Dielectric permittivity, conductivity and breakdown field of hexagonal boron nitride

In view of the extensive use of hexagonal boron nitride (hBN) in 2D material electronics, it becomes important to refine its dielectric characterization in terms of low-field permittivity and high-field strength and conductivity up to the breakdown voltage. The present study aims at filling this gap using DC and RF transport in two Au-hBN-Au capacitor series of variable thickness in the 10--100 nm range, made of large high-pressure, high-temperature (HPHT) crystals and a polymer derivative ceramics (PDC) crystals. We deduce an out-of-plane low field dielectric constant $\epsilon_\parallel=3.4\pm0.2$ consistent with the theoretical prediction of Ohba et al., that narrows down the generally accepted window $\epsilon_\parallel=3$--$4$. The DC-current leakage at high-field is found to obey the Frenkel-Pool law for thermally-activated trap-assisted electron transport with a dynamic dielectric constant $\epsilon_\parallel\simeq3.1$ and a trap energy $\Phi_B\simeq1.3\;\mathrm{eV}$, that is comparable with standard technologically relevant dielectrics.


I. INTRODUCTION
Hexagonal boron nitride (hBN) is a van der Waals crystal insulator introduced in graphene electronics a decade ago [1] and since then extensively used as encapsulant [2], tunnel barrier [3][4][5], or gate dielectric in 2D material electronics [6,7]. In view of its technological relevance, it is important to improve its characterization both in terms of low-field permittivity and high-field dielectric conductivity and breakdown field. Accepted values for the DC dielectric permittivity constant lie in the broad range ǫ = 3-4 [1,[8][9][10], whereas breakdown fields are more scattered, with E BD = 4-10 MV/cm [9,11], depending on material quality and breakdown-field definition criterion. The purpose of the present study, which is based on DC and RF transport in Au-hBN-Au capacitors made of two types of hBN crystals, is to narrow down the uncertainty in permittivity, to shed light onto the dielectric breakdown mechanism, and to use these characterizations to benchmark the two hBN crystal sources.
We have used two series of Au-hBN-Au capacitors made of large exfoliated hBN crystals.
Crystals are grown either under high-pressure high-temperature (HPHT samples) as described in Ref. [12], or with a polymer derivative ceramics (PDC) route described in Ref. [13].
Exfoliated hBN flakes, of thickness d ∼ 10-100 nm, are sandwiched between Au electrodes of lateral dimensions L × W = 10 × 10 µm. A significant fraction of samples (17 HPHTs and 11 PDCs of the total 41 capacitors) follows the parallel-plate capacitance law with a dielectric constant ǫ ≃ 3.4 ± 0.2. The other 14 samples deviate from from this law with lower capacitance values, presumably due to process imperfections involving spurious air gaps, due to dust or bubbles between the hBN flake and the bottom electrode. Besides, we do not see significant differences between HPHT and PDC crystals in terms of permittivity.
The dielectric strength is characterized by monitoring the leakage current at high bias which is analyzed in terms of a bulk conductivity. This analysis is carried out on a subset of 7 capacitors having survived the high-bias training. As shown in Ref. [9], dielectric conductivity is a three step process, starting by a threshold-less exponential current growth, followed by a quasi-saturation at a breakdown field E BD and culminating by an irreversible current runaway for E > ∼ E BD usually leading to sample breakdown. Here we focus on the prebreakdown regime E < ∼ E BD where moderate current densities (J < J BD ≃ 0.5 A/cm 2 ) are applied that secure device integrity. In these conditions, we observe a bias-reversible, reproducible and polarity-independent behavior. The leakage current obeys a standard exponential growth with voltage which precludes unambiguous determination of a dielectric breakdown voltage (see Fig.2-a). However, when using an arbitrary breakdown current criterion J < ∼ J BD , we find an overall increase of the breakdown voltage with thickness which suggests the relevance of a breakdown field, i.e. a bulk scenario, and justifies our breakdown analysis in terms of conductivity, e.g. J/E(E) below a breakdown conductivity σ BD ∼ 1 µΩ/cm. We find that the conductivity, σ = J/E obeys the Frenkel-Pool (FP) law (Eq.(1) below), corresponding to a trap-assisted, thermally-activated, bulk Schottky transport [19,20]. Its signature lies in the linear dependence, ln σ/σ BD = f ( E/ǫ T 2 ) (see The FP activation scenario is confirmed by the temperature dependence measured in one representative sample (inset of Fig.2-b). The robustness of the field dependence contrasts with the large variability (within a factor 10 6 ) of the conductivity prefactor. We assign the latter to a variability in the deep-level donor energy Φ B ≃ 0.9-1.3 eV, with a logarithmical precision in the σ BD prefactor. Remarkably, five devices show very similar Frenkel-Pool plots with deep-level traps energy Φ B,hBN ≃ 1.27 ± 0.03 eV, suggesting the existence of a quasi-intrinsic limit. This value is quite typical of that of technology-relevant insulators, such as Si 3 N 4 where Φ B ≃ 1.3 ± 0.2 eV [21], or SiO 2 where Φ B ≃ 1 eV [22] are reported.
