi level in the p-type material, relative to the Fermi level in the n-type material, making it possible for electrons to flow from the power supply, through the ntype semiconductor to the dipole interface. At the interface, they are pulled through the dipole, by the applied field within it, into the p-type semiconductor, through which they can flow back to the power supply. (b) In the case of electron-extraction from an NEA, n-type substrate, the dipole is sandwiched between the n-type semiconductor and the vacuum. The latter has no free carriers and is thus an insulator. Thus, the applied field, generated by an anode potential F; appears over both the dipole and the region between the anode and the electron charge layer. This effectively lowers the vacuum level outside the electron-charge layer. It should now be recalled that the reason why the dipole formed, in the first place, was to screen the field that had been caused by the potential energy offset w relative to the vacuum level. Thus, any additional, applied field will have a similar effect. In order to reach a state of thermodynamic equilibrium, both the depletion layer and the electron-charge layer will increase further in width from wn to wnf ; and we to wef ; respectively, until the applied field is also screened. The energy band diagram, after this has been achieved, is shown in Fig. 3. It should be noted that an extraction field only appears after the applied potential has exceeded a critical value FW ¼ W DEF ; where W is the work function of the anode metal, and DEF is the difference in energy between the conduction band edge and the Fermi level of the n-type substrate [15]. The potential FW
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Initial energy position of the conduction band
Surface
χ χeff conduction band
++ + + ++ ++ Fermi level
n-type NEA semiconductor
e(Φ -ΦW)
-vacuum level -with field -vacuum -Fermi level
a n o eΦW d e
of anode electron-charge layer
valence band depletion layer
-wnf
0
z
wef
Fig. 3. The electron-energy band diagram, when applying an electric field to an NEA, n-type semiconductor, in an attempt to extract electrons into the vacuum. Owing to the workfunction of the anode, an extraction field only appears after the applied potential F exceeds a critical magnitude FW : The applied field then lowers the vacuum level outside of the semiconductor surface, and the dipole at the surface expands to screen this additional offset eFVAC ¼ eðF FW Þ in the potential energy. In other words, the applied potential effectively increases the original NEA w to have a value of weff ¼ w þ eFVAC :
is needed to initially align the vacuum levels of the substrate and the anode. Accordingly, the field, which needs to be screened, is generated by the potential FVAC ¼ F FW ; as shown in Fig. 3.
4. Electron tunnelling into the vacuum Because the Fermi level of an NEA, n-type semiconductor will always be at a higher energy than the Fermi level of any metal, the only way to establish an ohmic contact to a metal would be to generate a tunnelling junction between the two materials. One method is to dope the surface and near-surface region with a very high density of shallow donors before depositing the metal. Although an ‘‘ohmic contact’’ to an insulator does not make any sense, my research has led to the postulate that the only way to ensure that electrons keep on flowing easily from the conduction band of an NEA, n-type semiconductor into
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the vacuum, would be to, similarly, generate an ‘‘ohmic’’-tunnelling region or ‘‘contact’’ [15]. For example, doping the surface region to a high enough density ensures that the depletion layer, which has to develop at the surface of such a semiconductor, must be very narrow, such that when applying an extraction field, by means of an anode, the electrons are able to easily tunnel through it into the electron-charge layer. By increasing the applied voltage to the anode, more electrons should keep on adding to the electron-charge layer, in this way causing it to expand towards the anode. At some critical applied potential FC ; the electron-charge layer should just fill the whole distance dgap between the semiconductor surface and the anode, as shown in Fig. 4. A catastrophic, metastable equilibrium point is then reached, and this will be disturbed when the potential F is increased to be larger than FC : The self-consistent potential well will then expand into the anode. The anode is a conductor, and the self-consistent potential well cannot be maintained within it. Thus, the electrons, which enter the anode, become free to move. An electrical contact is established between the anode and the electron-charge layer, which must result in a field being established to attract electrons into the anode in order to equalize the Fermi level outside its surface to the one within the anode. This field will add to the field, which the dipole expansion is trying to neutralize. A direct field should, thus, be established owing to the offset between the Fermi levels at the surface of the diamond and the Fermi level in the anode. In other words, the potential, FVAC ¼ ðFC FW Þ; now adds to the offset FW between the Fermi levels, in order to increase this offset to FC : This is also shown in Fig. 4. At this point it should be pointed out that for a large enough gap distance dgap ; the applied potential might become so large, that the electron-charge layer breaks down before it can fill the whole gap [14]. However, if this happens, the electrons will still experience the field caused by the offset between the two Fermi levels. Electric current flow will initiate. At position z; as measured from the substrate surface, the potential increase FðzÞ is now equal to zEgap ; where Egap is the field between the diamond
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74 Initial energy position of the conduction band
JðzÞ may be written as
Surface
2eFðzÞ JðzÞ ¼ enðzÞvðzÞ ¼ enðzÞ m
original position of vacuum level
n-type NEA semiconductor gap filled with electrons
conduction band
e(ΦC - ΦW )
a n o d e
Fermi level
eΦW eΦC valence band
a n o d e
0
z
Fig. 4. The electron-energy band diagram, at the critical applied potential FC ; where current flow, between the semiconductor surface and the anode, initiates. The selfconsistent potential well, within which the electrons were bound within quantum states (see Figs. 2 and 3), has now expanded to fill the whole gap. A further slight increase in potential allows the electrons at the Fermi level of the electroncharge layer, to enter the metal anode. An electrical contact is established between the semiconductor surface and the anode. The whole applied potential FC now manifests itself as an offset between the Fermi levels of the semiconductor and the anode, which generates a field that makes current flow possible. The electrons are now not bound anymore and thus act as charge carriers. The depletion layer is shown with a very narrow width, because it is assumed that the region, just below the surface, has been doped to a high density in order to facilitate tunnelling of the electrons through this layer (see text).
