A hydrodynamic equivalent circuit model for the Gunn diode

Solid-State Electronics 46 (2002) 915–923

Short communication

A hydrodynamic equivalent circuit model for the Gunn diode Chia-Hsiung Kao *, Liang-Wei Chen Department of Electrical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan, ROC Received 16 February 2001

Abstract The 1961 Sah equivalent circuit model is extended to hydrodynamic transport in semiconductors for applications to high speed one-carrier devices such as the Gunn diodes. A transient simulation of the Gunn diode is given using the circuit simulator SPICE3 to demonstrate this hydrodynamic extension. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Equivalent circuit; Gunn diode; Hydrodynamic equations

1. Introduction In the conventional semiconductor device simulation, the Poisson equation, continuity equation and driftdiffusion current equation are the basic equations. According to Sah’s equivalent circuit model [1–4], these basic equations can be transformed into the equivalent circuit that consists of capacitors, conductors and current sources. In the conventional drift-diffusion current model, the mobility l and diffusion coefficient D are field-dependent parameters. In short channel devices, the drift-diffusion model must be extended to a hydrodynamic model to explicitly include the kinetic energy of the carriers [5]. In the hydrodynamic model, l and D are carrier kinetic energy or carrier temperature dependent parameters. The momentum and power conservation equations are solved to give the current density and carrier temperature or carrier average kinetic energy. These equations have been used for simulating the Gunn diodes and oscillator circuits [6,7]. The more exact Monte Carlo simulations have also been made to Gunn diode oscillator circuits [8,9]. However, device simulation using the Monte Carlo simulation is time consuming. A more

*

Corresponding author. Tel.: +886-7-525-2000x4123; fax: +886-7-525-4199. E-mail address: [email protected] (C.-H. Kao).

efficient approach is to extract the energy dependent parameters from Monte Carlo calculation and use the hydrodynamic equations for device simulation. In this paper, the equivalent circuit model from the hydrodynamic equations is developed and applied to the GaAs Gunn diode. In Section 2, the equivalent circuit model is derived. In Section 3, the Gunn diode is represented by the equivalent circuit model and solved by the circuit simulator SPICE3 to give an example to demonstrate this circuit model approach for the first time for the Gunn diode. 2. Model development 2.1. The Gunn diode The Gunn diode is a transfer electron device (TED) which has an nþ =n=nþ structure shown in Fig. 1. In the transferred electron mechanism, the conduction electrons in GaAs are shifted from the high mobility state to the low mobility state by a high electric field. This negative conductivity phenomenon is important, since it generates electrical domains in the diode. When the domain transports from the cathode to the anode, it produces current and voltage oscillations. The Gunn diode characteristics can be solved as a one-dimensional device using the one-dimensional hydrodynamic equations for hot electrons. The device is divided into many small elements, dx ¼ h, as shown in

0038-1101/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 8 - 1 1 0 1 ( 0 1 ) 0 0 3 4 7 - 1

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Fig. 1. The Gunn diode device structure.

Fig. 2. In this paper, the hydrodynamic partial differential equations are represented by their difference equations. The equivalent circuits are then synthesized from the difference equations following the method introduced by Sah [1–4]. 2.2. Basic semiconductor equations Gunn diode is a one-carrier (electron) semiconductor device. An equivalent single valley model is used for the electron. Since the hole concentration is negligible, hole contributions in the device equations can be discarded. Thus the basic semiconductor equations for the Gunn diodes are the Poisson equation, electron continuity equation, momentum conservation equation and power conservation equation [5]: o2 V q ¼  ðND  nÞ ox2 

ð1Þ

on 1 oJ ¼ ot q ox

ð2Þ

onp 2 onE np ¼ qne   ot 3 ox sp

ð3Þ

onE oF nðE  E0 Þ ¼ þ Je  ot ox sE

ð4Þ

where V is the voltage, q elementary charge,  dielectric constant, ND donor density, n electron concentration, J

current density, p average momentum, e electric field, E average kinetic energy, sp momentum relaxation time, sE energy relaxation time, and F energy flux, respectively. E0 is the average energy in thermal equilibrium and it is equal to (3/2)kB T0 where T0 is the lattice temperature. p is the product of electron mass m and average electron velocity v and the current density J can be written as: p ð5Þ J ¼ qnv ¼ qn m In steady state, J is equal to the equilibrium current density J 0 from the momentum conservation equation as [5]: 2 onE J 0 ¼ qnle þ l 3 ox where qsp l¼ m

ð6Þ

ð7Þ

The energy flux F in the power conservation equation can be written as [5]: F ¼

5E J þ Qh 3q

ð8Þ

where Qh is the heat flow and it is assumed to be zero for the constant lattice temperature. 2.3. Equivalent circuit model With the above equations, the hydrodynamic equivalent circuit model can be developed. 2.3.1. The Poisson equation Applying the time derivative to the Poisson equation and using the continuity equation, we have o ot




o2 V ox2

 ¼

q on  ot

Fig. 2. Discretization of the one-dimensional device.

