## Optimal doping profiles via geometric programming (2005)

Venue: | IEEE Transactions on Electron Devices |

Citations: | 2 - 1 self |

### BibTeX

@ARTICLE{Joshi05optimaldoping,

author = {Siddharth Joshi and Stephen Boyd and Robert W. Dutton},

title = {Optimal doping profiles via geometric programming},

journal = {IEEE Transactions on Electron Devices},

year = {2005}

}

### OpenURL

### Abstract

Abstract—We first consider the problem of determining the doping profile that minimizes base transit time in a (homojunction) bipolar junction transistor. We show that this problem can be formulated as a geometric program, a special type of optimization problem that can be transformed to a convex optimization problem, and therefore solved (globally) very efficiently. We then consider several extensions to the basic problem, such as accounting for velocity saturation, and adding constraints on doping gradient, current gain, base resistance, and breakdown voltage. We show that a similar approach can be used to maximize the cutoff frequency, taking into account junction capacitances and forward transit time. Finally, we show that the method extends to the case of heterojunction bipolar junction transistors, in which the doping profile, as well as the profile of the secondary semiconductor, are to be jointly optimized. Index Terms—Base doping profile, base transit time minimization, cutoff frequency maximization, geometric programming, Ge-profile optimization, optimal doping profile. I.

### Citations

3667 | L.: Convex Optimization - BOYD, VANDENBERGHE - 2004 |

88 |
Ning, Fundamentals of modern VLSI devices
- Taur, H
- 1998
(Show Context)
Citation Context ...insic carrier concentration and carrier diffusion coefficient are functions of the doping concentration . The intrinsic carrier concentration depends on the effective bandgap reduction as [17], [19], =-=[32]-=- where is the intrinsic carrier concentration in undoped silicon, is the Boltzmann constant, and is the temperature (Kelvin). In a homojunction BJT the effective bandgap reduction is only due to dopin... |

67 |
A tutorial on geometric programming
- Boyd, Kim, et al.
- 2007
(Show Context)
Citation Context ...cial type of mathematical optimization problem. Recently developed numerical methods for GPs can solve even large scale problems very efficiently, always guaranteeing that the global optimum is found =-=[6]-=-, [7], [15]. In particular, the method described in this paper finds the globally optimal doping profile (for the models used). The method described here can in addition handle a variety of practical ... |

51 | Optimal design of a cmos op-amp via geometric programming
- Hershenson, Boyd, et al.
- 2001
(Show Context)
Citation Context ...ts relation to convex optimization, we refer the reader to [7, Sect. 4.5]. Geometric programming has been used to solve a variety of circuit design problems, including sizing of analog circuits [10], =-=[13]-=-, [21], [33], digital circuits [5], [27], mixed-signal circuits [8], [12], and RF circuits [14], [22] (for a more complete list, see, e.g., [6]). Geometric programs can be transformed to convex optimi... |

36 | Semiconductor Device Fundamentals - Pierret - 1996 |

33 |
An infeasible interior-point algorithm for solving primal and dual geometric programs
- Kortanek, Xu, et al.
- 1997
(Show Context)
Citation Context ...of mathematical optimization problem. Recently developed numerical methods for GPs can solve even large scale problems very efficiently, always guaranteeing that the global optimum is found [6], [7], =-=[15]-=-. In particular, the method described in this paper finds the globally optimal doping profile (for the models used). The method described here can in addition handle a variety of practical constraints... |

33 | Simple accurate expressions for planar spiral inductances
- Mohan, Hershenson, et al.
- 1999
(Show Context)
Citation Context ...been used to solve a variety of circuit design problems, including sizing of analog circuits [10], [13], [21], [33], digital circuits [5], [27], mixed-signal circuits [8], [12], and RF circuits [14], =-=[22]-=- (for a more complete list, see, e.g., [6]). Geometric programs can be transformed to convex optimization problems, and therefore solved globally and efficiently, for example by recently developed int... |

17 | Silicon-Germanium heterojunction Bipolar Transistors, Artech House - Cressler, Niu - 2003 |

16 |
Cmos op-amp sizing using a geometric programming formulation,” Computer-Aided Design of Integrated Circuits and Systems
- Mandal, Visvanathan
- 2001
(Show Context)
Citation Context ...ation to convex optimization, we refer the reader to [7, Sect. 4.5]. Geometric programming has been used to solve a variety of circuit design problems, including sizing of analog circuits [10], [13], =-=[21]-=-, [33], digital circuits [5], [27], mixed-signal circuits [8], [12], and RF circuits [14], [22] (for a more complete list, see, e.g., [6]). Geometric programs can be transformed to convex optimization... |