and the anode caused by the offset between their respective Fermi levels. The gain in kinetic energy Tkin ðzÞ for an electron, with mass m; which enters the gap, may be written as 1 pðzÞ2 Tkin ðzÞ ¼ mvðzÞ2 ¼ ¼ eFðzÞ ¼ ezEgap : 2 2m
ð1Þ
Because the electrons enter the gap from the minimum energy of the conduction band, their initial speeds are equal, or very near, to zero. Thus vðzÞ and pðzÞ can be taken as the actual speed and momentum of the electrons at position z: If the density of the electrons is nðzÞ; the current density
1=2 :
ð2Þ
This equation proves that, even with no atoms in the gap, a potential is still needed to drive the current through the gap, i.e. the gap with electrons has an effective resistivity rðn; FÞ: Thus, if after reaching the critical juncture, demonstrated in Fig. 4, the electron density nðzÞ increases, the resistivity within the gap will decrease. The applied potential will, accordingly, redistribute over the diamond and the gap, to drive an increased current around the circuit.
5. Electron extraction from n-type diamond with an ‘‘ohmic’’-tunnelling contact to the vacuum (experimental results) The following results are from the first, pioneering experiment using low-energy oxygen-ion implantation [15]. Since then, similar results have been repeatedly obtained when doping diamond with either oxygen- or nitrogen-ion implantation to generate shallow donors. This substrate was a high-purity (natural type IIa) diamond with an area of 3.6 3.6 mm2. It was decided to start off with ion energies well below 1 keV in order to just dope the very-near surface region. The ions were extracted through a hole, from a direct-current (DC) oxygen-plasma, into the diamond surface, using an extraction potential of 150 V, and the current density measured from the diamond, to the ground potential, E0.3 mA cm2. The diamond was then annealed up to E400 C, while monitoring its resistance with a potential of 10 V across it. After cooling, the sheet resistance was E5 106 O: After another anneal for 3 h at 400 C, in a high-purity argon atmosphere, without applying a potential, the sheet resistance was found to have increased to E1 1011 O, and the diamond conducted like an oxygen-doped diamond, which had been doped by keV ion implantation. This result indicates that the injected oxygen ions did not just form donor states very near to the surface, but also deeper lying donors owing to LODDI-type oxygen-interstitial
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in-diffusion, and that, during annealing, without an applied potential, the near surface states are less stable than these deeper-lying ones. The diamond was again treated with low-energy oxygen ions, and annealed by heating it in vacuum with an applied potential of 10 V. Afterwards, the sheet resistance was E1.75 106 O. From the results above, the deduction was made that the implanted surface then, most probably, consisted of a deeper lying, lighter-doped, n-type layer and an overdoped surface layer that may be able to form an ‘‘ohmic’’-tunnelling contact to the vacuum, provided the diamond is an NEA material. A typical result obtained when applying a field, with a gold-plated anode sphere of radius 0.5 mm, is shown in Fig. 5. The measurements were made while maintaining the distance between the diamond surface and the anode at 10 mm. At first, while increasing the positive potential on the 0.6 0.4
Current (mA)
0.2
Gap width: dgap = 10 µm Behaviour after current flow has stabilised
0.0 Critical potential at which current starts flowing
-0.2 -0.4 -0.6 -1000
-500
0
500
1000
Probe ("anode") potential (V) Fig. 5. Experimentally measured electrical current when a potential is applied to an anode, in order to extract electrons out of the surface of an n-type, diamond doped, to a very high donor density near its surface, by means of oxygen-ion implantation. The distance between the diamond surface and the spherical anode (with radius 0.5 mm) was 10 mm. Initially, no current flow occurred with increasing positive potential, until a critical potential of E420 V had been exceeded. After the current had then been allowed to stabilize, its behaviour was the same whether the potential was positive or negative. It is argued that the latter behaviour is only possible if the electrons, between the diamond and the anode, have formed a superconducting phase.
