title: Reflection of Electrons during Tunneling and an Intersubband Polaron in the 2D Electron System of a Delta-Layer in GaAs
authors: S. E. Dizhur I. N. Kotel'nikov E. M. Dizhur
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  Institute of Radioengineering and Electronics RAS, Moscow, Russia
  
  
  Institute of Radioengineering and Electronics RAS, Moscow, Russia
  
  
  Institute for High Pressure Physics RAS, Troitsk, Moscow, Russia
  
  
  
  Al-δ-GaAs structures are studied where tunneling to one or more subbands of the 2D electron system
  of a near-surface delta-doped layer is observed. 
  Reflection of electrons at the threshold of the emission of an LO-phonon is observed. 
  This phenomenon occurs when a new subband is involved in the tunneling process and when intersubband transitions of the electrons in a 2D-system with the emission of an LO-phonon are added
  to the inelastic tunneling within a single subband. 
  It is shown that, under the conditions of an intersubband polaron resonance, the reflection processes dominate during tunneling into the delta-layer. 
  If, however, tunneling from the delta-layer occurs, the reflection processes are observed when two subbands are occupied.
  
  
  
  delta-doping photoconductivity
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  § INTRODUCTION
  
  
  
  The technology of delta-doping (hereinafter, δ-doping) enables implementation of extremely inhomogeneous doping in GaAs by the method of molecular beam epitaxy (MBE). 
  Ideally, in this case, atoms of the base lattice are only replaced with doping atoms within
  a single lattice plane, where δ-doping is performed (see, e.g., review <cit.>). 
  As a result, a V-shaped well for electrons is created in this plane (for example, in the
  case of Si doping). 
  The electrons are confined by the charges of donors near the δ-doped layer. 
  The conductivity of the 2D electron system (2DES) in a δ-layer may be sufficiently high owing to both the high maximum concentration of carriers (≃ 10^13  cm^-2) and to the growth of their mobility that is observed as the 2D subband’s number increases. Such properties of the 2DES in
  δ-layers show that they potentially may be used as a field-effect transistor's channel <cit.>. If the distance between the δ-doped layer and the metal/GaAs interface is reduced to tens of nanometers, the onset of the carriers’ transitions between the metal and the δ-layer due to the tunnel effect is observed. 
  Such a structure with a high-quality Al/GaAs interface produced in an MBE chamber opens a new avenue for the use of the near-surface δ-layer. 
  An Al-δ-GaAs tunnel junction produced on the basis of this layer makes it possible to
  analyze effects of the density of states and many-particle interactions in a 2DES by the tunnel spectroscopy
  method <cit.>. 
  Such a tunnel system seems the most promising for observation of the resonance phenomena related to the polaron interaction between different levels <cit.>. 
  It is important that the distances between the quantum confinement levels in the δ-doped layer in
  GaAs prove to be close to the energy of longitudinal optical phonons, ε_LO = 36.5  meV. 
  The positions of levels in the 2D channel of the Al-δ-GaAs junction may be altered using, e.g., a diamagnetic shift to bring the energy between subbands to resonance with ε_LO.
  Such an experiment was carried out in <cit.>, where anticrossing of terms characteristic to interacting levels was observed in the case when E_01 = 2ε_LO. 
  Similar effects (anticrossing of terms) were observed in optical experiments involving tunnel-coupled quantum wells <cit.> and in magnetic tunneling in AlGaAs/GaAs structures <cit.>. 
  These effects are a manifestation of the interaction between the Landau levels of different subbands
  when the ratio of the energy between the subbands and the cyclotron energy is an integer.
  
