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Author: phonchi chung
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Description: Jupyter notebook SPC_4.ipynb
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import numpy as np import matplotlib.pyplot as plt %matplotlib inline

System Prototype

The basic Runto Run Controller

The W2W control scheme:

  • The yky_k is either form AVM or Metrology tool
  • AkA_k (or A) is typically chosen to be least square estimates of β1β_1 based on historical data.The control valuable is set to nullify the deviation from target.

CMP Example

  1. yky_k is the actual removal amount measured from the metrology tool and PostYkPostY_k is the actual post CMP thickness of run k. The specification of PostYkPostY_k is 2800±150 Angstrom (Å) with 2800 being the target value denoted by TgtPostYTgtPostY Deifned PreYkPreY_k be the one before CMP process, ARRkARR_k is the polish rate, μk\mu_k is the polish time that we can control!

The material removal model for CMP can be divided into two parts, mechanical model and chemical model. The chemical action of slurry is responsible for continuosly softening the silicon oxide. The fresh silicon oxide or metal surface is then rapidly removed by mechanical part.

  1. The AkA_k is the nominal removal rate, which is empirically simulated by a polynomial curve fitting of parts usage count between PMs (denoted by PU varying from 1 to 600)
  1. Process gain:
  1. Simulation parameters:
  1. Assumption:
  1. AkA_k is set to 300, and A is set to mean of ARRkARR_k (also 300)
  2. β0\beta_0 is set to 700 and b0b_0 is 500
  3. RI is simulated with 0.9 plus some variation, and assume controller can detect any value that is beyond UCL or LCL
  4. yk from AVM is virtual times yz(actual metrodlogy data), where virtual is a normal distribution variable
  5. ηk\eta_k is simulated as ARIMA(0 1 1)
  6. Control β\beta not η\eta

