The layer structure of the sample used in this study is given in Table 1. The sample, which was grown by molecular beam epitaxy (MBE) on semi-insulating GaAs substrate, consists of three 7 nm thick GaInNAs QWs, separated by 20 nm thick Be-doped GaAs barriers. These p-type-doped barriers are separated from the QWs by 5 nm undoped spacer layers to reduce the remote impurity scattering. The mole fraction of indium and nitrogen in the Ga_1-x In _x N _y As_1-y QWs is x = 0.3 and y = 0.015, respectively. The sample was fabricated in the shape of a simple bar for I-V measurement. Fabrication details are given somewhere else 1 .

The nonlinear transport processes depicted in Figure 1 are modeled by a set of dynamic equations relevant to current instabilities in semiconductors. We derive a set of nonlinear partial differential equations for the hole density in the wells (p_w ), and in the barriers (p_b ), the potential barrier in each of GaAs layers (Φ _B ), and the dielectric relaxation of the applied parallel field (ξ_II ).

The dynamics of the carrier density in the well and in the barrier are given by 5

∂ p w ∂ t = 1 q L w ( J w − b − J b − w )

∂ p b ∂ t = 1 q L b ( J b − w − J w − b )

where J_w-b and J_b-w are the thermionic currents flowing from the GaInNAs well layers to the GaAs barrier and from the barrier into the well, respectively, q is the positive electron charge, and L_w and L_b are the width of the GaInNAs QW and the GaAs barrier, respectively. The electric field parallel to the layer interface ξ_II can be derived from Poisson's equation and is given by

ε 0 ε s ∂ ξ I I ∂ y = q L w + L b [ ( N A − p b ) L b + p w L w ]

where ε ₀ and ε_s are the absolute and relative permittivity, respectively. Using Equations (1)-(3), the dielectric relaxation of the applied parallel field ξ_II as a function of the current flow (y-direction), the transverse space coordinator (x-direction), and the time t can be written as

ε 0 ε s ∂ d y ( ∂ ξ I I ∂ t ) = q L w + L b [ d p b d t L b + d p w d t L w ] = q L w + L b [ − [ 1 q L b ( J b − w − J w − b ) + μ b ∂ ( p b ξ I I ) ∂ x ] L b + [ 1 q L w ( J w − b − J b − w ) + μ w ∂ ( p w ξ I I ) ∂ x ] L w ]

= σ L ( ξ 0 − ξ I I ) − 1 L w + L b [ − q p b μ b L b ∂ ξ I I ∂ y ] + [ q p w μ w L w ∂ ξ I I ∂ y ]

Where ξ ₀ = U ₀/d is the applied field and U ₀ is the applied voltage, σ_L = d/⌊h(L_w +L_b )qμ_wR_LN_A ⌋ is connected to the load resistance R_L , d is the sample length, h is the width of the sample, μ_w and μ_b are the hole mobility in the QW and the GaAs barrier, respectively. By integrating both sides of Equation (5), we finally have the dielectric relation of the parallel electric field

∂ ξ I I ∂ t = J I I − 1 L w + L b [ q p w μ w L w + q p b μ b L b ] ξ I I

where J_II is the external current density flowing through the external circuit at applied bias voltage U ₀.

Here, we define the current density flowing through the sample as a function of applied parallel field, using

J I I = 1 L w + L b [ q p w μ w L w + q p b μ b L b ] ξ I I

The time-dependent potential barrier in the GaAs layer is given by

∂ Φ B ∂ t = q 2 ε 0 ε s [ − μ b N A q Φ B + q μ b L b 2 2 ε 0 ε s ( N A − p b ) 2 ]

Equations (1) and (2) represent particle continuity, where the thermionic current densities J_b-w and J_w-b can be calculated using Bethe's theory, by assuming that the width of the space charge is comparable to the mean free path L_m of the holes 4 5

J w − b = − q p w [ K B T w 2 π m w * ] 1 / 2 e ( − Δ E v K B T w )

J w − b = − q p b [ K B T b 2 π m b * ] 1 / 2 e ( − Φ B K B T b )

where m w * and m b * are the hole effective mass in the GaInNAs QW and GaAs barrier, respectively, and the hole temperatures in the well and barrier are approximately given by

T w ≈ T L + τ E w q μ w ξ I I 2   T b ≈ T L + τ E b q μ b ξ I I 2

Since the number of holes in the well and the barrier are related to each other, their total number is conserved 1 :

p 0 L w = p w L w + p b L b

where p ₀ is the 3 D hole density in the well at low field.

The steady-state can be evaluated by setting Equations (1), (2), (6), and (8) to zero, and using the parameters listed in Table 2. The resulting static current density characteristic as a function of the static electric field is shown in Figure 2. The measured I-V curve obtained with the same sample in our previous study is placed in the figure inset for comparison 1 . Simulation results predict that the RST of hot holes leads to an N-shaped characteristic with a regime of negative differential resistance 4 7 8 . The critical field for the onset of NDM is the order of 6 kV/cm, which agrees well with our experimental results.

The time-dependent nonlinear Equations (1), (2), (6), and (8) have been numerically resolved using Euler's methods. The simulation reveals that the instability of the dynamic system is strongly dependent on the applied dc bias field, ξ ₀ = U ₀/d. We found that self-generated nonlinear oscillation appears in a range of applied dc electric fields where the load line lies in the NDM regime, as shown in Figure 3a. Figure 3b shows the corresponding current-density oscillations with frequency of 44 GHz, for ξ I I * = 10.1 kV/cm and N_A = 2.2 × 10¹⁶ cm^-3.

It is interesting to find that the oscillation frequency is strongly dependent on the dopant concentration in the barrier and the barrier thickness, as shown in Figure 4. The oscillation frequency increases from 29 to 50 GHz as the dopant concentration in the barrier increases from 1.9 × 10¹⁶ cm ^-3 to 2.4 × 10¹⁶ cm ^-3, accompanied by gradually reduced oscillation amplitude. Finally, the periodic oscillation damps out when the dopant concentration is above 2.4 × 10¹⁶ cm ^-3, as shown in Figure 5. The oscillation shows similar behavior as the barrier thickness increases. The fact that the self-generated oscillation frequency can be tuned by the doping concentration and the layer width can be explained by the nonlinear combination of the effective thermionic emission time, τ 0 = e 3 / 2 L w 3 π m w * / Δ E v and the dielectric relaxation time, τ _r = ε₀ε_s/qμ _b N _A as suggested by Döttling and Schöll 9 . The hysteretic switching transitions between the stable stationary state and the periodic oscillation in a uniform dynamic system depend on the ratio of the effective thermionic emission time and the dielectric relaxation time, γ. In our case, τ ₀ = 0.21ps, the change in dopant concentration from 1.8 × 10¹⁶ cm^-3 to 2.5 × 10¹⁶ cm^-3 leads to γ increases from 0.076 to 0.12 resulting in phase transition in dynamic system.