75
anode from zero to higher values, no current flow occurred, until at E420 V, current flow suddenly initiated. From the discussion above, one may conclude that, at this point, the self-consistent potential well, and, thus, the electron-charge layer, broke down. Once current flow had initiated and stabilised at higher voltages, the potential was changed between +1000 V (the maximum of the power supply) in steps down to 1000 V, while monitoring the current. Current could now be measured for potentials lower than 420 V, as well as negative potentials (see Fig. 5). For the same magnitude of the potential, the same current flow occurred in both directions when the polarity was changed from positive to negative. It was also found that one could now switch off the potential, even for a week and longer, without the conducting material, within the gap, disappearing. When subsequently switching on, at any potential between 1000 and 1000 V, the current started flowing. Furthermore, when pushing the probe against the surface, exactly the same curve was measured; i.e. the current was, within experimental error, the same, whether there was a gap or not. Even when breaking the vacuum, and removing the system from the vacuum chamber, the stable conducting material could persist. One could then actually observe the gap between the diamond and anode probe. No extraneous material could be seen within this gap. Furthermore, subsequent, careful microscopic examination of the diamond and probe surfaces showed no evidence, whatsoever, of any contamination, like, for example, a filamentary material, which could account for the observed conduction. The experiment has been repeated on a large number of alternative substrates. Not in one of the latter cases could the same, or even similar, results be obtained. Only when diamond substrates were employed, where their surfaces had been highly doped with shallow donors, could commensurate results be measured. Furthermore, in all the latter cases the current flow was not a function of the gap size between the diamond substrate and the anode, but related directly to the resistivity of the diamond. This means that the voltage drop over the gap was negligibly small: within experimental error, zero.
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After exhaustive, multiple experiments, the only acceptable conclusion that could be reached has been that the stable, highly conducting phase, which formed within the gaps between the n-type diamonds and their respective anodes, has to consist entirely of electrons.
6. The requirement for superconduction to establish a steady-state current As already discussed above, the situation depicted in Fig. 1 is not physically possible. The dipole structures, shown in Figs. 2 and 3, have to form in order to establish equilibrium, as required by the laws of Thermodynamics. The reason, why these dipoles form, is the presence of unscreened fields at the surface. Thus, when a field appears, for any reason, at the surface, the depletion layer will again start to inject more electrons, into the vacuum, in an attempt to also eliminate this field. Current flow initiates after the re-appearance of a field at the surface, as shown in Fig. 4. The depletion layer at the surface will, thus, react to eliminate this field, and the electron density within the gap will, in concert, start to increase further. According to Eq. (2), an increase in the density of the electrons causes either the current to increase or the potential over the gap to decrease. In fact both happens [15]. Thus, as long as the depletion layer adds electrons, steady-state current flow, as thermodynamically mandated, cannot be achieved. This leads to the unequivocal conclusion that a steady-state condition can only be generated when the field, within the gap, becomes zero while, at the same time, a current still flows between the diamond and anode. The only material that can fulfil both conditions at the same time is a superconductor. Accordingly, the second law of Thermodynamics has to fail, or Mother Nature has to find a way to form a superconducting phase, which has to consist entirely of electrons. In this respect, it is relevant to consider the BCS theory for superconduction [16] within metals at low temperatures. The electrons are believed to interact via phonons to form so-called Cooper pairs. Because the individual electrons are fermions with half-integral
spin, the Cooper pairs have integral (zero) spins. They can thus act like bosons, which are able to form a collective state represented by a single, coherent wave function: akin to Bose–Einstein Condensation of atoms during cooling to temperatures near absolute zero [17]. It is this state that is superconducting.