yz is atual measurement value, and A is used internal by R2R

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from statsmodels.tsa.arima_model import ARIMA import sklearn.linear_model as skl_lm
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# Simulated environment Error = np.random.normal(0, 25, 600) PM1 = np.random.normal(0, 50, 600) PM2 = np.random.normal(0, 3.5, 600) Stress1 = np.random.normal(1000, 55,600) Stress2 = np.random.normal(0, 4, 600) Rotspd1 = np.random.normal(100, 5.5, 600) Rotspd2 = np.random.normal(0, 0.4, 600) Sfuspd1 = np.random.normal(100, 5.5, 600) Sfuspd2 = np.random.normal(0, 0.4, 600) PreY = np.random.normal(3800, 70 ,600) TgtPostY = np.array([2800]*600) # Assumption Virtual = np.random.normal(1, np.sqrt(0.01), 600) Ak0 = 1100 # Not sure, does not know PU Ak = np.array([1100]*600) RI = 0.9 TgtY = PreY - TgtPostY def ARR (stress1, stress2, rotspd1, rotspd2, sfuspd1, sfuspd2, pm1, pm2, error): return Ak0*((stress1+stress2)/1000)*((rotspd1+rotspd2)/100)*((sfuspd1+sfuspd2)/100)+pm1+pm2+error ARRk = ARR(Stress1, Stress2, Rotspd1, Rotspd2, Sfuspd1, Sfuspd2, PM1, PM2, Error) # beta1 actual process gain # ARIMA (0 1 1) assume error model = ARIMA(Error, order=(0, 1, 1)) results_AR = model.fit(disp=-1) beta0 = 0 RI=RI*np.sqrt(np.sqrt(Virtual)) alpha1 = 0.35 betak = np.zeros(len(PreY)) mu = np.zeros(len(PreY)) PostY = np.zeros(len(PreY)) yz = np.zeros(len(PreY)) mu[0] = (TgtY[0] - beta0)/Ak[0] yz[0] = ARRk[0]*mu[0]#+ results_AR.fittedvalues[0] # Actual value PostY[0] = PreY[0] - yz[0] betak[1] = alpha1*(yz[0]-Ak[0]*mu[0])+(1-alpha1)*beta0 mu[1] = (TgtY[1] - betak[1])/Ak[1] yz[1] = ARRk[1]*mu[1]#+ results_AR.fittedvalues[1] # Actual value PostY[1] = PreY[1] - yz[1] betak[2] = alpha1*(yz[1]-Ak[1]*mu[1])+(1-alpha1)*betak[1] mu[2] = (TgtY[2] - betak[2])/Ak[2] for k in range(2,599): yz[k] = ARRk[k]*mu[k]#+ results_AR.fittedvalues[k] ## ARRk[k], Toub[k] approximate beta_1, beta_0(not tou!!) PostY[k] = PreY[k] - yz[k] betak[k+1] = alpha1*(yz[k]-Ak[k]*mu[k])+(1-alpha1)*betak[k] mu[k+1] = (TgtY[k+1] - betak[k+1])/Ak[k+1] # Regression coefficients (Ordinary Least Squares) mu = mu.reshape((600,1)) yz = yz.reshape((600,1)) regr = skl_lm.LinearRegression() regr.fit(mu,yz) # Plotting x = range(1, 20) U = np.empty(19) L = np.empty(19) UCL = 2950 LCL = 2650 #U.fill(2950) #L.fill(2650) plt.plot(x,U) plt.plot(x,L) plt.plot(x,ARRk[1:20]) #plt.ylim([2600, 3000]) np.mean(PostY[10:590]) print(regr.intercept_) print(regr.coef_)
[-1.88585635] [[ 1095.86374957]]
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x = range(1, 600) U = np.empty(599) L = np.empty(599) UCL = 2950 LCL = 2650 U.fill(2950) L.fill(2650) plt.plot(x,U) plt.plot(x,L) plt.ylim([2600, 3000]) fig = plt.plot(x,PostY[1:600]) from matplotlib.backends.backend_pdf import PdfPages pp = PdfPages('foo1.pdf') pp.savefig(fig)
--------------------------------------------------------------------------- TypeError Traceback (most recent call last) <ipython-input-20-94514a95eb0b> in <module>() 13 from matplotlib.backends.backend_pdf import PdfPages 14 pp = PdfPages('foo1.pdf') ---> 15 pp.savefig(fig) /usr/local/lib/python2.7/dist-packages/matplotlib/backends/backend_pdf.pyc in savefig(self, figure, **kwargs) 2434 figureManager = Gcf.get_active() 2435 else: -> 2436 figureManager = Gcf.get_fig_manager(figure) 2437 if figureManager is None: 2438 raise ValueError("No such figure: " + repr(figure)) /usr/local/lib/python2.7/dist-packages/matplotlib/_pylab_helpers.pyc in get_fig_manager(num) 42 figure and return the manager; otherwise return *None*. 43 """ ---> 44 manager = Gcf.figs.get(num, None) 45 if manager is not None: 46 Gcf.set_active(manager) TypeError: unhashable type: 'list'
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x = range(2, 590) plt.plot(x,betak[2:590])
[<matplotlib.lines.Line2D at 0x7fbbc0c4bc50>]
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x = range(2, 590) plt.plot(x,mu[2:590])
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RI=RI*np.sqrt(np.sqrt(Virtual)) RI

Round 1

First two rounds, we use actual metrodlogy value

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alpha1 = 0.35 betak = np.zeros(len(PreY)) mu = np.zeros(len(PreY)) PostY = np.zeros(len(PreY)) yz = np.zeros(len(PreY))
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mu[0] = (TgtY[0] - beta0)/Ak[0] yz[0] = ARRk[0]*mu[0]+ beta0 + results_AR.fittedvalues[0] # Actual value PostY[0] = PreY[0] - yz[0] betak[1] = alpha1*(yz[0]-Ak[0]*mu[0])+(1-alpha1)*beta0 mu[1] = (TgtY[1] - betak[1])/Ak[1]

Round2

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yz[1] = ARRk[1]*mu[1]+ betak[1] + results_AR.fittedvalues[1] # Actual value PostY[1] = PreY[1] - yz[1] betak[2] = alpha1*(yz[1]-Ak[1]*mu[1])+(1-alpha1)*betak[1] mu[2] = (TgtY[2] - betak[2])/Ak[2]