Some PDC-grown devices show however traps with a smaller energy, which opens routes for improvement of this less mature growth process.

II. CAPACITOR FABRICATION AND SETUP
We have fabricated 41 Au-hBN-Au capacitors, 27 of them with HPHT-hBN from NIMS and 14 with the PDC-hBN from LMI. The growth technics, as well as structural and optical characterizations, of these high-quality crystals are detailed in [12] and [13,14] respectively.
The capacitors were deposited on high-resistivity Si substrates, suitable for RF measurement, that are covered with a 285 nm-thick SiO2 layer. The bottom electrode and the RF coplanar waveguide structure are first deposited using laser lithography and thermal evaporation. The bottom 110/5 nm Au/Cr electrode is buried into SiO2 to ensure a planar surface for hBNflakes transfer. Planarisation is completed by mechanically polishing the small metallic pitches at the gold edges with isopropyl alcohol (IPA). This process minimizes air gaps between hBN and the bottom gold electrode which are ultimately limited by Au roughness of amplitude 2δ ≃ 1.5 nm, as measured by atomic force microscopy (AFM). hBN crystals were mechanically exfoliated with polydimethylsiloxane (PDMS). We use poly-propylene carbonate PPC-PDMS stamps to dry transfer the hBN flakes on the bottom electrode. A second lithography step allows covering the hBN flake with a second Au/Cr electrode, which is conformal to the hBN dielectric and therefore airgap-free. The obtained structures form a nominally 10 × 10 µm capacitor in a parallel-plate configuration. After annealing (1h at 240 Celsius under N 2 ) the capacitor's hBN thickness is determined by AFM and falls in the range d = 6-98 nm. This range exceeds the minimal thickness (10-20 nm) for mobilitypreserving encapsulation, is relevant for gate dielectric applications, and reaches the value (∼ 100 nm) for fully-developed radiative cooling [16,17]. It is however not relevant for tunnelbarrier applications which are described elsewhere [3][4][5]. The lateral size of capacitors was deliberately maximized at the limits of our exfoliation technique to increase experimental resolution, minimize spurious edge effects, and address the homogenous properties of hBN.
It is precisely measured by scanning electron microscopy. We have indifferently used colinear and perpendicular source and drain electrodes, the later geometry being shown in the optical image of Fig.1-a.
High frequency admittance measurements were carried out in a Janis (cryogenic) probe station under vacuum at room temperature (see sketch in Fig.1-a). The two-port scattering parameters S ij of the capacitor were measured using an Anritsu MS4644B vectorial network analyzer (VNA) in the 10 MHz-10 GHz range. A short-open-load-reciprocal protocol was used to calibrate the wave propagation until the probe tips. As explained in Ref. [23], the wave propagation in the coplanar access of the capacitor is de-embedded by calculating the ABCD matrix from the S parameters of a symmetric thruline reference structure. To correct for residual parasitic stray capacitance effects, we convert the previous ABCD matrices into complex admittance matrices and subtract the contribution of a dummy reference structure of same geometry but devoid of the central Au-hBN-Au capacitor. The total capacitance is directly deduced from the low frequency (sub-GHz) imaginary part Y 12 = jωC. The same procedure has been applied at low temperature on a representative capacitor to check the activation mechanism at stake in the dielectric breakdown. Capacitance data, deduced from VNA measurements, are first compared with complementary data obtained by sub-MHz Lock-In techniques (not shown) to ascertain their frequency independence. Data are scattered, but we can still find a significant fraction of devices, of both HPHT and PDC sources, showing capacitance-data accumulation along an upper limit given by the parallel-plate capacitance formula C = ǫ 0 ǫ LW/d * (red and blue symbols in Fig.1-b), where ǫ ≃ 3.4 ± 0.2 and d * = d + ǫ δ is an effective dielectric thickness accounting for spurious air-gap contributions due to (bottom) metal roughness 2δ ≃ 2 nm. One third of the series (14 devices, black symbols in Fig.1-b), exhibiting lower capacitance values caused by process imperfections including spurious air gaps, presumably due to dust or bubbles between the hBN flake and the bottom electrode, are discarded. From the selected devices (17 HPHTs and 11 PDCs), we are able to narrow down the dielectric constant window and provide a recommended value of the hBN dielectric constant ǫ = 3.4 ± 0.2. This value exceeds by 13% the ǫ ≃ 3.0 reported in metal-hBN-graphene quantum-Hall devices [10,16].