7. Electron pair formation without phonon interaction In the present situation, the electrons cannot form pairs by the Cooper mechanism, because there are no vibrating atoms within the gap between the diamond cathode and the anode, and, thus, also no phonons. One may consider the possibility of magnetic coupling between two electrons owing to their spins. However, it is well known that this would be a very weak interaction [18]. Except if one wants to re-consider the concept of ‘‘hidden variables’’, the needed interaction, which could lead to the required pair formation, should rather be sought as a logical consequence of the existing interpretation of Quantum Mechanics. The Pauli Exclusion Principle manifests itself as a much stronger, apparent force, which causes two electrons to pair (spin up and spin down) when they are occupying the same quantum orbital. In fact, the strongest chemical bond, namely the covalent bond, is formed by this interaction. For an average electron density ngap ; between the n-type diamond and the anode, the average volume per electron becomes n1 gap : Thus, the average distance between electrons in the z1=3 direction follows as Dz ¼ ngap : Because, according to the Pauli Exclusion Principle, electron pairs, with spin-up and spin-down, can occupy the same limited space, the uncertainty /DzS of the position of an electron, may be written as 2 1=3 /DzS ¼ : ð3Þ ngap From Eq. (1) the quadratic uncertainty in the momentum of such an electron may be written as /Dp2z S ¼ 2me/DFðzÞS ¼ 2meEgap /DzS:
ð4Þ
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Thus, the Heisenberg Uncertainty Relationship may be written as [19] /Dp2z S/Dz2 S ¼ 4meEgap n1 gap X
h2 16p2
ð5Þ
where h is Planck’s constant. But, as concluded above, the average density ngap must keep on increasing as long as Egap a0: A point must, thus, be reached at which ngap becomes so large, for the prevailing value of Egap ; that the left-hand side becomes equal to the right-hand side in Eq. (5): i.e. a further increase in ngap has to violate Eq. (5). At this critical point, each electron pair is forced to stay within the coordinate constraints imposed by the latter equation: i.e. each pair occupies a new type of spatially restricted ‘‘orbital’’, and the latter ‘‘orbitals’’ fill the space between the diamond surface and anode completely. If one of the electrons attempts to move as an independent entity, it implies that it has to exit the volume it shares with its partner and enter an adjoining volume. However, the latter volume will now already be occupied by an electron pair. The Pauli Exclusion Principle will thus prevent a third electron from entering. Thus, any current within the gap has to, henceforth, involve the combined motion of electron pairs; i.e. each pair, or ‘‘orbital’’ containing two electrons, has to move as a single charge carrier with a charge of 2e; and these charge carriers have integral (zero) spin; they are boson-like entities.
8. Formation of a Bose–Einstein-like Condensate After extending the statistics developed by Bose, for photons, to particles with mass, and before it was known that there are two types of particles with half-integral and integral spins, Einstein [17] postulated the formation of (what is today known as) a Bose–Einstein Condensate, as follows: ‘‘When a given number of particles approach each other sufficiently closely, and move sufficiently slowly, they will together convert to the lowest energy state possible’’. It is now known that these particles have to be, or act like bosons. It is generally accepted in the literature that the formation of such a condensate requires the
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constituent particles to be cooled to extremely low temperatures near absolute zero. One should now note that, in the present case, these exact conditions are approached for the charge carriers injected into the gap between the ntype diamond substrate and the anode. And this occurs without cooling! Furthermore, because a threshold condition can be reached where the electrons form pairs, a Bose–Einstein-like Condensate becomes possible at room temperature. This implies that the speeds of the electron pairs are now, within the limits of the Heisenberg Uncertainty Relationship (as applied to bosons for which the uncertainty in position is the same as the gap width), the same at the cathode as at the anode. No acceleration takes place anymore, which, in turn, implies that, although a current is flowing, there is no field within the gap. A superconducting state has formed, and steadystate current flow can proceed, as required by the laws of Thermodynamics.
9. Summary and conclusion Diamonds can be doped by ion implantation to generate metastable donor flaws with donorelectron energy levels situated near to the conduction band edge. Experiments on electron extraction from such n-type diamonds, where their surfaces had been doped to a high density with shallow donors, consistently show that a stable, highly conducting material, consisting entirely of electrons, form between the diamond’s surface and the anode. Furthermore, the resistance of this material is independent of the gap width between the anode and the diamond. In fact, it is the same as when the anode touches the diamond, thus indicating that it might be zero. A theoretical analysis based on the accepted principles, used in the band theory of solids, leads to the unequivocal conclusion that a steady-state current can only flow into the anode, if the field, within the gap between the diamond and the anode, becomes zero. Furthermore, the conditions required for the formation of such a superconducting material follow logically when the Pauli Exclusion Principle and Heisenberg Uncertainty Relationship are
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applied to the situation. The electrons are forced to pair and form boson-like entities, which then, in turn, form a Bose–Einstein-like Condensate at room temperature. The experimental results, as well as the theoretical analysis, independently and jointly, provide compelling evidence that a superconducting phase can form, and actually has formed, at room temperature.
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