ð9Þ

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Fig. 3. The equivalent circuit of the Poisson equation at xi . In;i is the current in the capacitor Cn;i .

where the finite difference Eq. (9) at xi is,   oðViþ1  Vi Þ oðVi  Vi1 Þ  ¼ In;i Cv ot ot

where the current I is equal to AJ. The charge Qn;i is related to the node voltage Vn;i by ð10Þ

with

ð14Þ

Then

A Cv ¼ h

ð11Þ Cn

where Cv is the capacitance, A is the cross-section area. Vi is the voltage at xi and the current In;i is equal to oQn;i =ot with Qn;i ¼ Ahqni

ð12Þ

Eq. (10) can be represented by the equivalent circuit shown in Fig. 3. 2.3.2. The continuity equation The finite difference continuity equation at xi is, 

Cn Vn;i ¼ Qn;i

oQn;i ¼ In;i ¼ Iiþ12  Ii12 ot

ð13Þ

oVn;i ¼ Ii12  Iiþ12 ot

ð15Þ

which is represented by the equivalent circuit shown in Fig. 4. The choice of Cn is arbitrary. The capacitor Cn used is the same as Cv in this study. 2.3.3. The momentum conservation equation The change of current density during the momentum relaxation time sp is included in new model. The momentum conservation equation is used to determine the accurate value of current density. By selecting mv ¼ m0 vs , where m0 is the electron mass in vacuum and vs is the pseudo velocity for the pseudo-current Is , the momentum conservation equation at xi12 becomes

Fig. 4. The equivalent circuit of the continuity equation at xi . Ii12 is related to Is;i12 in Eq. (19).

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0 Ii 1 m0 oIs;i12 m0 I 1 ¼ 2 þ q ot li12 qsp;i12 s;i2

where ð16Þ B1 ¼

2 ð1 þ ai ÞðEi  Ei1 Þ  1þai 3 1  EEi1i

ð21Þ

B2 ¼

2 ð1 þ ai ÞðEi  Ei1 Þ  ð1þai Þ 3 1  EEi1i

ð22Þ

with Is;i12 ¼ Aqnvs;i12

ð17Þ

and 0 0 Ii 1 ¼ AJi1 2

2

ð18Þ with

From Eq. (5), the current Ii12 can be obtained directly from Is;i12 , as: m0 I 1 ð19Þ Ii12 ¼ AJi12 ¼ Aqnvi12 ¼ m s;i2 The result of Eq. (16) can be represent by the equivalent circuit as shown in Fig. 5 with L~p;i1 ¼ m0 =q, R~p;i12 ¼ m0 = 2 0 qsp;i12 and V~p;i12 ¼ ðIi 1 Þ=ðli1 Þ. 2 2 The modified Gummel–Scharfetter difference scheme 0 in [10] is used to approximate the current density Ji 1: 2

0 Ji 1 ¼ 2

li12 h

ðB1 ni1 þ B2 ni Þ

ð20Þ

3q ai ¼ 2




Vi1  Vi Ei  Ei1

 ð23Þ

2.3.4. The power conservation equation The power conservation equation at xi is, ni

oEi oFi ni ðEi  E0 Þ oni ¼ þ J i ei   Ei sE;i ot ox ot

ð24Þ

Let C~E;i ¼ Ahqni , Fa ¼ AF , and VE;i ¼ Ei =q. The continuity equation (2) give the following difference equations: oVE;i ¼ Fa;i12  Fa;iþ12 þ I~E;i C~E;i ot

ð25Þ

where ðVE;i  Vw0 Þ þ VE;i In;i I~E;i ¼ hIi ei þ Cn Vn;i sE;i

ð26Þ

with hIi ei ¼

Fig. 5. The equivalent circuit of the momentum conservation equation at xi12 .

Ii12 ðVi1  Vi Þ þ Iiþ12 ðVi  Viþ1 Þ 2

ð27Þ

The result of Eq. (25) can be represent by the equivalent circuit as shown in Fig. 6. The upwind difference scheme [11] for the energy flux Fa at xi12 may be written as:

Fig. 6. The equivalent circuit of the power conservation equation at xi .

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( Fa;i12 ¼

5VE;i Ii12 3 5VE;i1  3 Ii12



if Ii12 < 0 otherwise

and ð28Þ VE;1 ¼ VE;kþ1 ¼

Since electrons flow from xi to xi1 , the current Ii12 is always negative. 2.3.5. The equivalent circuit The length L of the Gunn diode is separated into k partitions. The equivalent circuit of the Gunn diode with a resistor R0 and a DC bias Vapplied is shown in Fig. 7(a)– (d) for the Poisson equation, continuity equation, momentum conservation equation, and energy conservation equation respectively. The boundary conditions on both sides are qh2 NDþ 

ð29Þ

3 kB T0 2 q

ð30Þ

The initial conditions inside the device are Vn;i ¼ 

Vn;1 ¼ Vn;kþ1 ¼ 

919

VE;i ¼

qh2 ND;i 

3 kB T0 2 q

ð31Þ

ð32Þ

2.4. Energy dependent parameters The momentum relaxation time sp , mobility l, effective mass m, and energy relaxation time sE are energy dependent. The energy dependence of sp , and m can be obtained from Monte Carlo calculations, and l can be