9 | Optimization of phase-locked loop circuits via geometric programming
- Colleran, Portmann, et al.
- 2003
(Show Context)
Citation Context ... 4.5]. Geometric programming has been used to solve a variety of circuit design problems, including sizing of analog circuits [10], [13], [21], [33], digital circuits [5], [27], mixed-signal circuits =-=[8]-=-, [12], and RF circuits [14], [22] (for a more complete list, see, e.g., [6]). Geometric programs can be transformed to convex optimization problems, and therefore solved globally and efficiently, for... |

7 | Optimal allocation of local feedback in multistage amplifiers via geometric programming
- Dawson, Boyd, et al.
(Show Context)
Citation Context ... and its relation to convex optimization, we refer the reader to [7, Sect. 4.5]. Geometric programming has been used to solve a variety of circuit design problems, including sizing of analog circuits =-=[10]-=-, [13], [21], [33], digital circuits [5], [27], mixed-signal circuits [8], [12], and RF circuits [14], [22] (for a more complete list, see, e.g., [6]). Geometric programs can be transformed to convex ... |

6 |
Two integral relations pertaining to the electron transport through a bipolar transistor with a nonuniform energy gap in the base region
- Kroemer
- 1985
(Show Context)
Citation Context ... space variable over the interval 0 , wheresJOSHI et al.: OPTIMAL DOPING PROFILES VIA GEOMETRIC PROGRAMMING 2661 is the base width. A model for the base transit time in a homojunction BJT is given by =-=[16]-=- where is the intrinsic carrier concentration and is the carrier diffusion coefficient. This model assumes low-level injection, and neglects velocity saturation and carrier recombination in the base r... |

4 |
Digital Circuit Sizing via Geometric Programming
- Boyd, Kim, et al.
- 2005
(Show Context)
Citation Context ... we refer the reader to [7, Sect. 4.5]. Geometric programming has been used to solve a variety of circuit design problems, including sizing of analog circuits [10], [13], [21], [33], digital circuits =-=[5]-=-, [27], mixed-signal circuits [8], [12], and RF circuits [14], [22] (for a more complete list, see, e.g., [6]). Geometric programs can be transformed to convex optimization problems, and therefore sol... |

3 |
SiGe Heterojunction Bipolar Transistors
- Ashburn
- 2003
(Show Context)
Citation Context ...scretization, and therefore can be globally and efficiently solved. The current gain can be expressed as ratio of Gummel numbers (30) where is the emitter Gummel number, and is the base Gummel number =-=[2]-=-, [32]. The emitter Gummel number depends on the emitter doping profile and not on the base doping profile, and therefore can be treated as a positive constant for our purposes. The base Gummel number... |

3 | Automated design of operational transconductance amplifiers using reversed geometric programming
- Vanderhaegen, Brodersen
- 2004
(Show Context)
Citation Context ...to convex optimization, we refer the reader to [7, Sect. 4.5]. Geometric programming has been used to solve a variety of circuit design problems, including sizing of analog circuits [10], [13], [21], =-=[33]-=-, digital circuits [5], [27], mixed-signal circuits [8], [12], and RF circuits [14], [22] (for a more complete list, see, e.g., [6]). Geometric programs can be transformed to convex optimization probl... |

2 |
Automated optimal design of switched-capacitor filters
- Hassibi, Hershenson
- 2002
(Show Context)
Citation Context .... Geometric programming has been used to solve a variety of circuit design problems, including sizing of analog circuits [10], [13], [21], [33], digital circuits [5], [27], mixed-signal circuits [8], =-=[12]-=-, and RF circuits [14], [22] (for a more complete list, see, e.g., [6]). Geometric programs can be transformed to convex optimization problems, and therefore solved globally and efficiently, for examp... |

2 | Profile design considerations for minimizing the base transit time in SiGe HBTs
- Patri, Kumar
- 1998
(Show Context)
Citation Context ...1109/TED.2005.859649 0018-9383/$20.00 © 2005 IEEE voltage. The method also extends to the problem of minimizing base transit time in heterojunction bipolar transistors (HBTs), as studied in [3], [4], =-=[17]-=-, [18]. In the case of SiGe HBT, our approach applies to several problems: determining the base doping profile, given the Ge-profile; determining the Ge-profile, given the base doping profile; and the... |