  Al-δ-GaAs structures exhibit one more important feature. 
  In addition to the low-density 2DES, a 2DES with a very high density of electrons can be created, where Fermi energy ε_Fδ  is on the order of the LO-phonons’ energy or even exceeds it. 
  Similar to tunneling between 3D electrodes, the 2DES spectrum in the latter case should exhibit singularities in the self-energy (related to polarons) at distances ±ε_LO
  from the Fermi surface <cit.>. 
  Effects of inelastic tunneling during interaction with an LO-phonon <cit.> may be observed as
  well and should not depend on the ratio between ε_Fδ and ε_LO. 
  The lines of LO-phonons have been, indeed, observed in Al-δ-GaAs structures at the potential bias
  U = ±ε_LO/e, where e is the electron charge. 
  However, their form strongly depends on the variation of the intersubband energies near the polaron resonance <cit.> or on the variations in the spectrum of levels in a 2DES.
  Additionally, it has been found that, when two subbands are involved in the tunneling process, a new type
  of phonon lines is observed <cit.>, which corresponds to a decrease in tunnel conductance (reflection
  of electrons). 
  When the concentration is increased in the 2DES and, hence, the number of the subbands
  involved in tunneling increases, the process turns out to be more complicated: reflection of electrons at the threshold of the emission of LO-phonons may prove to be the dominant effect in the conditions of the polaron resonance between levels E_1 and E_2 (E_12 = ε_LO) when
  tunneling into a 2DES occurs. 
  Under certain conditions, this reflection may be observed during tunneling from a 2DES.
  
  
  
  § SAMPLES AND THE MEASUREMENT PROCEDURE
  
  
  
  The Al-δ-GaAs structures produced at the Institute of Radio Engineering and Electronics of the Russian Academy of Sciences were studied. 
  The samples were made by the molecular beam epitaxy described in <cit.>.
  Owing to the technology where aluminum is deposited directly in the MBE chamber, the interface between Al
  and GaAs had the best possible quality. 
  Distance z between this interface and the δ-doped (Si) layer was ∼ 20  nm. 
  The tunnel spectra of the Al-δ-GaAs junction were measured at a temperature of 4.2  K. 
  The measurement procedure is described in <cit.>. 
  The dependence of the logarithmic derivative S = d lnσ / dU of tunnel conductance
  σ = dI/dU on U was used as a tunnel spectrum.
  Such a choice made it possible to reduce the tunnel characteristics of junctions with different values of
  σ to a unified scale. 
  The tunnel conductance in the case of tunneling from a 3D- to a 2D-state is known <cit.> to
  be proportional to the density of states ρ_i(E) in the 2DES:
  
      σ(U)≃∑_i D_i(U)ρ_i(U),
  
  where D_i(U) is the tunnel barrier transparency for level E_i. 
  This is clearly seen in Fig. <ref>, where potential profile E(z) for an Al-δ-GaAs tunnel junction is shown. 
  
  This figure also displays measured dependence σ(E_Fm - E_Fδ), which exhibits step-like changes in the density of 2DES states when the Fermi level in metal E_Fm crosses the
  energy levels in the δ-layer. 
  This behavior is more pronounced in the tunnel spectrum S(E) = dlnσ/dU, where
  positions of levels E_i correspond to smooth minima on curve S(E). 
  The arrows indicate step-like changes in the 2D density of states ρ(E) observed in the tunnel conductance displayed as a function of the difference E = E_Fm - E_Fδ between the Fermi levels of the electrodes.
  They also show the dips in the tunnel spectrum whose location in the energy spectrum corresponds to the
  position of the energy levels in the δ-layer. 
  In addition, the tunnel spectrum exhibits features that are typical of many-particle effects: phonon lines located at E = ±ε_LO and a zero-bias anomaly. 
  The curves shown in Fig. <ref> show the high quality of the Al-δ-GaAs tunnel junction,
  whose spectrum exhibits one occupied (E_0 < 0) and nine empty quantum confinement subbands that
  can be observed with high reliability. 
  The energy E_iF = E_i - E_Fδ can be observed using the tunnel spectra <cit.>,
  and the value of |E_0F| = 20  meV corresponds to the Fermi energy of the 2DES.
  