Fig. 7. The equivalent circuit of the Gunn diode. (a) The equivalent circuit of the Poisson equation for the Gunn diode and external connections. Dashed line represent the repetition of the circuit from x3 to xk1 . (b) The equivalent circuit of the continuity equation for the Gunn diode. Dashed line represent the repetition of the circuit from x3 to xk1 . (c) The equivalent circuit of the momentum conservation equation for the Gunn diode. Dashed line represent the repetition of the circuit from x112 to xk12 . (d) The equivalent circuit of the energy conservation equation for the Gunn diode. Dashed line represent the repetition of the circuit from x3 to xk12 .

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calculated from Eq. (7). A Monte Carlo program for GaAs is developed using [12,13] with doping concentrations of 1016 cm3 and 2  1017 cm3 . The equivalent single valley energy band model is extracted by least square fit and the approximation relations are: s0 ð33Þ sp ¼ 1 þ bp0 ðE  E0 Þ þ ebp1 ðEEp1 Þ =ð1 þ ebp2 ðEEp2 Þ Þ m 1 þ bm0 ðE  E0 Þ þ ebm1 ðEEm1 Þ ¼ m1 1 þ ebm2 ðEEm2 Þ m0 sE ¼ s1

ð34Þ

1 þ bE0 ðE  E0 Þ 1 þ ebE1 ðEEE1 Þ =ð1 þ ebm2 ðEEE2 Þ Þ bE3 ðEE0 Þ

ð35Þ

þ s2 e

ND (cm3 )

1  1016

2  1017

s0 (ps) bp0 (eV1 ) bp1 (eV1 ) bp2 (eV1 ) Ep1 (eV) Ep2 (eV)

0.291 4.58 27.5 31.0 0.233 0.345

0.176 0.0 19.5 28.9 0.227 0.375

Table 2 Coefficients of m=m0 (eV1 ) (eV1 ) (eV1 ) (eV) (eV)

tE1 (ps) tE2 (ps) bE0 (eV1 ) bE1 (eV1 ) bE2 (eV1 ) bE3 (eV1 ) EE1 (eV) EE2 (eV)

0.583 12.9 13.8 67.4 55.5 960 0.314 0.324

The coefficients are listed in Tables 1–3 and the results are shown in Figs. 8–10.

3. Device simulation results and discussion

Table 1 Coefficients of sp

m1 bm0 bm1 bm2 Em1 Em2

Table 3 Coefficients of sE

0.0732 0.885 46.4 44.8 0.287 0.307

In order to test the new model, a GaAs Gunn diode in the transit time mode is simulated to check the spontaneous oscillations as well as domain generations and propagations. This diode has an nþ  n  nþ doping profile with NDþ ¼ 2  1017 cm3 in the contact region and ND ¼ 1016 cm3 in the active region. The crosssection area A is 0.003 mm2 . The diode length is 4 lm and and the active region length is 2.8 lm. The contact region on both side have the same length: 0.6 lm. The device is divided into 60 partitions for simulations. The lattice temperature T0 is 300 K. The Gunn diode model is operated in a simple circuit with a resistance R0 ¼ 0:01 X and a bias voltage Vapplied ¼ 4 V. In the beginning, this bias voltage is increased from 0 V to its final value within 25 ps to avoid a non-physical simulation of the differential equation system by a step function. The model is implemented

Fig. 8. The Energy dependent momentum relaxation time sp .

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Fig. 9. Energy dependent effective mass ratio m=m0 .

Fig. 10. Energy dependent energy relaxation time sE .

with SPICE3 build-in circuit elements and the simple circuit is shown in Fig. 7(a)–(d). The results shown in Figs. 11 and 12 indicate the success in the simulation of the spontaneous Gunn diode oscillations. As shown in Fig. 11, the domain is generated near the cathode, propagating from the cathode to the anode, growing up during the transport, and vanished near the anode. The process is repeated, as a result, there is an oscillating current at the terminal as shown in Fig. 12. The oscillating frequency is about 66 GHz which is different

from the numerical simulation result of 47 GHz in [6] under similar conditions. The difference may be due to different Monte Carlo models used for parameter extraction. During the simulations, we find that different discretization methods for the energy flux equation have strong effects on the numerical stability and simulation result. In Eq. (28), the upwind method, which is widely used in fluid dynamics, has been adopted to solve this problem.

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Fig. 11. The transport of domain in the active region.

Fig. 12. Current oscillation of the Gunn diode.

4. Conclusion The new model has the advantage that it combines easily with the equivalent circuits of other devices and with the external circuits in a circuit simulator. And the circuit elements of the new Gunn diode equivalent circuit model connects directly with the device and material physics. It has been successfully applied to the Gunn diode simulation and it can be incorporated easily to any circuits. The same technique can also be applied to devices which operate in the hydrodynamic hot carrier transport range.

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