2 | Scaling issues and Ge profile optimization in advanced UHV/CVD SiGe HBT’s - Richey, Cressler, et al. - 1997 |

2 |
Base transit time of shallow-base bipolar transistors considering velocity saturation at base-collector junction
- Suzuki, Nakayam
- 1992
(Show Context)
Citation Context ...bination of these GP compatible constraints. A. Velocity Saturation A model for the base transit time in a homojunction BJT, taking velocity saturation at the base–collector into account, is given by =-=[31]-=- 1 (16) where is the saturation velocity of electrons, which is a constant. Using the simple discretization described above, the (discretized, approximate) base transit time can be expressed as (17) T... |

1 |
An analytical approach to the modeling of intrinsic base sheet resistance in SiGe HBT and optimal profile design considerations for its minimization
- Biswas, Basu
- 2002
(Show Context)
Citation Context ...tifier 10.1109/TED.2005.859649 0018-9383/$20.00 © 2005 IEEE voltage. The method also extends to the problem of minimizing base transit time in heterojunction bipolar transistors (HBTs), as studied in =-=[3]-=-, [4], [17], [18]. In the case of SiGe HBT, our approach applies to several problems: determining the base doping profile, given the Ge-profile; determining the Ge-profile, given the base doping profi... |

1 | Si/SiGe epitaxial-base transistors - Harame, Comfort, et al. - 1995 |

1 |
Ge-profile design for high-speed SiGe HBTs: Modeling and analysis
- “Novel
- 1999
(Show Context)
Citation Context ...ED.2005.859649 0018-9383/$20.00 © 2005 IEEE voltage. The method also extends to the problem of minimizing base transit time in heterojunction bipolar transistors (HBTs), as studied in [3], [4], [17], =-=[18]-=-. In the case of SiGe HBT, our approach applies to several problems: determining the base doping profile, given the Ge-profile; determining the Ge-profile, given the base doping profile; and the joint... |

1 |
the iterative schemes to obtain base doping profiles for reducing base transit time in bipolar junction transistor
- “On
- 2001
(Show Context)
Citation Context ...t time is a function of the doping profile; minimizing base transit time, by proper choice of doping profile, is a well studied problem [29]. Methods that have been proposed include iterative schemes =-=[19]-=-, variational calculus [30], and optimal control [25], [26], which are compared in [30]. None of these methods, however, can guarantee (global) optimality of the resulting doping profiles. In this pap... |

1 | A closed-form analytic forward transit time model considering specific models for bandgap-narrowing effects and concentration-dependent diffusion coefficients for BJT devices operating at 77K - Lu, Kuo - 1993 |

1 |
Minimization of the base transit time in semiconductor devices using optimal control
- Rinaldi, Schättler
- 2002
(Show Context)
Citation Context ...g base transit time, by proper choice of doping profile, is a well studied problem [29]. Methods that have been proposed include iterative schemes [19], variational calculus [30], and optimal control =-=[25]-=-, [26], which are compared in [30]. None of these methods, however, can guarantee (global) optimality of the resulting doping profiles. In this paper we introduce a new method for finding the doping p... |

1 |
optimal control problem with state space constraints arising in the design of bipolar transistors
- “An
- 2004
(Show Context)
Citation Context ... transit time, by proper choice of doping profile, is a well studied problem [29]. Methods that have been proposed include iterative schemes [19], variational calculus [30], and optimal control [25], =-=[26]-=-, which are compared in [30]. None of these methods, however, can guarantee (global) optimality of the resulting doping profiles. In this paper we introduce a new method for finding the doping profile... |

1 |
Optimal base profile design for minimum base transit time
- Suzuki
- 1991
(Show Context)
Citation Context ...response of a bipolar junction transistor (BJT). The base transit time is a function of the doping profile; minimizing base transit time, by proper choice of doping profile, is a well studied problem =-=[29]-=-. Methods that have been proposed include iterative schemes [19], variational calculus [30], and optimal control [25], [26], which are compared in [30]. None of these methods, however, can guarantee (... |

1 |
base–doping profile for minimum base transit time considering velocity saturation at base–collector junction and dependence of mobility and bandgap narrowing on doping concentration
- “Optimum
- 2001
(Show Context)
Citation Context ... doping profile; minimizing base transit time, by proper choice of doping profile, is a well studied problem [29]. Methods that have been proposed include iterative schemes [19], variational calculus =-=[30]-=-, and optimal control [25], [26], which are compared in [30]. None of these methods, however, can guarantee (global) optimality of the resulting doping profiles. In this paper we introduce a new metho... |