  Figure <ref> shows the tunnel spectrum of sample 1 (4.2  K), which is cut from the same substrate material as the sample shown in Fig.r̃ef1; however, the area of its tunnel gate is 0.5, mm^2. Voltage U across the junction, which is considered positive when electrons tunnel into
  metal (E_Fm < E_Fδ), is equal to eU = E_Fδ - E_Fm. 
  In the sample 1, when tunneling in the 2DES occurs at the threshold of the emission of an LO-phonon, i.e., at U = -36.5  mV, electrons can transit to one or two subbands under the effect of the external influence described below. 
  The Fermi level in metal, E_Fm, at the threshold of the emission of an LO-phonon proves to be higher
  than the Fermi level in the δ-layer (E_Fδ) by the value of ε_LO. 
  Energies E_iF of three lower subbands can be determined using the tunnel spectrum shown in Fig. <ref>: E_0F = -20, E_1F = 36, and E_2F = 76  meV. 
  Figure<ref> also shows the lines of LO-phonons separated from the background.
  
  
  To determine background curve S_0 and to separate the lines of optical phonons from the tunnel spectra,
  spectrum S_bg outside the phonon-related singularity has been approximated by cubic splines (Fig. <ref>). 
  This procedure, which was successfully applied for the first time in <cit.>, is described in detail in <cit.>. 
  The part of curve S_bg(U) containing the singularity (range of the bias values ± 6  mV around the line center for U = -36.5 mV) is taken into account by a weight that is significantly
  smaller than that of the other points in the approximation region. 
  Background line Δ S_bg is separated as the difference Δ S_bg = S_bg - S_0 = d(lnσ_bg - lnσ_0)/dU. 
  Assuming that the tunnel conductance represented by the formula σ_bg = Δσ_bg + σ_0, where Δσ_bg≪σ_0, is the variation of the conductance due to the electron–phonon interaction, we arrive at Δ S_bg≃ d(Δσ_bg/σ_0)/dU. Therefore, integration of Δ S_bg over dU yields the relative variation of tunnel conductance
  Δσ_bg/σ_0 in the vicinity of the phonon line (Fig. <ref>c).
  When separating the phonon line, the position of the line's center is chosen at |U| = 36.5  mV. Therefore, the separated line Δσ_bg/σ_0 depends weakly on the method
  used to determine σ_0 under reasonable assumptions of the width of the area where the phonon singularity is
  significant.
  
  
  The phonon line for sample 1 (Fig. <ref>) was measured after having exposed the sample to light from an LED for 3.2  s. 
  This procedure corresponds to the unusual processes of interaction between electrons and phonons
  during of tunneling <cit.> because the conductance decreases when the threshold |eU| = ε_LO is attained. 
  This phenomenon is associated with a negative step in the dependence of Δσ_bg/σ_0 on U (the magnitude of step Δσ^* < 0 in Fig. <ref>). 
  Inelastic processes involving emission of an LO-phonon are known (see, e.g., <cit.>) to
  increase conductivity in the region above the threshold, so that Δσ^* > 0. 
  Lines of such type were observed in sample 1 after it had been cooled to 4.2  K in darkness
  (see Fig. <ref>). 
  Below, we consider the dependence of the height of step Δσ^* in the tunnel conductance on the
  position of the bottom of the subband, which is closest to the energy E_Fδ + ε_LO. 
  It will be shown that inversion of the sign of Δσ^* can be observed in a sample by
  changing the positions of 2D subbands under the effect of external influence.
  