CASE1 R2R with in-situ metrology

set to 0.35, and all actual metrology data are available

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for k in range(2,599): yz[k] = ARRk[k]*mu[k]+ betak[k] + results_AR.fittedvalues[k] ## ARRk[k], Toub[k] approximate beta_1, beta_0(not tou!!) PostY[k] = PreY[k] - yz[k] betak[k+1] = alpha1*(yz[k]-Ak[k]*mu[k])+(1-alpha1)*betak[k] mu[k+1] = (TgtY[k+1] - betak[k+1])/Ak[k+1]
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x = range(2, 590) plt.plot(x,betak[2:590])
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# Plotting x = range(10, 590) U = np.empty(580) L = np.empty(580) UCL = 2950 LCL = 2650 U.fill(2950) L.fill(2650) plt.plot(x,U) plt.plot(x,L) plt.plot(x,PostY[10:590]) #plt.ylim([2600, 3000]) np.mean(PostY[10:590])

CpK(Process Capability)

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Cpk=np.zeros(len(PreY)) MAPEp =np.zeros(len(PreY))
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for k in range(2,599): Cpk[k] = min((UCL-np.mean(PostY[1:k]))/(3*np.std(PostY[1:k], dtype=np.float64)), (np.mean(PostY[1:k])-LCL)/(3*np.std(PostY[1:k], dtype=np.float64))) plt.plot(x,Cpk[10:590]) plt.ylim([0.3, 2.0])

Mean-absolute-percentage error (MAPEp)

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for k in range(2,599): MAPEp[k] = sum(np.absolute((PostY[1:k]-2800)/2800))/k*100 plt.plot(x,MAPEp[10:590]) plt.ylim([0.7, 2.5])

CASE2 R2R+VM without RI

, apply to following wafers

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# Reinitialized alpha1 = 0.35 betak = np.zeros(len(PreY)) mu = np.zeros(len(PreY)) PostY = np.zeros(len(PreY)) PostYv = np.zeros(len(PreY)) y = np.zeros(len(PreY)) yz = np.zeros(len(PreY)) # Round1 mu[0] = (TgtY[0] - betak0)/Ak[0] yz[0] = ARRk[0]*mu[0]+beta0+results_AR.fittedvalues[0] # Actual value PostY[0] = PreY[0] - yz[0] betak[1] = alpha1*(yz[0]-Ak[0]*mu[0])+(1-alpha1)*betak0 mu[1] = (TgtY[1] - betak[1])/Ak[1] # Round2 yz[1] = ARRk[1]*mu[1]+beta0+results_AR.fittedvalues[1] # Actual value PostY[1] = PreY[1] - yz[1] betak[2] = alpha1*(yz[1]-Ak[1]*mu[1])+(1-alpha1)*betak[1] mu[2] = (TgtY[2] - betak[2])/Ak[2]
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for k in range(2,599): yz[k] = ARRk[k]*mu[k]+results_AR.fittedvalues[k]+beta0 # Metrology data if(k%25)==1: y[k] = yz[k] else: y[k] = yz[k]*Virtual[k] # Assume VM value ok ARR=ARR_bar PostY[k] = PreY[k] - yz[k] betak[k+1] = alpha1*(y[k]-Ak[k]*mu[k])+(1-alpha1)*betak[k] mu[k+1] = (TgtY[k+1] - betak[k+1])/Ak[k+1]
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x = range(2, 590) plt.plot(x,betak[2:590])
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x = range(10, 590) U = np.empty(580) L = np.empty(580) U.fill(2950) L.fill(2650) plt.plot(x,U) plt.plot(x,L) plt.plot(x,PostY[10:590]) np.mean(PostY[10:590]) plt.ylim([2600, 3000])
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Cpk=np.zeros(len(PreY)) MAPEp =np.zeros(len(PreY))
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for k in range(2,599): Cpk[k] = min((UCL-np.mean(PostY[1:k]))/(3*np.std(PostY[1:k], dtype=np.float64)), (np.mean(PostY[1:k])-LCL)/(3*np.std(PostY[1:k], dtype=np.float64))) plt.plot(x,Cpk[10:590]) plt.ylim([0.3, 2.0])
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for k in range(2,599): MAPEp[k] = sum(np.absolute((PostY[1:k]-2800)/2800))/k*100 plt.plot(x,MAPEp[10:590]) plt.ylim([0.7, 2.5])