This apparent disagreement can be lifted when considering the series quantum capacitance of the graphene electrode in these thin hBN samples [10,18], as well as roughness-induced air-gaps at the bottom electrodes. Our measured permittivity turns out to be consistent with the theoretical prediction ǫ = 3.38 of Ref. [15], following an ab-initio approach which already quantitatively predicts correctly the optical permittivity and its anisotropy [25].  Fig.2-a, I-V curves are reproducible and bias symmetric (inset of Fig.2-a).
As seen in the figure, the breakdown voltage V BD shows a tendency to increase with hBN thickness d, suggesting a bulk origin of breakdown.
A deeper insight into the breakdown mechanism is provided by Fig.2 leads to full ionization of traps whenever (Φ B − eE/πǫ BD ǫ 0 ) < ∼ k B T , which defines a breakdown electric field E BD . In this picture, E BD = Φ B /r 0 is on the order of the impurity field, where r 0 = e/πǫ BD ǫ 0 Φ B is the screening length. The FP conductivity writes [20]: where the breakdown conductivity σ BD = J BD /E BD ≃ N T eµ corresponds to the band conductivity (mobility µ) for fully ionized traps (concentration N T ). Eq.(1) holds in the field range E F N < ∼ E < ∼ E BD , or J = 10 −4 -10 −1 µA/cm 2 , where E F N is the Fowler-Nordheim tunneling limit for a defect-free thin triangular barrier [21].
As seen in Fig.2 With this procedure, we extract Φ B = 0.9-1.3 V in Fig.2-b. Remarkably, we observe an accumulation of conductivity data (4 HPHT and 1 PDC capacitors) along an upper limit represented by the Φ B = 1.27 ± 0.03 V line. This suggests the existence of a dielectric strength limit in high-quality hBN crystals. Dispersion among that data-subset of highestquality samples corresponds to ∆Φ B = 0.06 V, or a variation of σ BD in the range 0.01-1 µS/cm. Taking a typical insulator mobility µ < ∼ 1 cm 2 /Vs, this translates into a trap density N T ∼ 10 12 -10 13 cm −3 and a trap number N T dLW ∼ 10-100.
To further establish the FP mechanism of breakdown we have added, in the insert of The relevance of the 3D Frenkel-Pool mechanism of conductivity in the 2D hBN was not granted as 2D materials may sustain specific mechanisms. We demonstrate here that it works for c-axis transport, but the situation can be different for in-plane electric fields with the opening of new leakage channels associated with charges gliding in-between hBN planes. The identification of deep-level traps responsible for breakdown is beyond the scope of our work, especially as leakage current alone cannot identify the acceptor/donor nature of the levels. We find Φ B ≃ 1.27 ± 0.03 eV in hBN, which is larger than the 1 eV reported in SiO 2 [22], and comparable with the 1.3 ± 0.2 eV in Si 3 N 4 [21]. In the latter case, deep traps are attributed to silicon-dangling-bond centers [26]. This trap energy determines a maximum breakdown field E BD ≃ 5 MV/cm, defined as the threshold for current quasi-saturation.
Finally we conclude on the strong similarity of HTHP and PDC hBN crystals in terms of dielectric permittivity and strength, with however a better yield in terms of intrinsic dielectric breakdown for the NIMS samples, which can to a large extent be attributed to a longer maturity of the growth technique.