  
  
  § RESULTS AND DISCUSSION
  
  
  
  
  
   §.§ Reflection of Electrons during Tunneling into a Low-Density 2DES
  
  
  In sample 1, the Fermi energy in a single occupied band is only 20  meV, a value that corresponds to a low-density 2DES (a concentration of 2D electrons of ≈ 5 × 10^11  cm^-2). 
  The case of the low density of 2D electrons in a δ-layer is the most convenient for identifying the
  mechanism of reflection of electrons (Δσ^* < 0) during tunneling to a 2DES since, according to Fig. <ref>, no more than two subbands are involved in the tunnel current at the threshold of the emission of an LO-phonon. 
  Let us use Fig. <ref> to analyze the behavior of Δσ^* in sample 1 as a function of the position of subband's bottom E_1F. 
  Initially, the subband's bottom is located at E_1F = 36  meV, i.e., near the threshold of the phonon emission, corresponding to point B. 
  For U = -36.5  mV, the tunnel current in this case mainly depends on the transition to
  unoccupied states in subband E_0. 
  Under these conditions, Δσ^* > 0 corresponding to usual processes of inelastic tunneling: opening of an additional channel of the current from the Fermi level when an LO-phonon is emitted.
  
  
  With the use of external influence, it is possible to move apart the subbands in a δ-layer with the diamagnetic shift in a parallel magnetic field <cit.> or to condense them owing to the effect of persistent tunneling photoconductivity (PTPC) discovered in <cit.>. 
  When, owing to the diamagnetic shift, bottom E_1F is pushed up over the threshold to 43  meV (Fig. <ref>, closed circles), Δσ^* remains positive. 
  Such a situation, when electrons from the Fermi surface of metal tunnel to empty states in the
  2DES lower subband, is schematically shown in Fig. <ref>a. 
  An additional channel then opens for tunneling within the same subband with the emission of an
  LO-phonon in the entire area of the barrier between the metal and the 2DES. 
  Therefore, tunnel conductance should increase and the contribution from inelastic electron-phonon scattering within subband Δσ_int should correspond to a positive step on the conductance
  curve. 
  In this case, Δσ_bg/σ_0 may be represented as Δσ_int/σ_0, where 
  Δσ_int > 0.
  
  
  After a sample has been exposed to light from an LED during time t, energies of empty (E_i - E_Fδ > 0) levels ”thicken” to the ground state. 
  When the PTPC effect is saturated, the value of E_1F(t) in sample 1 falls to 22  meV, i.e., to a value that is significantly lower than the threshold (Fig. <ref>, open circles). 
  Electrons now tunnel to two subbands of the 2DES. 
  As follows from Fig. <ref>b, when second subband E_1 (E_1 - E_0 > ε_LO) is involved
  in the tunnel charge transfer, electrons can, after having emitted an LO-phonon with a nonzero component
  of momentum k_z in the direction of tunneling, experience a transition to subband E_0. 
  As a result, additional contribution to conductance Δσ_ext related to such a
  transition of E_01 proves to be negative. 
  This result may be explained in qualitative terms by back scattering (reflection) of tunneling electrons as a result of collision with the LO-phonons emitted during transitions between the subbands in the 2DES. 
  In this case, the relative conductance in the vicinity of the threshold is contributed by two mechanisms: Δσ_bg/σ_0 = (Δσ_int + Δσ_bg)/σ_0. 
  When |Δσ_ext| exceeds Δσ_int , Δσ^* becomes negative, as is observed in the experiment (Fig. <ref>, curve 1). 
  Note that Δσ^* crosses the energy axis at E_1F≃ 33  meV, i.e., somewhat below the threshold, since contribution of Δσ_int > 0 has to be compensated.
  
  The obtained results show that the emergence of the phonon line with Δσ^* < 0, which corresponds to reflection of electrons, is related to a new subband being involved in the tunneling process and to the emission of LO-phonons when electrons experience transitions from this subband to the ground state in the 2DES region. 
  This conclusion is in line with calculations of a 3D tunnel system carried out in <cit.>, where an analysis of the imaginary part of the electron-phonon system’s self-energy has indicated the existence of the reflection effect during tunneling. 
  Additionally, it has been found in <cit.> that Δσ^* < 0 when relaxation processes are
  localized on LO-phonons in one of the tunnel system's electrodes. 
  Note that, in sample 1, only usual processes of inelastic tunneling with Δσ^* > 0 are observed during tunneling of electrons from the 2DES single occupied subband to a metal electrode. 
  This conclusion follows from the shape of the phonon line in the spectrum for this sample, which is shown in Fig. <ref> for U > 0. 
  The situation changes drastically in the case of sample 2, where reflection effects are observed for both directions of the tunneling process (Fig. <ref>).
  