CASE3: R2R+VM with RI

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# Reinitialized alpha1 = 0.35 betak = np.zeros(len(PreY)) mu = np.zeros(len(PreY)) PostY = np.zeros(len(PreY)) PostYv = np.zeros(len(PreY)) y = np.zeros(len(PreY)) yz = np.zeros(len(PreY)) # Round1 mu[0] = (TgtY[0] - betak0)/Ak[0] yz[0] = ARRk[0]*mu[0]+beta0+results_AR.fittedvalues[0] # Actual value PostY[0] = PreY[0] - yz[0] betak[1] = alpha1*(yz[0]-Ak[0]*mu[0])+(1-alpha1)*betak0 mu[1] = (TgtY[1] - betak[1])/Ak[1] # Round2 yz[1] = ARRk[1]*mu[1]+beta0+results_AR.fittedvalues[1] # Actual value PostY[1] = PreY[1] - yz[1] betak[2] = alpha1*(yz[1]-Ak[1]*mu[1])+(1-alpha1)*betak[1] mu[2] = (TgtY[2] - betak[2])/Ak[2]
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for k in range(2,599): yz[k] = ARRk[k]*mu[k]+beta0+results_AR.fittedvalues[k] # process output how to filter out if(k%25)==1: y[k] = yz[k] alpha = alpha1 else: y[k] = yz[k]*Virtual[k] alpha = RI[k]*alpha1 PostYv[k] = PreY[k] - yz[k]*Virtual[k] if PostYv[k]>2950 or PostYv[k]<2650: alpha = 0 betak[k+1] = alpha*(y[k]-Ak[k]*mu[k])+(1-alpha)*betak[k] #Toub canceled out mu[k+1] = (TgtY[k+1] - betak[k+1])/Ak[k+1] PostY[k] = PreY[k] - yz[k] # Process output
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x = range(2, 590) plt.plot(x,betak[2:590])
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x = range(10, 590) U = np.empty(580) L = np.empty(580) U.fill(2950) L.fill(2650) # plot(x, PostY[3:600], type="o") plt.plot(x,U) plt.plot(x,L) plt.plot(x,PostY[10:590]) np.mean(PostY[10:590]) plt.ylim([2600, 3000])
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Cpk=np.zeros(len(PreY)) MAPEp =np.zeros(len(PreY))
In [ ]:
for k in range(2,599): Cpk[k] = min((UCL-np.mean(PostY[1:k]))/(3*np.std(PostY[1:k], dtype=np.float64)), (np.mean(PostY[1:k])-LCL)/(3*np.std(PostY[1:k], dtype=np.float64))) plt.plot(x,Cpk[10:590]) plt.ylim([0.3, 2.0])
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for k in range(2,599): MAPEp[k] = sum(np.absolute((PostY[1:k]-2800)/2800))/k*100 plt.plot(x,MAPEp[10:590]) plt.ylim([0.7, 2.5])

CASE4: R2R+VM with (1-RI)

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# Reinitialized alpha1 = 0.35 betak = np.zeros(len(PreY)) mu = np.zeros(len(PreY)) PostY = np.zeros(len(PreY)) PostYv = np.zeros(len(PreY)) y = np.zeros(len(PreY)) yz = np.zeros(len(PreY)) # Round1 mu[0] = (TgtY[0] - betak0)/Ak[0] yz[0] = ARRk[0]*mu[0]+beta0+results_AR.fittedvalues[0] # Actual value PostY[0] = PreY[0] - yz[0] betak[1] = alpha1*(yz[0]-Ak[0]*mu[0])+(1-alpha1)*betak0 mu[1] = (TgtY[1] - betak[1])/Ak[1] # Round2 yz[1] = ARRk[1]*mu[1]+beta0+results_AR.fittedvalues[1] # Actual value PostY[1] = PreY[1] - yz[1] betak[2] = alpha1*(yz[1]-Ak[1]*mu[1])+(1-alpha1)*betak[1] mu[2] = (TgtY[2] - betak[2])/Ak[2]
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for k in range(2,599): yz[k] = ARRk[k]*mu[k]+beta0+results_AR.fittedvalues[k] # process output how to filter out if(k%25)==1: y[k] = yz[k] alpha = alpha1 else: y[k] = yz[k]*Virtual[k] alpha = (1-RI[k])*alpha1 PostYv[k] = PreY[k] - yz[k]*Virtual[k] if PostYv[k]>2950 or PostYv[k]<2650: alpha = 0 betak[k+1] = alpha*(y[k]-Ak[k]*mu[k])+(1-alpha)*betak[k] #Toub canceled out mu[k+1] = (TgtY[k+1] - betak[k+1])/Ak[k+1] PostY[k] = PreY[k] - yz[k] # Process output
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x = range(2, 590) plt.plot(x,betak[2:590])
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x = range(10, 590) U = np.empty(580) L = np.empty(580) U.fill(2950) L.fill(2650) # plot(x, PostY[3:600], type="o") plt.plot(x,U) plt.plot(x,L) plt.plot(x,PostY[10:590]) np.mean(PostY[10:590]) plt.ylim([2600, 3000])
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Cpk=np.zeros(len(PreY)) MAPEp =np.zeros(len(PreY))
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for k in range(2,599): Cpk[k] = min((UCL-np.mean(PostY[1:k]))/(3*np.std(PostY[1:k], dtype=np.float64)), (np.mean(PostY[1:k])-LCL)/(3*np.std(PostY[1:k], dtype=np.float64))) plt.plot(x,Cpk[10:590]) plt.ylim([0.3, 2.0])
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for k in range(2,599): MAPEp[k] = sum(np.absolute((PostY[1:k]-2800)/2800))/k*100 plt.plot(x,MAPEp[10:590]) plt.ylim([0.7, 2.5])