  
  
  
   §.§ Reflection of Electrons in the Process of Tunneling to a High-Density 2DES and Near a Polaron Resonance
  
  
  As follows from the tunnel spectrum displayed in Fig. <ref>, the energies of the first three levels for sample 2 are E_0F = -61, E_1F = -7.5, and E_2F = 28  meV. 
  Here, subbands E_1 and E_0 are occupied, a result that corresponds to a concentration of ≈ 2 × 10^12  cm^-2, i.e., to a high-density 2DES with the Fermi energy exceeding ε_LO. 
  In addition, the phonon lines in this sample differ significantly from those in the case of a low-density 2DES; namely, the values of Δσ^* < 0 for both directions of tunneling
  (compare the phonon lines in Fig. <ref> with the data for sample 1 shown in Fig. <ref>). 
  Let us consider the behavior of the phonon lines in sample 2 starting with the process of tunneling in the 2DES. 
  In this case, similar to the case of sample 1, one can assume that the main tunnel current on the threshold of the emission of LO-phonons is related to the tunnel transfer of charges from the metal to the subbands E_2F = 28  meV and E_1F = -7.5  meV. 
  The analysis shows that the probability of tunneling to level E_0 is significantly smaller than that
  for upper subbands. 
  The situation with additional reflection is then implemented in sample 2 and an additional reflection 
  (Δσ^* < 0) (see Fig. <ref>b) is expected to be observed near the threshold. 
  Indeed, this is seen in Fig. <ref>, where the closed squares correspond to the initial
  tunnel spectrum for E_2F = 28  meV and Δσ^* = -0.08.
  With the exposure of the sample to magnetic field B_|| parallel to the δ-layer, it is possible to transfer level E_2F above the threshold located at E_Fδ + ε_LO with the diamagnetic
  shift and by moving along the curve shown by closed squares (see Fig. <ref>).
  
  It is seen that Δσ^* vanishes when E_2F(B_||) = ε_LO, i.e.,
  when the tunneling process is directed toward occupied subband E_1 alone (see Fig. <ref>a). 
  The value of Δσ^* then proves to be positive and reaches ∼ 1% only when E_2F≃
  45  meV. 
  However, in contrast to sample 1, the processes of inelastic tunneling within the subband with
  Δσ_int > 0 are found to be suppressed strongly. 
  This phenomenon is also indicated by the shift of the zero-crossing point to the value E_2F(B_||) = ε_LO, which corresponds to equal contributions of intersubband scattering |Δσ_ext| and of intrasubband scattering Δσ_int by LO-phonons.
  Recall that, for sample 1, this point corresponds to E_1F(t) = 33  meV < ε_LO, where contributions |Δσ_ext| and |Δσ_int| are on the same order of magnitude. 
  The specific features of the behavior of the phonon contribution to tunnel conductance discovered in sample 2 require additional study of the reflection processes in the 2DES where density is as high as in this sample. 
  (The concentration of 2D electrons is ≈ 2 × 10^12  cm^-2.) 
  Here we only highlight the most pronounced features of the interaction between electrons and optic phonons under such conditions.
  