CASE5: R2R+VM with RI.S.(1-RI)

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# Reinitialized alpha1 = 0.35 betak = np.zeros(len(PreY)) mu = np.zeros(len(PreY)) PostY = np.zeros(len(PreY)) PostYv = np.zeros(len(PreY)) y = np.zeros(len(PreY)) yz = np.zeros(len(PreY)) # Round1 mu[0] = (TgtY[0] - betak0)/Ak[0] yz[0] = ARRk[0]*mu[0]+beta0+results_AR.fittedvalues[0] # Actual value PostY[0] = PreY[0] - yz[0] betak[1] = alpha1*(yz[0]-Ak[0]*mu[0])+(1-alpha1)*betak0 mu[1] = (TgtY[1] - betak[1])/Ak[1] # Round2 yz[1] = ARRk[1]*mu[1]+beta0+results_AR.fittedvalues[1] # Actual value PostY[1] = PreY[1] - yz[1] betak[2] = alpha1*(yz[1]-Ak[1]*mu[1])+(1-alpha1)*betak[1] mu[2] = (TgtY[2] - betak[2])/Ak[2]
In [ ]:
for k in range(2,599): yz[k] = ARRk[k]*mu[k]+beta0+results_AR.fittedvalues[k] PostYv[k] = PreY[k]- yz[k]*Virtual[k] if k>= 25: if PostYv[k]>2950 or PostYv[k]<2650: # should be replace with RI, GSI alpha = 0 else: alpha = (1-RI[k])*alpha1 else: if PostYv[k]>2950 or PostYv[k]<2650: alpha = 0 else: alpha = RI[k]*alpha1 if(k%25)==1: y[k] = yz[k] alpha = alpha1 else: y[k] = yz[k]*Virtual[k] betak[k+1] = alpha*(y[k]-Ak[k]*mu[k])+(1-alpha)*betak[k] #Toub canceled out mu[k+1] = (TgtY[k+1] - betak[k+1])/Ak[k+1] PostY[k] = PreY[k] - yz[k] # Process output
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x = range(2, 590) plt.plot(x,betak[2:590])
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x = range(10, 590) U = np.empty(580) L = np.empty(580) U.fill(2950) L.fill(2650) # plot(x, PostY[3:600], type="o") plt.plot(x,U) plt.plot(x,L) plt.plot(x,PostY[10:590]) np.mean(PostY[10:590]) plt.ylim([2600, 3000])
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Cpk=np.zeros(len(PreY)) MAPEp =np.zeros(len(PreY))
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for k in range(2,599): Cpk[k] = min((UCL-np.mean(PostY[1:k]))/(3*np.std(PostY[1:k], dtype=np.float64)), (np.mean(PostY[1:k])-LCL)/(3*np.std(PostY[1:k], dtype=np.float64))) plt.plot(x,Cpk[10:590]) plt.ylim([0.3, 2.0])
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for k in range(2,599): MAPEp[k] = sum(np.absolute((PostY[1:k]-2800)/2800))/k*100 plt.plot(x,MAPEp[10:590]) plt.ylim([0.7, 2.5])
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