  Above all, note that the data obtained in a magnetic field for sample 2 show resonance (with respect to the
  LO-phonon) interaction between levels E_1 and E_2. 
  The dependence of the positions of the energy levels in the δ-layer on (Fig. <ref>) shows that level E_2 follows after E_1 + ε_LO (pinning), thus indicating the existence of the
  intersubband polaron interaction <cit.>. 
  Indeed, as has been observed already for δ-layers <cit.>, the diamagnetic shift of the ith level is Δ E_i∼Δ Z_i , where Δ Z_i is the wave function’s characteristic width in the ith state.
  Even though Δ Z_2/Δ Z_1≃ 1.5 according to estimates, the diamagnetic shift of the levels remains unchanged up to B_||c≃ 7  T. 
  Thereafter, when grows, variation Δ E_2 increases to the expected values. 
  Depletion of subband E_1 with the increase of the magnetic field seems to result in a ”collapse” of the resonance polaron state E_2 - E_1 = ε_LO. 
  If B_|| > B_||c, level E_1 is depleted and turns out to be higher than E_Fδ (Fig. <ref>, filled circles). 
  A selfconsistent calculation (neglecting interaction between electrons and phonons) shows that, for U = -36.5  mV, this phenomenon may occur earlier, even at B_||≃ 5  T (see Fig. <ref>, curve 1). Note that, interestingly, the reflection effects are only observed in the region where level
  E_2 is pinned to E_1 + ε_LO. 
  In the region of repulsion of levels (collapse of the resonance polaron state), the contribution of the inelastic-tunneling processes with Δσ^* > 0 proves to be negligible as compared to those in
  sample 1. 
  This may indicate that a relation exists between the intersubband polaron interaction, effects of 
  inelastic tunneling, and reflection on the threshold of the emission of an LO-phonon.
  
  
  
  The obtained data show that the effects of electron reflection can be observed during tunneling from the
  2DES, when electrons experience transitions to the metal from two occupied subbands. 
  This is the case for sample 2. 
  Curves Δσ_bg/σ_0(U) (see Fig. <ref>, insert) show how the conductance decreases for U > 0 beyond the threshold. 
  Additional experiments are needed to determine the 2DES density for which Δσ^* will change its
  sign as a result of tunneling from the δ-layer. 
  Remember that Δσ^* > 0 is observed reliably for sample 1, where the concentration is ≈ 5 × 10^11  cm^-2. 
  Note also that in the experiments presented in Fig. <ref> the maximum magnetic field of 8  T proved to be insufficient for pushing up subband E_1 from below the 2DES’ Fermi level. 
  Although the measured position of E_1F is somewhat higher than the Fermi level already at B_||c≃ 7  T, when U = 36.5  mV (the value near which the phonon line is measured), the expected position of E_1F should be several millielectron volts lower owing to enrichment of the channel containing the 2DES for the positive bias values on the metal electrode. 
  Estimates (a self-consistent calculation with the electron-phonon interaction neglected) show that this downward shift is ∼ 4  meV. 
  This yields a new position of E_1F (Fig. <ref>, curve 2) that turns out to be lower than the Fermi level in the 2DES.
  
  
  
  § CONCLUSIONS
  
  
  
  Thus, when the concentration of 2D electrons in the near-surface δ-doped layer increases, an Al-δ-GaAs tunnel system exhibits new features of the effects discovered earlier: reflection of electrons at the threshold of the emission of LO-phonons <cit.> and intersubband polaron interaction <cit.>. 
  To study these features further, it is necessary to make special samples whose concentration
  of electrons in the δ-layer and population of levels may be varied over a broad range. 
  Experiments with such structures are planned for the near future.
  
  
  
  § ACKNOWLEDGMENTS
  
  
  We thank Yu.V. Fedorov and A.S. Bugaev for the preparation of samples, N.A. Mordovets for help with
  conducting measurements, V.A. Kokin for performing self-consistent calculations of the levels in the δ-layer, and S.N. Artemenko, V.A. Volkov, and A.Ya. Shul’man for helpful discussions.
  
  This study was supported by the Russian Foundation for Basic Research (project nos. 03-02-16756-a,
  05-02-16316-a, 05-02-17095-a, and 06-02-16955-a) and by the program of basic research of the Division of
  Physical Sciences, Russian Academy of Sciences.
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
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